Contents Chapters
Page Number
1. Percentage
02 - 07
2. Profit & loss
08 - 14
3. Ratio & Proportion and Variation
15 - 21
4. Partnership
22 - 29
5. Simple Interest & Compound Compound Interest
30 - 38
6. Averages and Mixture & Alligation
39 - 45
7. Time & Work
46 - 52
8. Time Speed and Distance
53 - 60
9. Number System
61 - 76
10. Permutation and Combination
77 - 84
11. Probability
85 - 91
12. Set Theory
CHAPTER 1 PERCENTAGES Percentage: The word percentage means per hundred, for example, if a person saves of his income it means he saves 30 part for every 100 100 part he earns. earns. Ex. 1 : quantity of water in milk constitutes 5 parts every 15 parts of t he mixture what is the percentage percentage of water in the mixture? Solution:
Percentage of Water = No. of parts of water/No. of parts of mixture = 5/15 x100=33.33% Percentage increase increase or decrease is calculated with respect to the base value unless mentioned otherwise. Percentage increase or decrease = increase or decrease/base value x 100 Ex. 2 : if A’s income is 25% more than B’s income then by what percentage is B’s income less than A’s? Solution:
Let B’s income =100 A’s income =100+25% of 100=125 B”s income is less than A’s income by 25 Percentage decrease= decrease/base value x100 = 25/125x100=20%
Note here b’s income is compared compared with A’s income, so A’s A’s income should be taken as base value. value. If a quantity is increased by x% then the final quantity is obtained by multiplying original quantity with (100+x/100) If a quantity is reduced by r%, then the final quantity is obtained by multiplying original Quantity with (100+r/100)
Ex. 3 : if the price of petrol is increased by 20% and subsequently subsequently by 40% , what is the final price if the original price is Rs. 25? Solution:
Final price= original price x (100+a/100) x (100+b/100) Therefore Final price = 25 x (100+25/100) x (100+40/100) = 42 Rs.
CHAPTER 1 PERCENTAGES Percentage: The word percentage means per hundred, for example, if a person saves of his income it means he saves 30 part for every 100 100 part he earns. earns. Ex. 1 : quantity of water in milk constitutes 5 parts every 15 parts of t he mixture what is the percentage percentage of water in the mixture? Solution:
Percentage of Water = No. of parts of water/No. of parts of mixture = 5/15 x100=33.33% Percentage increase increase or decrease is calculated with respect to the base value unless mentioned otherwise. Percentage increase or decrease = increase or decrease/base value x 100 Ex. 2 : if A’s income is 25% more than B’s income then by what percentage is B’s income less than A’s? Solution:
Let B’s income =100 A’s income =100+25% of 100=125 B”s income is less than A’s income by 25 Percentage decrease= decrease/base value x100 = 25/125x100=20%
Note here b’s income is compared compared with A’s income, so A’s A’s income should be taken as base value. value. If a quantity is increased by x% then the final quantity is obtained by multiplying original quantity with (100+x/100) If a quantity is reduced by r%, then the final quantity is obtained by multiplying original Quantity with (100+r/100)
Ex. 3 : if the price of petrol is increased by 20% and subsequently subsequently by 40% , what is the final price if the original price is Rs. 25? Solution:
Final price= original price x (100+a/100) x (100+b/100) Therefore Final price = 25 x (100+25/100) x (100+40/100) = 42 Rs.
Note: use negative negative sign in case case of price reduction. reduction.
Successive increase or decrease of percentage:
Let ‘a’ be the first percentage change and ‘b’ be the second, the net change is given by [a+b+axb/100]%
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
LEVEL 1 1) The population of a t own has increased from 133575 to 138918. The percent increase in population is. (A) 2.5%
(B) 3.5%
(c) 3%
(D) 4%
2) Mr. Smith credits 20% of his salary into a fixed deposit account and spends 35% of the remaining on groceries, if cash in hand is Rs.2600 what is his salary? (A) 4000
(B) 4500
(C) 5000
(D) 5500
3) The price of an article has been reduced by 30% , in order to restore the original price the current price must be increased by? by? (A) 30%
(B) 42 6/7%
(C) 23 1/3%
(D) 42 1/9%
4) If the denominator is increased by 20% and the numerator is diminished by 10% the value of fraction is 21/6, the original fraction is ? (A) 5/4
(B) 7/4
(C) 4/7
(D) 14/3
5) If a number is increased by 12% and then decreased by 18%, then find the net % change in the number. (A) 8.16% decrease decrease
(B) 8.42 % increase
(C) 8.44% decrease decrease
(D) 8.18% increase
6) The price per Kg of rice increases by 20% by what percentage should the consumption be decreased such that expenditure remains the same? (A) 20%
(B) 16.67%
(C) 25%
(D) 16.33%
7) Daniel bought a calculator at the store with the tax of 7.5%.The tax amount was 187.Find the calculator price before tax? (A)Rs.174
(B) Rs. 171
(C) Rs. 170
(D) Rs. 179
8) The value of a machine depreciates at the rate of 20% every year. It was purchased 2 years ago. If its present value is 6400, its purchase price was (A) Rs.9240
(B) Rs.7920
(C) Rs.6400
(D) Rs.10000
9) A reduction of 12.5% in the price of a dining table brought down its price to Rs.4375, the original price (in Rs.) of the table was.` (A) 5000
(B) 4900
(C) 5100
(D) None of These
10) Avinash spends 30% of his income on scooter petrol, ¼ o the remaining on house rent and the balance on food. If he spends Rs.300 on petrol, then what is the expenditure on house rent? (A) Rs.180
(B) Rs.175
(C) Rs.160
(D) Rs.165
LEVEL-2 1) In an examination, A got 10% marks less than B, B got 25% marks more than C and C got 20% less than D. If A got 360 marks out of 500, the percentage of marks obtained by D was: (A) 85%
(B) 74%
(C) 82
(D) 80%
2) A rainy day occurs once in every 25 days. Half of the rainy days produce rainbows. The percentage of days having no rainbow is : (A) 2%
(B) 12.5%
(C) 98%
(D) 87.5%
3) 35.In an examination, 35% of total student failed in Hindi, 45% failed in English and 20% in both. The percentage of those who passed in both subjects is? (A) 45%
(B) 40%
(C) 42.5%
(D) 35%
4) A batsman scored 110 runs which included 3 boundaries and 8 sixes. What percent of his total score did he make by running between the wickets? (A) 45%
(B) 45 5 %
(C) 54 6 %
(D) 55%
5) Two students appeared at an examination. One of them secured 9 marks more than the other and his marks was 56% of the sum of their marks. The marks obtained by them are: (A) 39, 30
(B) 41, 32
(C) 42, 33
(D) 43, 34
6) What percentage of numbers from 1to 70 have 1 or 9 in the unit's digit? (A) 16%
(B) 14%
(C) 20%
(D) 21%
7) In an election between two candidates, one got 55% of the total valid votes, 20% of the votes were invalid. If the total number of votes was 7500, the number of valid votes that the other candidate got, was: (A) 2700
(B) 2900
(C)3000
(D) 3100
8) Rajeev buys good worth Rs. 6650. He gets a rebate of 6% on it. After getting the rebate, he pays sales tax @ 10%. Find the amount he will have to pay for the goods. (A) Rs. 6876.10
(B) Rs. 6999.20
(C) Rs. 6654
(D) Rs. 7000
9) Two candidates fought an election. One of them got 84% of the total votes and won with 476 votes. What was the total number of votes polled ? (A) 800
(B)7 00
(C) 550
(D) None of these
10) In a test minimum passing percentage for girls and boys is 35% and 40% respectively. A boy scored 483 marks and failed by 117 marks. What are the minimum passing marks for girls ? (A) 425
(B) 525
(C) 500
(D) 450
11) A positive number is by mistake divided by 6 instead of being multiplied by 6. What is the % error on the basis of correct answer? (A) 3%
(B) 97%
(C) 17%
(D) 83%
12) What 30% of a number is added to another number the second number is increases to its 140%. The second number is =x% of the first number. The value of x is: (A) 30%
(B) 75%
(C) 133.33%
(D) 33.33%
13) In a class 52 students, 25% are rich and other are poor. There are 20 females in the class, of whom 55% are poor. How many rich males are there in the class ? (A) 200
(B) 150
(C) 300
(D) data inadequate
14) Puneet scored 175 marks in a test and failed by 35 marks. If the passing percentage of the test is 35%, what are the maximum marks of the test ? (A) 650
(B) 700
(C) 750
(D)600
15) After three successive equal percentage rise in the salary the sum of 100 rupees turned into 140 rupees and 49 paise. Find the percentage rise in the salary. (A) 12%
(B) 22%
(C) 66%
(D) 82%
COMPANY SPECIFIC QUESTIONS
1. If the radius of a circle is increased by 20% then the area is increased by: (A) 44 %
(B) 144 %
(C) 120
(D) 40 %
(Capgemini)
2. Which of the following is the greatest? (A) 40% of 30 (B) 3/5 of 25
(C) 6.5% of 200
(D) five more than the square of 3 (Capgemini)
3. A number, when 35 is subtracted from it, reduces to its 80 percent. What is four-fifth of that number? (A) 70
(B) 90
(C) 120
(D) 140
4. If 20% of A = B and 40% of B = C, then 60% of (A + B) is:
(L&T Infotech)
(A) 30% of C
(B) 60% of C
(C) 75% of C
(D) None of these
(L&T Infotech)
5. A businessman sold 2/3 of his stock at a gain of 20% and the rest at a gain of 14%. The overall percentage of gain to the businessman is: (A) 12% (B) 17% (C) 18% (D) 20% (Tata-Elxsi) 6. A candidate who gets 30% of the marks in a test fails by 50 marks. Another candidate who gets 320 marks fails by 30 marks. Find the maximum marks. (A) 900
(B) 1000
(C) 1200
(D) 800
(PERSISTENT)
7. If 24% of 395 is y, then what is 360% of 79. (A) y
(B) 2y
(C) 3y
(D) 3.5 y
(SAMSUNG)
8. A jogger wants to save 1/4th of his jogging time. He should increase his speed by how much percentage. (A) 25% (B) 36.95% (C) 33.33 % (D) 43.73% (BirlaSoft)
9. A merchant gained 25% by selling a stock of grains. In maintaining the stock, he has spent 10,000 Rupees. If the total cost of grains sold is Rs. 60,000, how much has he gained? (A) Rs. 5, 000
(B) Rs. 15,000 (C) Rs.10, 000 (D) None of these
(ZENSAR)
10. In a class of 50 students, the score of 30% of students is above Shahrukh. What percent of students have score below Shahrukh? (ZENSAR) (A) 70% (B) 68% (C) 66%s (D) None of these
ANSWER KEY LEVEL 1:
1D
2C
3B
4D
5A
6B
7A
8D
9A
10 B
6C
7A
8A
9B
10 B
7C
8A
9A
10 B
ANSWER KEY LEVEL 2:
1D
2C
3B
4B
5 C
11 B
12 B
13 D
14 D
15 A
ANSWER KEYCOMPANY SPECIFIC QUESTIONS:
1A
2B
3D
4D
5C
6B
CHAPTER 2 PROFIT AND LOSS
1.1 Profit and Loss:
The difference between selling price and cost price is known as profit or loss. Profit = selling price – cost price Loss = cost price – selling price Profit % = profit / cost price x 100 Loss % = loss / cost price x 100 Selling price = [1 + gain% / 100] x cost price Selling price = [1 + loss% / 100] x cost price
Ex 1: Find the profit % , if an article worth 300% is sold for 312 ? Solution:
Profit= selling price-cost price = 312 – 300 = 12 Profit % = profit / cost price x100 = 12 / 300 x 100 = 4% Percentage reduction in consumption If the rate of a commodity is increased, then the consumption should be reduced to maintain the same expenditure. Percentage reduction in consumption = (% change / 100% + change) x100
Ex 2: If the rate of rice is increased by 25% then what should be the consumption to maintain the same expenditure? Solution:
= (25 / 100 + 25) x 100 = 20% So the quantity of rice consumption should be reduced by 20% to maintain the same expenditure. Successive Discount: If two successive discounts of a% and b% are offered then the net discount offered is [a+b-axb/100]%
Ex 3: Find the net discount offered on two successive discounts of 20% and 30%? Solution:
Net discount = [a + b – a x b / 100] = [20 + 30 – 20 x 30 / 100] = 44 % Discount is the reduction of price. if a% discount is offered then the article would be sold for (100 - a)% of the cost price. Selling price = (100-a) / 100 x Cost price 1.2 False Weight:
In case a false weight which is less than the actual weight is used then the transaction ends in a profit. % profit = [error / true value – error x 100] Ex 4: If a shop keeper sells 800 gm at cost price claiming it to be 1 kg, find the gain %? Solution:
% profit = [ error / true value – error x 100] = 200 / 1000 - 200 x 100 = 25% In case where selling price of two articles is the same and one is sold at a loss of x% t hen after at a profit of x% then this transaction always leads to a loss of x^2/100 1.3 Marked Price:
MRP of an article is known as Marked Price or labeled price or listed price and denoted by MP. Discount always carried on MP (MRP). Marked Price is always 100 in the case of discount.
--------------------------------------------------------------------------------------------------------------------------
LEVEL 1
1) A retailer buys 40 pens at the marked price of 35 pens from a wholesaler. If he sells these pens giving a discount of 1%, what is the profit percent? (A)13%
(B) 10 %
(C) 15 %
(D) 16 %
2) John bought 15 apples for Rs.10 and sold them at the rate of 12 apples for Rs.12. What is the percentage of profit made by him? (A) 55%
(B) 60 %
(C) 50 %
(D) 36 %
3) A person incurs 5% loss by selling a bat for Rs 1140. At what price should the watch be sold to earn 5% profit? (A) 1260
(B) 1255
(C) 1270
(D) 1250
4) The price of an article including the sales tax is Rs 616.The rate of sales tax is 10%, if the shopkeeper has made a profit of 12%, then the cost price of the article is? (A) 490
(B) 530
(C) 500
(D) 600
5) Mayank Bothra purchased 20 dozens of toys at the rate of 375 Rs per dozen He sold each one of them at the rate of Rs 33.What was his percentage profit? (A) 5.6%
(B) 4.8%
(C) 6 %
(D) 6.8%
6) A person earns 15% on investment but loses 10% on another investment .If the ratio of the two investments be 3:5, what is the gain or loss on the two investments taken together? (A) 65%
(B) 55%
(C) 60 %
(D) 63%
7) An article costing Rs. 200 is marked 25% higher than its C.P. and is sold at a discount of 10% for cash payment. A customer is ready to pay t he complete money on the spot . What is the shopkeeper’s percentage profit? (A) 15%
(B) 12.5%
(C) 20 %
(D) None of these
8) In a store a dress tagged at RS 800 was offered at a discount of 12.5% when it did not sell at owner price an additional discount of 10%was offered .what was the final selling price? (A) 630
(B) 650
(C) 550
(D) None of these
9 ) A man buys 5 horses and 7 bulls for Rs 1950 he sells the horses at a profit of 10%and bulls at a profit of 16% and on the whole his gain is Rs 237 what price does he pay for a horse? (A) 230
(B) 250
(C) 300
(D) 225
10) If a commission of 10% is given on the marked price of a book than the publisher gains 20%.If the commission is increased to 15% than what is t he gain percent?
(A) 15
(B) 11.22
(C) 12.5
(D) 13.33
LEVEL 2
1) If toffees are bought at the rate 18 for a rupee than how many of them must be sold for a rupee to gain 20%? (A) 16
(B) 12
(C) 14
(D) 15
2) In order to increase revenue a dealer announces 20% reduction in the unit price of an article as a result his sales volume increases by 20% .what is the overall gain / loss to the dealer? (A) 5% loss
(B) 8% loss
(C) 4% loss
(D) 8.5% loss
3) A dishonest dealer pretends to sell at the cost price but earn a profit of 25% by under weighing .what weight must he be using for a kg? (A) 700gm
(B) 800gm
(C) 750gm
(D) 600gm
4) A man buys two goats at Rs. 120 each. He sells one at 25% gain and other at 25% loss. How much is his profit or loss? (A) No loss no gain
(B) 1% Loss
(C) 1% Profit
(D) None of these.
5) Profit made by selling an article at Rs. 425 is the same as the loss incurred by selling it at RS. 375. What is the cost price of the article? (A) 500 6)
(B) 400
(C) 600
(D) 550
A shopkeeper bought 22kg of certain commodity of type “A” at rate of Rs. 8.40 per kg and 28 kg of the same commodity of type “B” at rate of Rs. 6.30 per kg. He mixed the two types and sold the mixture at the rate of Rs 7.8 per kg. What is his profit or loss? (A) 28.8 profit
(B) 27 profit
(C) 30 profit
(D) 25 profit
7) A shopkeeper offers a discount of 15% after making up 70% on cost price. As a result ,his profit drops by Rs. 127.5 .Find the cost price? (A) Rs.500
(B) Rs. 450
(C) Rs.600
(D) Rs. 550
8) A person purchases 50 dozen eggs at Rs.4 per dozen. Of these, 40 eggs were found broken. At what price should he sell the remaining eggs in order to make a profit of 5%? (A) Rs. 5.00
(B) Rs. 4.50
(C) Rs. 6.50
(D) Rs. 4.00
9) Two articles sold at Rs. 198 each such that a profit of 10% is made on the first while a loss of 10% is made on the other. What would be the net profit/loss on the two transactions combined? (A) 27%
(B) Rs. 4
(C) No change
(D) None of these
10) If the income of Ram is more than that of Shyam by 37.5%, then by how much % Shyam’s income is less than that of Ram? (A) 27% these
(B) 25%
(C) No change
(D) None of
11) If books bought at price ranging from Rs.200 to Rs.325 are sold at prices ranging from Rs.300 to Rs.450, what is the greatest possible profit that might be made in selling eight books? (A) Rs.400
(B) Rs. 2000
(C) Rs.600
(D) Cannot be determined
12) Saif purchased 20 dozens of toys at the rate of Rs.375 per dozen. He sold each one at the rate of Rs.33. What was his percentage profit ? (A) 3.5%
(B) 5%
(C) 5.6%
(D) 6.5%
13) A sells an article which costs him Rs.500 to B at a profit of 20%. B then sells it to C, making a profit of 10% on the price he paid to A. How much does C pay B ? (A) Rs.472
(B) Rs.476
(C) Rs.528
(D) Rs.660
14) The profit earned by selling an article for Rs.832 is equal to the loss incurred when the same article is sold for Rs.448. What should be the sale price for make 50% profit? (A) Rs.920
(B) Rs.960
(C) Rs.1060
(D) Rs.1160
15) A man purchased a box full of pencils at the rate of 7 for Rs.9 and sold all at the rate of 8 for Rs.11. In this transaction, he gained Rs.10. How many pencils did the box contain ? (A) 100
(B) 112
(C) 114
(D) 115
COMPANY SPECIFIC QUESTIONS
1. A vendor sold two things at Rs.12 each, with one item at 25% profit and other at 20% loss. By this transaction he made profit or loss and by how much? (A) loss 40%
(B) loss 60%
(C) profit 27%
[L&T Infotech] (D) loss 38%
2. A trader, frauds by 10% while buying and 10% while selling the same. What is the total gain he obtained during the transaction? (A) 13% 3.
(B) 20%
[COGNIZANT] (C)10%
(D) None of these
The prices of Gold suddenly rose from Rs. 3000 per gram to Rs. 3298 per gram. Mrs. Verma was lucky enough to buy 340 grams yesterday only. How much Mrs. Verma will gain if she sells the gold today?
[BIRLA SOFT] (A) Rs. 98260
(B) Rs. 101320
(C) Rs. 62880
(D) Rs.70520
4. A man sold two cows for Rs. 210 at a total profit of 5%. He sold one cow at a l oss of 10% and another at a profit of 10%. What is the price of each cow (in Rs.)? (A) 130 & 60
(B) 120 & 80
(C) 150 & 50
[INFOSYS]
(D) None of these
5. If the price of an item is decreased by 10% and then increased by 10%, the net effect on the price of the item is?
[ TCS ]
(A) A decrease of 99% (B) No change
(C) A decrease of 1% (D) An increase of 1%
6. A merchant sells an item at a 20 percent discount, but still makes a gross profit of 20 percent of the cost. What percent of cost would be the gross profit on the item have been if it had been sold without the discount? (A) 20%
(B) 40%
[HCL] (C) 50%
(D) 60%
7. Ravi's salary was reduced by 25%. Percentage increase to be effected to bring the salary to the original level is: (A) 33.33%
[WIPRO] (B) 25%
(C) 20%
(D) 30%
8. The total population of a village is 5000. The number of males and females increases by 10% and 15%, respectively and consequently the population of the village becomes 5600. What was the number of males in the village? (A) 2000
[WIPRO]
(B) 2500
(C) 3000
(D) 4000
9. A shopkeeper labels the price of article 15% above t he cost price. If he allow Rs 51.20 discount on an article of Rs 1024, find his profit percent. (A) 10%
(B) 8% (C) 12%
[ACCENTURE]
(D) 9.25%
10. The profit earned by selling an article for Rs. 832 is equal to the loss incurred when the same article is sold for Rs. 448. What should be the sale price for making 50% profit? [L&T INFOTECH] (A) Rs. 920
(B) Rs. 960
(C) Rs. 1060
(D) Rs. 1200
ANSWER KEY LEVEL 1:
1B
2C
3A
4C
5A
6D
7B
8A
9B
10 D
6A
7A
8B
9B
10 D
ANSWER KEY LEVEL 2:
1D
2C
3B
4C
5B
11 B
12 C
13 D
14 B
15 B
ANSWER KEYCOMPANY SPECIFIC QUESTIONS:
1B
2D
3B
4C
5C
6C
7A
8C
CHAPTER 3 RATIO, PROPORTION & VARIATION 3.1 Ratio: A ratio is a comparison of two numbers by division. The ratio of a to b is expressed as follows: a:b=
Here ‘a’ is called antecedent and ‘b’ is called a consequent 3.2 Proportion: Proportion is the equality of two ratios. E.g. 4/20= 1/5 is a proportion If a, b, c, d are in proportion Then
=
Ad = cb Product of extremes = Product of means
3.3 Properties of Proportion: If a/b = c/d then, Invertendo
b/a=d/c
Alternendo
a/c=b/d
Componendo
(a+b) / b = (c+d) / d
Dividendo
(a-b) / b = (c-d) / d
Componendo & Dividendo
(a+b) / (a-b) = (c+d) / (c-d)
Here, ‘a’ is the 1st proportional and ‘b’ is the 2nd proportional ‘c’ is the 3rd proportional ‘d’ is the 4th proportional
3.4 Duplicate Ratios:
Duplicate ratio of (a : b) is (a2 : b2).
9D
10 B
Sub-duplicate ratio of (a : b) is (a : b). Triplicate ratio of (a : b) is (a3 : b3). Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).
3.5 Continued Proportion: Three quantities a, b, c are said to be in continued proportion.
If a:b=b:c=>b=√ac Here, b is called the mean proportional of a and c E.g. The mean proportional of 0.32 and 0.02 is X=
0.08
Third proportional of a and b is b²/a E.g. Find the third proportional of 16 and 24 X=24²/16=36
LEVEL 1 1. The number of marbles with A and B are in the Ratio of 10:11. Which of t he following cannot be a possible number of marbles with A and B together? (A) 189
(B) 210
(C) 231
(D) 153
2. Two numbers are in the ratio of 2:5. If the difference between these numbers is 24, then find the sum of the numbers. (A) 52
(B) 46
(C) 48
(D) None of these
3. If a:b :: 1:6 and b:c :: 5:7, then, a:b:c = ? (A) 2:14:56
(B) 3: 25:46
(C) 5:30:42
(D) None of these
4. The sum of three numbers which are in ratio 5:7:13 is 1250. The difference between the greatest number and the least number is (A) 50
(B) 100
(C) 400
(D) 300
5. The salaries A, B, C are in the ratio 2 : 3 : 5. If the increments of 15%, 10% and 20% are allowed respectively in their salaries, then what will be the new ratio of their salaries? (A) 3:3:10
(B) 10:11:20
(C) 23:33:60
(D) can’t be determined
6. A's money is to B's money as 4:5 and B's money is to C's money as 2:3. If A has Rs.800, C has (A) Rs. 1000
(B) Rs. 1200
(C) Rs. 1500
(D) Rs. 2000
7. The mean Proportional between two numbers is 9 and the t hird proportional of the two numbers is 243. Find the larger of the two numbers? (A) 27
(B) 81
(C) 9
(D) 54
8. If 93 is divided into two parts such that thrice the first part and twice the second part are in the ratio 25: 4. Find the first part? (A) 60
(B) 75
(C) 50
(D) 70
9. The students in three classes are in the ratio 2:3:5. If 20 students are increased in each class, the ratio changes to 4:5:7. The total number of students before the increase were : (A) 100
(B) 90
(C)10
(D) none of these
10. What least number must be subtracted from each of the numbers 14, 17, 34 and 42 so that the remainders are proportional? (A) 10
(B)5
(C) 7
(D) 2
LEVEL 2 1. The monthly Expenses of Peter on his bike is partly fixed and partly vary with number of kilometers he travels in a month. If he travels 100 kms in a month his total car expenses will be Rs 3500. If he travels 200 kms in a month, his t otal expenses will be Rs 4000. If he travels 500 kms in a month, what will be his total expenses? (A) 5500
(B) 5200
(C) 5300
(D) 5450
2. Three friends divide Rs.624 among themselves in the ratio 1/2:1/3:1/4. The share of the third friend is : (A) Rs. 288
(B) Rs.192
(C) Rs. 148
(D) Rs. 144
3. The ratio of flow of water in pipes varies inversely as the square of the radius of the pipes. What is the ratio of the rates of flow in two pipes of diameters 2 cm and 4 cm? (A) 3:1
(B) 6:1
(C) 7:15
(D) 4:1
4. Rs. 170 is generated using a combination of 10 paise, 25 paise and 50paise coins, if the ratio of 10paise, 25 paise and 50 paise coins is 5:10:11, then the total number of coins is: (A) 100
(B) 200
(C) 520
(D) 220
5. A, B and C do a work in 20, 25 and 30 days, respectively. They undertook to finish the work together for Rs.2220, then the share of A exceeds that of B by : (A) Rs. 120
(B) Rs.180
(C) Rs.300
(D) Rs. 600
6. The ratio of the age of a man and his wife is 4:3. After 4 years, this ratio will be 9:7. If at the time of marriage, the ratio was 5:3, then how many years ago they were married? (A) 12 years
(B) 8 years
(C) 10 years
(D) 15 years
7. A, B, C and D share a property worth Rs. 77500.If A:B = 3:2,B:C = 5:4 and C:D=3:7. Find the share of B. (A) Rs. 20000
(B) Rs.15000
(C) Rs. 25000
(D) Rs. 22000
8. 243 have been divided into three parts such that half of the first part, one third of t he second part and one fourth of the third part are equal. The largest part is: (A) 102
(B) 108
(C) 100
(D) 110
9. The proportion of milk and water in two samples is 5:2 and 7:5 if a mixture comprising of equal quantities of two samples is made, the proportion of milk and water in the mixture is (A) 12:7
(B) 7:12
(C) 109:59
(D) 59:109
10. The annual salary of Mr. John, Mr. Adam and Mr. Joe is in the ratio 2:3:5. If the salary of Mr. Joe is 90,000 more than that of Mr. John, then the monthly salary of Mr. Adam is (A) 7,500
(B)75,000
(C) 90,000
(D) None of these
11. If the work done by (x – 1) men in (x + 1) days is to the work done by (x + 2) men in ( x – 1) days is in the ratio 9 : 10, then the value of x is (A) 10
(B) 12
(C) 8
(D) 15
12. The duration of a railway journey varies as the distance and inversely as the velocity; the velocity varies directly as the square root of the quantity of coal used, and inversely as the number carriages in the train. In a journey of 50 km in half an hour with 18 carriages, 100 kg of coal is required. How much coal will be consumed in a journey of 42 km in 28 minutes with 16 carriages? (A) 64 kg
(B) 49 kg
(C) 25 kg
(D) 36 kg
13. The weight of a circular disc varies as the square of the radius when the thickness remains the same; it also varies as the thickness when the radius remains the same. Two discs have their thicknesses in the ratio of 9: 8; the ratio of the radii if the weight of the first is twice that of the second is (A) 4 : 3
(B) 5 : 2
(C) 2 : 1
(D) 1 : 2
14. If a and b are positive integers then always lies between: (A) (a + b)/( a – b) and ab
(B) a/ b and (a + 2b)/( a + b)
(C) a and b
(D) ab/( a + b) and (a – b)/ ab
15. The cost of digging a pit was Rs 1,347. How much will it cost (approximately) if the wages of workmen per day had been increased by 1/ 8 of the former wages and length of the working day increased by 1/ 20 of the former period? (A) Rs 1443
(B) Rs 1234
(C) Rs 1439
(D) Rs 1000
COMPANY SPECIFIC QUESTIONS
1. The ratio of daily wages of two workers is 4:3 and one gets daily Rs. 9 more t han the other. What are their daily wages? [ACCENTURE] (A) Rs. 32 and Rs.24 (B) Rs.60 and Rs. 45 (C) Rs. 80 and Rs. 60 (D) None of these 2. In a business P and Q invested amounts in the ratio 3:4, whereas the ratio between amounts invested by P and R was 6:7. If Rs 106501.50 was their profit, how much amount did Q receive? [ACCENTURE] (A) Rs. 40572 (B) Rs 30429 (C) Rs 35500.50 (D) Rs 34629 3. If Rs. 1260 is divided amongst A, B and C in the ratio 2:3:4. What is C’s share? [TCS] (A) 850 (B) 560 (C) 620 (D) 460 4. The ratio of incomes of C and D is 3:4. The ratio of their expenditures is 4:5. Find the ratio of their savings if the savings of C is one fourths of his income? [TCS] (A) 13/17 (B) 8/11 (C)12/19 (D) None of these 5. If
and
(A) 3/5
, find the value of (B) 3/17
[ACCENTURE] (C) 9/25
(D) 8/17
6. The ratio of boys and girls in a school is 370:356, and that of teachers to boys is 105:37. If the number of girls in school is 6764, how many teachers are there in the school? [SAMSUNG] (A) 6764 (B) 19950 (C) 20292 (D) 20342 7. The age of Ram and Sam are in the ratio 5:6 and after 4 years their ratios are 7:8 then what is the present age of Sam? [TCS] (A) 13 yrs. (B) 12 yrs. (C) 11 yrs. (D) none of these 8. 729 ml of a mixture contains milk and water in ratio 7:2. How much of the water is t o be added to get a new mixture containing half milk and half water? [HCL] (A) 79 ml
(B) 81 ml
(C) 72 ml
(D) 91 ml
9. Two solutions have milk & water in the ratio 7:5 and 6:11. Find the proportion in which these two solutions should b e mixed, so that the resulting solution has 1 part milk and 2 parts water? [COGNIZANT] (A) 35:3 (B) 21:36 (C) Not possible (D) none of these 10. The sides of a triangle are in the ratio (1/2):(1/3):(1/4) and its perimeter is 104 cm. The length of the longest side is: [L & T ] (A) 48
(B) 52
(C) 32
(D) 36
ANSWER KEY LEVEL 1:
1D
2D
3C
4C
5C
6C
7A
8B
9A
10 D
6A
7B
8B
9C
10 C
8B
9C
10 A
ANSWER KEY LEVEL 2:
1A
2D
3D
4C
5B
11 C
12 A
13 A
14 B
15 A
ANSWER KEY : COMPANY SPECIFIC QUESTIONS:
1D
2A
3B
4C
5D
6B
7B
Chapter 4 PARTNERSHIP 4.1 Partnership
If two or more people jointly run a business, the “profit or loss is shared in the ratio of the investments done by the people involved. They are called partners and the deal is known as partnership. In short, Partnership is an association of two or more parties, for a common business.
4.2 Types of Partnership Partnership is of two types: Simple Partnership: Simple partnership is the one in which the capitals of each of the partners are invested for the same time and profit or loss in a partnership are divided among the partners in t he ratio of their investments. Suppose A and B invest Rs. X and Rs. Y , respectively, for a year in a business, then at the end of the year: (A's share of profit): (B's share of profit) = x : y. Compound Partnership: Compound partnership is one wherein the periods of investment are unequal. And equivalent capital for a unit of time is calculated by multiplying the capital with the number of time units it was in business. Suppose A invests Rs. x for p months and B invests Rs. y for q months then, (A's share of profit) : (B's share of profit)= xp : yq. A partner who manages the business is known as a working partner and the one who simply invests the money is a sleeping partner.
Ex. Three partners A, B and C invested Rs. 1000, Rs. 1200 and Rs. 1500 respectively in business for one year. How should they divide a profit of Rs. 1295? Solution: As investment period is same so, profit should be divided in ratio of capitals as, 10 : 12 : 15, also 10 + 12 + 15 = 37
Ex. A, B and C enter into a business. A put Rs. 1000 for 6 months, B puts Rs. 1200 for 8 months and C puts Rs. 1400 for 10 months. Their gain was Rs. 666. Find out the share of each partner.
Solution: Ratio of profits will be 10*6 : 12*8 : 14*10 = 60 : 96 : 140 = 15 : 24 : 35
Ex. Three friends Ram, Shyam and Mohan enter into partnership. Ram put one forth of capital for one fourth of the time. Shyam puts one third of capital for one half of time. Shyam puts remaining capital for full period of time. Find the division of profit Rs. 806 among three friends. Solution: Ram's share : Shyam's share : Mohan's share =
Now LCM of 16, 6, 12 is 48, so multiplying the equation by 48 we get ratio as 3 : 8 : 20 , also 3 +8 + 20 = 31
Ex. A, B and C enter into a partnership and their shares are in ratio 1/2 : 1/3 : 1/4, after 2 months, A withdraws half of his capital and after 10 months, a profit of Rs 378 is divided among them. What is B's share? Solution : Ratio of investments =1/2 : 1/3 : 1/4 , now LCM of 2, 3, 4 is 12 on multiplying the ratio with 12 we get 6 : 4 : 3 , also we assume their initial investment be 6x, 2x and 3x so, we can write: A : B :C
Ex. A and B are partners in a business. A contributes 1 / 4 of the capital for 15 months and B received 2 / 3 of the profit, for how long B's money was used? Solution : Let total profit is x
Let total capital invested be Rs P and A's money was used for 15 months while B's money was used for b
months then we can write the equation as-
So, B's money was used for 10 months Ex. A began a business with Rs. 21, 000 and is joined afterwards by B with Rs. 42,500. For how much period does B join, If the profits at the end of the year are divided in the ratio 3 : 1? Solution: Let B joined for P months
then we can write equation as - (85,000 * 12 ) : ( 42,500 * P ) = 3 : 1
Ex. In a business A and C invested amounts in the ratio 2 : 1, whereas the ratio between amounts invested by A and B was 3 : 2 . If Rs. 56,914 was their profit, how much amount did B receive? Solution : A : B = 3 : 2 , A : C = 2 : 1
B:A=2:3,B:A=4:6 Now, A : C = 2 : 1, A : C = 6 : 3 , So B : A : C = 4 : 6 : 3 We can write as A : B : C = 6 : 4 : 3
LEVEL 1 1. Raj invested Rs 76000 in a business. After few months Monty joined him and invests Rs 57000. At the end of year both of them share the profits at the ratio of 2:1. After how many months Monty joined Raj ? (A) 4 Months (B) 8 Months (C) 5 Months (D) 6 Months 2. A and B started a business by investing money in ratio of 5:6. C j oined them after few months by
sharing an amount equal to B's share. At the end of year 20% profit was earned which was equal to Rs 98,000. How much money was invested by C? (A) Rs 200000
(B) 210000
(C) 205000
(D) 215000
3. A, B and C shared profits in ratio of 5:7:8. They partnered for 14 months, 8 months and 7 months
respectively. What was the ratio of their investments? (A) 21:49:56
(B) 20:49:56
(C) 20:49:64
(D) 24:49:64
4. X and Y invest Rs.21000 and Rs.17500 respectively in a business. At the end of the year, they
make a profit of Rs.26400. What is the share of X in t he profit? (A) Rs.14400
(B) Rs.26400
(C) Rs.12000
(D) Rs.12500
5. A, B and C enter into a partnership investing Rs 35000, Rs 45000 and 55000. Find the their
respective shares in annual profit of 40,500 (A) 10500, 13500, 19500
(B) 10500, 13500, 18500
(C) 10500, 13500, 17500
(D) 10500, 13500, 16500
6. A, B and C start a business each investing Rs. 20,000. After 5 months A withdrew Rs. 5000, B
withdrew Rs. 4000 and C invests Rs. 6000 more. At the end of the year, a total profit of Rs. 69,900 was recorded. Find the share of A? (A) 21200
(B) 28200
(C) 20500
(D) 26000
7. A and B started a partnership business investing some amount in the ratio of 3 : 5. C joined then
after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed among A, B and C? (A) 3 : 5 : 2
(B) 3 : 5 : 5
(C) 6 : 10 : 5
(D) Data inadequate
8. Sumit and Ravi started a business by investing Rs 85000 and 15000 respectively. In what ratio the
profit earned after 2 years be divided between Sumit and Ravi respectively? (A) 17:1 (B) 17:2 (C) 17:3 (D)17:4 9. A,B and C enter into a partnership investing Rs 35000, Rs 45000 and 55000. Find the their
respective shares in annual profit of 40,500 (A) 10500, 13500, 19500 (B) 10500, 13500, 17500
(C) 10500, 13500, 18500 (D) 10500, 13500, 16500
10. Rs. 700 is divided among A, B, C so that A receives half as much as B and B half as much as C.
Then C's share is (A) Rs 200
(B) Rs 300
(C) Rs 400
LEVEL 2
(D) Rs 500
1. A and B started a partnership business investing some amount in the ratio of 3 : 5. C joined them after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed amount A, B and C (A) 3:7:5 (B) 6:10:5 (C) 6:10:7 (D) 6:7:5 2. Three partners A,B and C shared the profit in a software business in the ratio 5:7:8. They had partnered for 14 months, 8 months and 7 months respectively. Find the ratio of their investments? (A) 19:49:64 (B) 20:49:64 (C) 20:49:65 (D) 20:50:64 3. A starts business with Rs. 3500 and after 5 months, B joins with A as his partner. After a year, the profit is divided in the ratio 2:3. What is B's contribution in the capital? (A) Rs 9000 (B) Rs 7000 (C) Rs 5000 (D) Rs 4000 4. Anand and Deepak started a business investing Rs.22,500 and Rs.35,000 respectively. Out of a total profit of Rs. 13,800. Deepak's share is (A) Rs 9600 (B) Rs 8500 (C) Rs 8450 (D) Rs 8400 5. A and B enter in to a partnership and A invests Rs.10, 000 in the partnership. At the end of 4 months he withdraws Rs.2000. At the end of another 5 months, he withdraws another Rs.3000. If B invests a certain sum in the partnership at the beginning of the year and leaves it intact and receives Rs.9600 as his share of the total profit of Rs.19, 100 for the year, how much did B invest in the company? (A) Rs. 12000
(B) Rs. 96000
(C) Rs. 8000
(D) Rs. 6000
6. A, B and C started a business by investing Rs. 1,20,000, Rs. 1,35,000 and ,Rs.1,50,000 respectively. Find the share of each, out of an annual profit of Rs. 56,700? (A) Rs. 16800, 18900, 21000
(B) Rs. 16800, 21000, 18900
(C) Rs. 18900, 16800, 21000
(D) Rs. 18900, 21000, 16800
7. Alfred started a business investing Rs. 45,000. After 3 months, Peter joined him with a capital of Rs. 60,000. After another 6 months, Ronald joined them with a capital of Rs. 90,000. At the end of the year, they made a profit of Rs. 16,500. Find the lire of each. (A) Rs. 3300, 6600, 3300
(B) Rs. 3300, 6600, 6600
(C) Rs. 6600, 6600, 3300
(D) Rs. 6600, 3300, 3300
8. A, B and C start a business each investing Rs 20,000 After 5 months A withdrew Rs.6000 B withdrew Rs. 4000 and C invests Rs. 6000 more. At the end of the year, a tot al profit of Rs. 69,900 was recorded. Find the share of each? (A) Rs. 28200, 21200, 20500
(B) Rs. 20500, 28200, 21200
(C) Rs. 21200, 20500, 28200
(D) Rs. 20500, 21200, 28200
9. A, B and C enter into partnership, A invests 3 times as much as B and B i nvests two-third of what C invests. At the end of the year, the profit earned is Rs. 6600. What is the share of B? (A) Rs. 1500
(B) Rs. 1200
(C) Rs. 1000
(D) Rs. 800
10. Four milkmen rented a pasture. A grazed 24 cows for 3 months; B 10 for 5 months; C 35 cows for 4 months and D 21 cows for 3 months. If A's share of rent is Rs. 720, find the total rent of the field? (A) Rs. 3000
(B) Rs. 3250
(C) Rs. 3500
(D) Rs. 3750
11. Nirmal and Kapil started a business investing Rs 9, 000 and Rs 12, 000 respectively. After 6 months, Kapil withdrew half of his investment. If after a year, the total profit was Rs 4, 600 what
was Kapil’s share in it? (A) Rs. 2000
(B) Rs. 2600
(C) Rs. 1900
(D) Rs. 2300
12. A, B and C enter into a partnership investing Rs. 3800, Rs. 4200 and Rs. 4000, resp. A profit of Rs. 1800 is divided among them. So what is the share of B? (A) Rs. 520
(B) Rs.780
(C) Rs. 600
(D) Rs. 630
13. A started a business with a capital of Rs. 10000 and 4 month later, B joined him with a capital of Rs. 5000.What is the share of A in the total profit of Rs. 2000 at the end of the year?
(A) Rs. 1500
(B) Rs. 1800
(C) Rs. 1600
(D) Rs. 1700
14. A, B and C start a business. A invests 3 times as much as B invests and B invests 2/3 of what C invests. If the total profit is Rs. 1320, fi nd the share of A.
(A) Rs. 720
(B) Rs. 620
(C) Rs. 600
(D) Rs. 800
15. B is a sleeping partner and A is the working partner. A puts in Rs. 5000 and B puts in Rs. 6000. A received 12
% of profit for managing the business and the rest is divided in proportion to their
capitals. What is the share of A in a profit of Rs. 880?
(A) Rs. 770
(B) Rs. 460
(C) Rs. 520
(D) None of these
COMPANY SPECIFIC QUESTIONS 1.
In a partnership, A invests 1/6 of the capital for 1/6 of the time, B invests 1/3 of the capital for 1/3
of the time and C, the rest of the capital for the whole time. Out of a profit of Rs. 4600, B’s share is: [L & T] (A) Rs. 650
(B) Rs. 800
(C) Rs. 960
(D) Rs. 100
2. Ram and Shyam invested Rs 3000 and Rs 4000 respectively in a business. If Ram doubles his
capital after 6 months, then in what proportion should Ram and Shyam divide that Year’s profit? [WIPRO]
(A) 3:4
(B) 4:3
(C) 16:9
(D) 9:8
3. P started a business by investing Rs 2,700 after sometime Q joined him by investing Rs 2,025. At the end of one year, the profit was divided in the ratio 2:1 After how many months did Q joined the business? [TCS] (A) 3 (B) 4 (C) 6 (D) 9 4. A, B and C are partners. A receives 2/3 of the preofit, B and C divide the remainder equally. A’s income increased by Rs 400 when the rate of profit rises from 5 t o 7 percent. The capital of B is? [VEDANTA] (A) Rs 30,000 (B) Rs 5,000 (C) 6,000 (D) 15,000
ANSWER KEY LEVEL 1: 1A
2B
3C
4A
5D
6C
7C
8C
9D
10 C
6A
7C
8D
9B
10 B
7
8
9
10
ANSWER KEY LEVEL 2: 1B
2B
3A
4D
5C
11 D
12 D
13 A
14 A
15 B
ANSWER KEY COMPANY SPECIFIC QUESTIONS: 1B
2D
3B
4B
5
6
Chapter 5 SIMPLE INTEREST COMPOUND INTEREST The lending and borrowing of money has been happening since thousands of years. Any sum of money, borrowed for a certain period, will invite an extra cost t o be paid on the money borrowed; this extra cost at a fixed rate is called the interest. The money borrowed is called the principal. The sum of interest and principal is called the amount. The time for which money is borrowed is called the period. Amount = Principal + Interest
The interest paid per hundred (or percent) for a year is called the rate percent per annum. The rate of interest is almost always taken as per annum; in calculations we will always consider it per annum unless indicated. The interest is of two types; one is simple, the other is compound. 5.1 Simple Interest (S.I)
It is the interest paid as it falls due, at the end of decided period (yearly, half yearly or quarterly), the principal is said to be lent or borrowed at simple interest. Simple Interest, SI = PRT / 100 Here P = Principal, R = Rate per annum, T = Time in years. Therefore Amount, A = P +
= P [1 +
]
If T is given in months, since rate is per annum, the time has to be converted in years, so the period in months has to be divided by 12. If T = 2 months = 2/12 years E.g. 1: Find the amount on S.I., when Rs. 4000 is lent at 5 % p.a. for 5 years.
By the formula, A = P (1 + RT/100) = 4000(1 + 5 x 5/100 ) = Rs. 5000 5.2 Compound Interest (C.I)
The compound interest is essentially interest over interest. The interest due is added to the principal and that becomes the new principal for the interest to be levied. This method of interest calculation is called compound interest. This can be for an y period (yearly, half yearly or quarterly) and will be called “Period compounded” like yearly com pounded or quarterly compounded and so on.
First period’s principal + first period’s interest = second period’s principal )time - Principal
Compound interest = principal (1 + CI = P { 1 +
}T – P
Here Amount = principal (1 +
)time
E.g. 2: Find the compound interest on Rs. 4500 for 3 years at 6 % per annum Using the formula, A = P (1 + R/100) T = 4500(1 + 6/100) 3 = 4500 (1.06) 3 = 5360 Compound Interest = 5360 – 4500 = Rs 860
5.3 THE RULE OF 72 The rule of 72 is a quick way to show how long it will take to double your money. The equation for the rule of 72 is: Number of years for money to double = (72/Annual Interest Rate) interest rate At 8% interest, it will take 72/8 = 9 years for your money to double. Here are more examples: At 6%, it will take 12 years (72/6 = 12). At 12%, it will take 6 years (72/12 = 6).
The rule of 72 is a shortcut to estimate the magic of compound interest that makes your money grow.
• Remember that the rule of 72 is an approximation and its accur acy reduces as the interest rate becomes high. Important notes 1. In case interest is paid half yearly, then the interest is divided by 2, and used as (R/2) in the formula and the time is multiplied by 2, and used as 2T in t he formula, given by
A = P [1+
]2T
E.g.3: Find the compound interest on Rs. 5000 for 3 years at 6 % per annum compounded half yearly. Using the formula, A = P [1 + ( R / 200 ) ]2T = 5000(1 + 6/200) 3x2 = 5000 (1.03) 6 = 5971 Compound interest = 5971 – 5000 = Rs 971 2. In case, interest is paid quarterly, then the interest is divided by 4, and used as (R/4) in the formula and the time is multiplied by 4, and used as 4T in the formula, given by
A=P[1+
]4T payable quarterly (rate = R/4, time = 4T)
E.g.4: Find the compound interest on Rs. 5000 for 3 years at 6 % per annum compounded quarterly. Using the formula, A = P [1 + (R / 400 ) ] 4T = 5000(1 + 6/400) 3x4 = 5000 (1.015) 12 = 5978 Compound interest = 5978 – 5000 = Rs 978 3. In case the rates are different (R1, R2, R3….) for R1/100)(1 + R2/100)(1 + R3/100).
different years, the amount is given by P (1 +
E.g.5: Find the compound interest on Rs. 5000 for 3 years at 6 % per annum for first year, 7% for the second year and 8% for the third year. Using the formula, P{1 + R1/100}{1 + R2/100}{1 + R3/100}
= 5000(1 + 6/100) (1 + 8/100) (1 + 9/100) = 6125 Compound interest = 6125 – 5000 = Rs. 1125. 4. For population increase the formula to be used is P {1 + R/100 } T, and for decrease P { 1 R/100 }T. It can also be used for the depreciation factor. E.g.6: The death rate of a town with population of 100000 is 5 %, considering there are no new births, what is the population of town in next three years? Using the formula, P {1 - R/100 } T = 100000(1-5/100)3 = 100000(0.857) = 8573 6. The SI and CI earned during the first period remains the same. E.g.7: The compound interest on a certain sum of money in 2 years is 210 and the si mple interest on the same amount is 200, what are the principle and the rate of interest? Since SI and CI for first year is the same, and SI for each year is the same, so SI for the first year = 200/2 = 100, CI for year I = 100, that means CI f or the year II = 210 – 100 = 110. Here the excess of interest over year I = 10. Since the excess of interest in CI is interest over first years interest, assuming I is the interest, I/100 x 100 = 10, so I = 10, and the principal is obviously 1000.(Try calculating it yourself) E.g.8: A sum of money placed at Compound Interest doubles in every 5 years, then in how many years it will become 16 times? Now, it is given that the principle gets doubled in every 5 years. So, if we start from initial amount P, then in first 5 years it will become 2P. In the next 5 years 2P will become 4P, next 5 years 4P will become 8P and finally in the next 5 years 8P will become 16P. So, it will take (5+5+5+5) = 20 years 5.4 Net Present Value (NPV)
Money received or paid today is not the same as money received or paid after a period. This is because the money has an opportunity cost of interest in the same period. What it simply means is that you can earn interest on money if you have it now, and if you get the money later, you lose the opportunity to make interest on that. For example, if the going interest rate in the market is 10%, and someone has to pay me Rs. 1000, and he pays after an year, so he should pay, 1100 (100 has the interest), Here 1100 is called the future value and 1000 is called the present value. Here the Future value (FV) = Present value (PV) {1 + Rate/100 }time, which is the basic formula for amount in the case for compound interest, this is the formula to be used for calculating present value. From here, PV = FV / {1 + Rate/100 } time This is the same formula as of the compound interest; herein we are calculating principal from the
amount, which’s it.
5.5 Equal annual installment to pay the debt amount
Let the borrowed (debt) amount = Rs. B, rate of interest per annum= R, amount of each installment= Rs. A, and time = t yrs. Then, 2
)+
a[
+………..
t
]= Borrowed amount B
E.g.9: What annual payment will discharge a debt of Rs. 50,440 due in 3 years at 5% per annum compounded annually? Let each annual installment= Rs. A. Then, by the formula,
A[
2
)+
A[
+
2
+
+……….. 3
t
]=Borrowed amount B
]=50,440
A=Rs.18522
LEVEL 1 1. The SI on a sum of money is 25% of the principal, and the rate per annum is equal to the number of years. Find the rate per cent. (a) 4.5% (b) 6% (c) 5% (d) 8% 2. The rate of interest for first 3 years is 6% per annum, for the next 4 years, 7 per cent per annum and for the period beyond 7 years, 7.5 percentages per annum. If a man lent out Rs 1200 for 11 years, find the total interest earned by him? (a) 1002 (b) 912 (c) 864 (d) 948 3. A sum of money doubles itself in 12 years. Find the rate percentage per annum if interest is calculated at simple interest. (a) 12.5% (b) 8.33% (c) 10% (d) 7.51% 4. A certain sum of money amounts to Rs 704 in 2 years and Rs 800 in 5 years. Find the principal. (a) Rs 580 (b) Rs 600 (c) Rs 660 (d) Rs 640 5. A sum of money was invested at SI at a certain rate for 3 years. Had it been invested at a 4% higher rate, it would have fetched Rs 480 more. Find the principal. (a) Rs 4000 (b) Rs 4400 (c) Rs 5000 (d) Rs 3500 6. A certain sum of money trebles itself in 8 years. In how many years it will be five times? (a) 22 years (b) 16 years (c) 20 years (d) 24 years 7. If CI is charged on a certain sum for 2 years at 10% the amount becomes 605. Find the principal? (a) Rs 550 (b) Rs 450 (c) Rs 480 (d) Rs 500
8. If the difference between the CI and SI on a certain sum of money is Rs 72 at 12 per cent per annum for 2 years, then find the amount. (a) Rs 6000 (b) Rs 5000 (c) Rs 5500 (d) Rs 6500 9. The population of Pune increases by 10% in the first year, it increases by 20% in the second year and due to mass exodus, it decreases by 5% in the third year. What will be its population after 3 years, if today it is 10,000? (a) 11,540 (b) 13,860 (c) 12,860 (d) 12,540 10. David borrows a sum of Rs 1200 at the beginning of a year. After 4 months, Rs 1800 more is borrowed at a rate of interest double the previous one. At the end of the year, the sum of interest on both the loans is Rs 216. What is the first rate of interest per annum? (a) 9% (b) 6% (c) 8% (d) 12%
LEVEL 2 1) A sum of money invested at simple interest triples itself in 8 years at simple interest. Find in how many years will it become 8 times itself at the same rate? (a) 24 years
(b) 28 years
(c) 30 years
(d) 21 years
2) A sum of money invested at simple interest triples itself in 8 years. How many times will it
become in 20 years’ time? (a) 8 times
(b) 7 times
(c) 6 times
(d) 9 times
3) If Rs. 1100 is obtained after lending out Rs. x at 5% per annum for 2 years and Rs. 1800 is obtained after lending out Rs y at 10% per annum for 2 years, find x + y? (a) Rs 2500
(b) Rs 3000
(c) Rs 2000
(d) Rs 2200
4) Peter borrows Rs 10000 at 20 % p. a. for 5 years at simple interest. From fourth year onwards, on the entire amount due at the end of three years, the lender begins to charge 20% p. a. compounded annually. What is the amount repaid by Peter after five years from the beginning? (a) 23040
(b) 22500
(c) 21000
(d) 19900
5) Raj takes a loan at 100% p. a. interest. When he was repaying it after three years, he had t o pay Rs 952000 more because the loan was compounded every moment, instead of annually. What was the loan amount? [Take e=2.71 and (2.71)3=19.9] (a) 78000
(b) 80000
(c) 80500
(d) 79500
6) A sum of money when kept at simple interest doubled in 8 years & 4 months. If rate of interest is doubled and interest is calculated under compound interest in which year same sum will become twice? (a) 3rd Year
(b) 4th Year
(c) 5th Year
(d) 6th Year
7) The population of a city is 200,000. If the annual birth rate and the annual death rate are 6% and 3% respectively, then calculate the population of the city after 2 years? (a) 2, 12,090
(b) 2, 06,090
(c) 2, 12,000
(d) 2, 12,180
8) A part of Rs 38,800 is lent out at 6% per six months. The rest of the amount is lent out at 5% per annum after one year. The ratio of interest after 3 years from the time when first amount was lent out is 5 : 4. Find the second part that was lent out at 5%? (a) 26,600
(b) 28,800
(c) 27,500
(d) 28,000
9) If the simple interest is 10.5 % annual and compound interest is 10% annual, find the difference between the interests after 3 years on a sum of 1000? (a) 15
(b) 12
(c) 16
(d) 11
10) A sum of 1000 after 3 years at compound interest becomes a certain amount that is equal to the amount that is the result of 3 year depreciation from 1728. Find the difference between the rates of CI and depreciation. (Given CI is 10% p.a.)? (a) 3.33%
(b) 0.66%
(c) 3%
(d) 2%
11) The Value of a machine depreciates 10% annually. If the present value of the machine is Rs 1, 00, 000/- then the total depreciation during 2 years hence will be? (a) Rs 81, 000
(b) 21, 000
(c) Rs 19, 000
(d) Rs 72, 000
12) The present population of a village is 9, 261. If the annual birth rate is 81/2 % and the annual death rate is 3.5%, then calculate the population 3 years ago. (a) 10, 721
(b) 11, 363
(c) 11, 391
(d) 8, 000
13) A certain sum amounts to Rs 1, 452 in 2 years and to Rs 1, 597.20 in 3 years at compound interest, then rate percent is? (a) 10
(b) 11
(c) 13
(d) 9
14) The bacteria in a culture grows by 10% in first two hours, decreases by 10% in next on ehour and again increases by 5% in next two hours. If the original count of the bacteria in the sample is 40, 000, find the bacteria count at the end of 5 hours? (a) 48, 000
(b) 48, 025
(c) 48, 050
(d) 48, 075
15) The population of a town was 2, 50,000 three years ago. If it has increased by 3%, 4% and 6% in the last three years, find the present population of the town? (a) 2,83,868
(b) 2,81,686
(c) 2,82,868
(d) 2,80,168
COMPANY SPECIFIC QUESTIONS 1. What will Rs. 1500 amount to in three years if it is invested in 20% p.a. compound interest, interest being compound annually? [ACCENTURE]
(A) 2592
(B) 2569
(C) 2540
(D) 2678
2. The difference between the compound interest and the simple interest earned at the end of 3rd year on a sum of money at a rate of 10% per annum is Rs. 77.5. What is the sum? [ACCENTURE] (A) Rs. 3500 (B) Rs. 3000 (C) Rs. 2000 (D) Rs. 2500 3. At a certain rate of simple interest a certain sum doubles itself in t en years. It will become four times of itself in how many years? [PERSISTENT] (A) 20 years
(B) 15 years
(C) 10 years
(D) 5 years
4. Ajit invested Rs 35000 for 8 months and Manjit invested Rs 42000 for 10 months. On a profit of Rs 31570 Ajit share is [VEDANTA] (A) Rs.13548
(B) Rs.14234
(C) Rs.12628
(D) None of these
5. Anuj started a business by investing Rs.20, 000. Six months later, Pankaj joined him with a capital of Rs.15,000. After another three months Puneet joined the team by i nvesting Rs.50, 0000. Find the ratio in which the profit at the end of two years should be divided by the three. [ORACLE] (A) 8:3:5
(B) 3:2:1
(C) 16:9:25
(D) 4:2:1
6. Mr. A lends 40% of a sum at 15% pa, 50% of the rest sum at 10% pa and the rest at 18% pa rate of interest. What would be the rate of interest if the interest is calculated on the whole sum? [Tech Mahindra] (A) 13.4% pa
(B) 14.33% pa
(C) 14.4% pa
(D) 13.33% pa
7. In simple interest what sum amounts of Rs.1120/- in 4 years and Rs.1200/- in 5 years? [Tech Mahindra] (A) 500 (B) 600 (C) 800 (D) 900 8. If a sum of money compound annually amounts of thrice itself in 3 years. In how many years will it become 9 times itself? [Tech Mahindra] (A) 6
(B) 8
(C) 10
(D) 12
9. In a partnership, A invests 1/6 of the capital for 1/6 of t he time, B invests 1/3 of the capital for 1/3 of the time and C, the rest of the capital for the whole time. Out of a profit of Rs 4600, B’s share will be? [L & T] (A) Rs. 650
(B) Rs. 800
(C) Rs. 960
10. A sum of Rs. 370 is to be divided among A, B and C such that:
(D) Rs. 700
=
=
Then A’s share is
[TCS]
(A) Rs. 240
(B) Rs. 120
(C) Rs. 100
(D) Rs. 90
ANSWER KEY LEVEL 1:
1C
2B
3B
4D
5A
6B
7D
8B
9D
10 B
6B
7D
8B
9C
10 D
7D
8B
9A
10 D
ANSWER KEY LEVEL 2:
1B
2C
3A
4A
5B
11 C
12 D
13 A
14 B
15 A
ANSWER KEYCOMPANY SPECIFIC QUESTIONS:
1A
2D
3A
4C
5C
6C
CHAPTER 6 AVERAGES AND MIXTURES 6.1 Average: Average is the mean value of a set of numbers of Values
Therefore Average = (x1+x2+x3…….xn)/n Ex: if the ages of four (4) students are 20,22,18 and 24 years respectively, find the average age. Average = sum of ages No. of students
=20+22+18+24 = 84 = 21 4 4
6.2 Weighted Average : Weighted average is the average of two or more groups whose individual averages are known W.A = n1.a1 + n2.a2 n1+n2
6.3 Average Speed: Average speed is the ratio of Total distance to the time taken Average Speed = Total Distance Total Time
6.4 Age and Average:
If the average age of n persons is decreased by ‘a’ years then the total age decreases by ( n×a years and vice versa)
Ex:The average weight of six men decreases by 3 kgs when a man whose weight is 60 kg is replaced by another man. Find the weight of the man newly included. Solution:
Decrease in total weight = 6 x 3 = 18 Kg so , the weight of the newly included man is 60-18 i.e is 42 kgs. Note:- if a value which is less than the actual value is entered, then the average would reduce.
6.5 Rules of Allegation: If two quantity are mixed in a ratio , then Quantity of cheaper = cp of dearer – Kean price Quantity of Dearer Mean price- Cp of Cheaper
Cp of cheaper(c)
cp of dearer(d)
Mean Price(m)
(d-m) Ratio = x1
(m-c) :
x2
Hence , cheaper quantity : dearer quantity is (d -m): (m-c)
Container originally contains ‘v’ units of liquid and ‘K’ units of liquid is taken out of this operation is repeated ‘n’ times , then the final quantity of liquid in container is V × [1- ]n units
Ex: for a container with 120 lts of milk, 40 lts is taken out and replaced with water , if this process is repeated twice, find the quantity of milk in the container now? Solution:
Quantity of milk in container = 120 [1-40/120]² = 120[1-1/3]²
=120 x 4/9 =53.33 ltrs
LEVEL 1 1. An airplane travels distance 2500 km, 1200 km and 500 km at speed 500 km/hr., 400 km/hr. and 250 km/hr. resp. The average speed of plane is (A) 440 km/hr
(B) 340 km/hr
(C) 300 km/hr
(D) 420 km/hr.
2. The average expenditure of a man for the first five months is Rs.120 and for the next seven months it is Rs.130. If he saves Rs.290 in that year, his monthly average income is : (A) Rs.140
(B) Rs. 150
(C) Rs.160
(D) Rs. 170
3. The average age of 5 members is 21 years. If the age of the youngest member be 5 years, find the average age of the family at the birth of the youngest member (A) 20 years
(B) 19 years
(C)13 years
(D) None of these
4. The average cost of 4 apples and 7 bananas is Rs.16, The average cost of 7 apples and 4 bananas is Rs.24. The cost of 1 apple and 1 banana is Rs. (A) 11
(B) 40
(C) 30
(D) 8
5. There are two groups A and B consisting of 30 and 70 students respectively, if the average weight of Group A is 30 KGs and that of group B is 70 KGs find the average weight of all the students of Group A and B (A) 58
(B) 50
(C) 40
(D) 42
6. The average of marks obtained by 120 candidates was 35. If the average of marks of passed candidates was 39 and that of failed candidates was 15, the number of candidates who passed the examination is : (A) 100
(B)110
(C)
120
(D) 150
7. The average temperature of the first three days is 27 ∘C and that of the next three days is 29 ∘C. If the average of the whole week is 28.5∘C, the temperature of the last day of the week is :
(A) 10.5∘C
(B) 21∘C
(C) 31.5∘C
(D) 42∘C
8. The average monthly expenditure of a family for the first four months is Rs. 2750, for the next three months is Rs. 2940 and for the last five months is Rs. 3130.If the family saves Rs. 5330 during the whole year, find the average monthly income of the family during the year. (A) Rs. 3800
(B) Rs. 3500 (C) Rs.3400
(D) Rs. 4200
9. A car owner buys petrol at Rs.7.50, Rs. 8 and Rs. 8.50 per litre for three successive years. What approximately is the average cost per litre of petrol if he spends Rs. 4000 each year?
(A) Rs. 7.98
(B) Rs. 8
(C) Rs. 8.50
(D) Rs. 9
10. The ratio of milk and water-milk mixture is 2:3.How much water should be added to 60 litres of the mixture to make the ratio of milk and water as 1:3 ? (A) 45
(B) 50
(C) 55
(D) 60
LEVEL 2 1. Nine litres are drawn from a cask full of wine and it is then filled with water, Nine litres of the mixture are withdrawn and the cask is again filled with water. The ratio of quantity of wine now remaining in the cask to that of water in it is 16:9.How much does the cask hold? (A) 15 2.
(C) 24
(D) none of these
A mixture contains brandy and water in the ratio 8: x. When 33 litres of the mixture and 3 litres of water are mixed, the ratio of brandy and water becomes 2:1.The value of x is: (A) 5
3.
(B) 45
(B) 3
(C) 7
(D) none of these
A sum of Rs. 39 was divided among 45 boys and girls. Each girl gets 50 paise, where as a boy gets one rupee. Find the number of boys. (A) 27
(B) 12
(C) 33
(D) none of these
4. A container contains 1000 litres of Milk. From this container 100 litres of milk was taken out and replaced by water .If this process was repeated three times , how much water is now contained by the container? (A) 729 lts
(B) 700 lts
(C) 656.1 lts
(D) 271.0 lts
5. In what proportion must wheat at rs 11:00 per Kg be mixed with wheat at Rs. 16:00 per Kg so that the mixture is worth Rs.13 per Kg? (A) 2:3
(B) 3:2
(C) 11:16
(D) 16:11
6. If 3 Kg of metal , which is one third silver and rest aluminum is mixed with 7 Kg of another metal, which is two-seventh silver and rest aluminum. What is the ratio of silver is to aluminum in the mixture (A) 3:7
(B) 7:3
(C) 1:7
(D) 2:7
7. A ten litres of water were added to ten litres of a 30% strong solution of sulphuric acid. Find the strength of the resulting solution. (A) 10% 8.
(B) 12%
(C) 14.5%
(D) 15%
Gold is 19 times as heavy as water and copper is 9 times heavy. In what ratio must these metals be mixed so that the mixture may be 15 times as heavy as water? (A) 4:1
(B) 3:2
(C) 2:1
(D) 1:3
9. A trader has 100 kg of rice, a part of which he sells at 30% profit and the rest at 5% profit. He gains 25% on the whole. What is the quantity sold at 30% gain? (A) 80 kg
(B) 40 kg
(C) 75 kg
(D) 35 kg
10. In two mixtures, spirit and water are related in the ratios of 3:5 and 7:4. 24 gallons of mixture I, 44 gallons of mixture II and 25 gallons of spirit are mixed together. What is the final ratio of spirit and water? (A) 2:1
(B) 1:5
(C) 5:2
(D) 1:6
COMPANY SPECIFIC QUESTIONS 1. The proportion of milk and water in 3 samples is 2:1, 3:2 and 5:3. A mixture, comprising of equal quantities of all 3 samples is made. The proportion of milk and water in the mixture is [ACCENTURE] (A) 2:1
(B) 5:1
(C) 227:133
(D) 99:61
2. A certain quantity of petrol is found to be adulterated to the extent of 10%.What proportion of the adulterated petrol should be replaced with pure petrol to take the purity level t o 98% ? [CAPGEMINI] (A) 80% (B) 76% (C) 94% (D) None of these 3. There are 150 weights. Some are 1 kg weights and some are 2 kg weights .The sum of weights is 260. What is the number of 1 kg weights? [TCS] (A) 50 4.
(B) 30
(C) 65
(D) 40
Three friends A, B and C went for a weekend party to McDonald’s restaurant and there they measure t he weights in some order in 7 rounds. A, B, C, AB, BC, AC, ABC. Final round measure is 155 kg, then find the average weight of all 7 rounds? [TCS] (A) 29 kg
(B) 30 kg
(C) 28 kg
(D) 31 kg
5. One quality of wheat costing Rs.9.30 per kg is mixed with another quality of wheat costing Rs. 10.80 per kg, then what will be the ratio in which they must be mixed so that the net cost is Rs. 10 per kg? [VEDANTA] (A) 5:7 (B) 8:7 (C) 13:19 (D) None of these 6. A 5 litre jug contains 4 litres of a salt water solution i.e it constitutes 15 percent salt. If 1.5 litres of the solution spills out of the jug, and the jug is then filled to capacity with water, approximately what percent of the resulting solution in the jug is salt? [HCL] (A) 7.5% (B) 9.5%
(C) 10.5%
(D) 12%
7. Fresh fruit contains 68% water and dry fruit contains 20% water. How much dry fruit can be obtained from 100 kg of fresh fruits? [WIPRO] (A) 32 kg
(B) 40 kg
(C) 52 kg
(D) 80 kg
8. A trader mixes rice of two qualities, the cost prices of which differ by Rs. 6, in a certain ratio to get the resultant cost price as Rs.12. If he mixes them in the reverse ratio and sells them at Rs.12, he gets a profit of 20%. What is the cost (in Rs.) of the more expensive quality? [COGNIZANT] (A) 10 (B) 14 (C) 12 (D) 16 9. The average salary of 3 workers is Rs. 95 per week. If one earns Rs.115 and second earns Rs.65, how much is the salary of the 3rd worker? [CAPGEMINI] (A) Rs.120
(B) Rs. 100
(C) Rs. 105
(D) None of these
10. A pupil's marks were wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by half. The number of pupils in the class is [COGNIZANT] (A) 10 (B) 20 (C) 40 (D) 7
ANSWER KEY LEVEL 1:
1D
2B
3A
4D
5A
6A
7C
8C
9A
10 D
4C
5B
6A
7D
8B
9A
10 A
7B
8B
9C
10 C
ANSWER KEY LEVEL 2:
1B
2B
3C
ANSWER KEYCOMPANY SPECIFIC QUESTIONS:
1C
2A
3D
4D
5B
6A
CHAPTER 7 TIME AND WORK 7.1 Work: Quantity of work is directly proportional to the time taken i.e W∞T Ex: If a man can reap 100 coins in 4 hours find the time taken by him to reap 5 coins?
W∞T Therefore w/t is constant 100/4 = 5/9
? = 20/100 (or) 1/5 hours
If ‘A” can do a piece of work in ‘n’ days then in one day he can finish 1/n part of work in 1 day. Ex: If a person can finish a work in 3 days then he can finish 1/3 rd of work in 1 day.
If A can do a piece of work in ‘N’ days and ‘B’ in ‘M’ , then both of them can finish the work in N xM M+N Days Ex: If Rocia can do a piece of work in 4 days and Siya in 6 days , find the time taken by both of them to finish the work Solution:
Time taken to finish the work = n xm / m + n days = 6 x4 / 6 + 4 = 2.4 days
If A & B can do a work in ‘n’ days , B and C in ‘y’ days , C and A in ‘z’ days then A,B,C working together can finish the work in 2x ( X x Y x Z) / X x Y + Y x Z + Z x X days A can do it in y x d/ y-d days B can do it in Z x d / Z- d days C Can do it in X x d/ X- d days D = (2 x X x Y x Z) / X x Y + Y x Z + Z x X 7.2 Efficiency: 1
Efficiency is the rate of doing work 1. Efficiency (E) is Inversely proportional to time
E ∞ 1/t where E x T is constant
2. Efficiency E is directly proportional to Wages/ Works
E∞ W where E/W = constant
Ex: If A can finish a piece of work in 60 days find the time taken by B to finish the work given that B is 20% more efficient than A? Solution :
Let the efficiency of A be 100% Time taken by A is 60 days Efficiency of B is 100+20= 120%
Efficiency ∞ 1/time Where EaTa = EbTb Therefore 100 x 60= 120 x Tb Where Tb= 50 days.
If a pipe can fill a tank in ‘n’ hours and another pipe empties the same tank in ‘m’ hours (n
Time taken to fill the tank = m x n/ m-n = 6 x 4/ 6-4 = 12 hours ----------------------------------------------------------------------------------------------------
LEVEL 1 1. Mohan and Sohan can do a job in 12 days. Sohan alone can finish it in 28 days. In how many days can Mohan alone finish the half work? a. 10 days
b. 7.5 days
c. 11 days
d. 10.5 days
2. A person can do a job in 15 days. His uncle takes 12 days and his friend finishes it in 60 days. How long will they take to complete a job if they all work together? 2
a. Less than 6 days
b. Exactly 6 days
c. More than 6 days
d. Exactly 10 days
3. Ajay can do a piece of work in 24 days; Sunil can do the same work in 8 days. If they work at it on alternate days with Sunil beginning the work, then in how many the work will be completed? a. 12 days
b. 20 days
c. 16 days
d. none
4. A man can complete 3/8 of a work in 24 days . At this rate, how much more time is required to complete the work? a. 15 days
b. 40 days
c. 64 days
d. 28 days
5. Mr. X can complete a job in 18 days , Mr.Y in 20 days and Mr. Z in 30 days , Mr. Y and Mr.Z start the work and are forced to leave after 4 days . The time taken to complete the remaining work by Mr.X is a. 12 days
b. 18 days
c. 20 days
d. 26 days
6. Sharma and Shyam can do a piece of work in 24 days, Shyam and Shastri in 30 days , Shastri and Sharma in 40 days. Who will take the least time to finish it alone? a. Sharma
b. Shyam
c. Shastri
d. None of these
7. If 3 women working for 6 hours a day earns Rs.650 in 10 days then how much will 18 women working 9 hours a day earn in 10 days? a. Rs. 6500
b. Rs. 5850
c. Rs. 925
d. none of these
8. Two pipes A and B can fill a tank in 1 hour 12 minutes and 1 hour 30 minutes, respectively. Pipe C can empty the tank in 1 hour. Intially, Pipes A and B are opened and after 14 minutes C is also opened. In how much time will the tank be full? (L-2) a. 1 hour
b. 80 min.
c. 84 min.
d. 1 hr 32 min.
9. Two pipes X and Y can fill a cistern in 15 min. and 40 min. respectively. Both pipes are opened together but after 4 minutes, pipe X is turned off. What is the total time required to fill the cistern? a. 10 min 10 sec
b. 25 min 20 sec
c. 14 min 40 sec
d. 20 min 10 sec
10. Some students can complete an assignment in 12 days. How many days will be taken by two times the number of such students for 1/3 rd of this assignment? a. 6 days
b. 4 days
c. 2 days
d. 3 days
3
LEVEL 2 1) P is 30% more efficient than Q. P can complete a work in 23 days, If P and Q work together, how much time will it take to complete the same work? a. 9
b. 11
c. 13
d. 15
2) Peter, Qureshi and Ricky together earn Rs 1620 in 9 days. Peter and Ricky can earn Rs 600 in 5 days, Qureshi and Ricky in 7 days can earn Rs 910. How much amount does Ricky can earn per day? a. Rs 90
b. Rs 100
c. Rs 40
d. Rs 70
3) 3 men and 7 women can complete a work in 10 days. But 4 men and 6 women need 8 days to complete the same work. In how many days will 10 women complete the same work? a. 50 days
b. 40 days
c. 30 days
d. 20 days
4) Rohit and Mayank working together can finish a job in x days . If Rohit working alone takes 8 days more than x and Mayank working alone takes 18 days more than x , then what is the number of days taken by Rohit and Mayank to complete the work together? a. 10 days
b. 8 days
c. 12 days
d. None of these
5) One Pipe can fill a tank 6 times as fast as another pipe. If together the two pipes can fill the tank in 22 minutes , then the slower pipe will be able to fill the tank in: a. 164 min
b. 154 min
c. 144 min
d. 134 min
6) 25 men & 15 women do a piece of work in 12 days. They started the work , after 8 days women leaves off the remaining work done by 25 men in 6 days. In how many days, 15 women can do the same piece of work? a. 36 days
b. 24 days
c. 20 days
d. 24 days
7) An empty tank be filled by an inlet Pipe A in 42 minutes. After 12 minutes an outlet Pipe B is opened which can empty the tank in 30 minutes. After 6 minutes another inlet Pipe C opened into the same tank , which can fill the tank in 35 minutes and the tank is filled. Find the time taken to fill the tank? a. 55 min
b. 50.5 min
c. 58.5 min
d. 51.5 min
8) Two workers A and B working together completed a job in 5 days. If A worked twice as efficient as he actually did and B worked 1/3 as efficiently as he actually did, the work would have been completed in 3 days. Find the time taken by A to complete the job alone.
4
a. 37/3 days
b. 45/4 days
c. 25/4 days
d. 37/4 days
9) There is a leak in the bottom of the cistern, when the cistern had no leak it was filled in 2.5 hours, it now takes half hour longer. If the cistern is full of water , how long will it take in leaking itself empty, in case the water leaks out at double the rate after half the cistern becomes empty? a. 15 hrs
b. 11 hrs 15 min
c. 11 hrs 25 min
d. 7.5 hrs
10) Sahitya is thrice as good as Roshni and therefore is able to finish a job in 60 days less than Roshni. What is the time taken to do twice the work, when they are working together? a. 45 days
b. 22.5 days
c. 25 days
d. 30 days.
11) Pipe A can fill a tank in 10 hours, Pipe B can fill the tank in 12 hours, but if the pipe A is opened for 5 hours and then pipe B is opened for 6 hours, how much time in hours it is required to fill the rest of the tank by pipe A and pipe B together? a. 20 b. 10 c. 15 d. 0 12) A pipe can fill a tank in x hours and another can empty it in y hours. If the tank is 1/ 3rd full then the number of hours in which they will together fill it in is (A) 3xy/2(y-x) (b) 3xy/y-x
(c) xy/3(y-x)
(d) 2xy/3(y-x)
13) A finishes 6/ 7th of the work in 2z hours, B works twice as fast and finishes the remaining work. For how long did B work? (A) (2/3)z
(b) (6/7)z
(c) (6/49)z
(d) (3/18)z
14) Ajit can do as much work in 2 days as Baljit can do in 3 days and Baljit can do as much in 4 days as Diljit in 5 days. A piece of work takes 20 days if all work together. How long would Baljit take to do all the work by himself? (A) 82 days
(b) 44 days
(c) 66 days
(d) 50 days
15) Two pipes can fill a cistern in 14 and 16 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom of the cistern, it takes 32 minutes extra for the cistern to be filled up. When the cistern is full, in what time will the leak empty it? (A) 114 h (b) 112 h (c) 100 h (d) 80 h
---------------------------------------------------------------------------------------------------------------------
COMPANY SPECIFIC QUESTIONS
5
1. Working independently, Tina can do a certain job in 12 hours. Working independently, Ann can do the same job in 9 hours. If Tina works independently at the job for 8 hours and then Ann works independently, how many hours will it take Ann to complete the remainder of the jobs? [HCL] (A) 2/3 (B) 3/4 (C) 1 (D) 3 2. A and B can together complete a piece of work in 12 days. A alone can complete in 20 days. If B does the work only for half a day daily, then in how many days A and B together will finish the job? [L&T] (A) 10
(B) 11
(C) 20
(D) 15
3. 12 men can complete a piece of work in 4 days, while 15 women can complete the same work in 4 days. 6 men start working on the job and after working for two days, all of them stop working. How many women should be put on the job to complete the remaining work, if it is to be completed in 3 days? [COGNIZANT] (A) 22
(B) 15
(C) 24
(D) Data inadequate
4. Anil and Suman can complete a work in 10 and 15 days, respectively. Anil starts the work, and after 2 days, Suman joins him. But Suman leaves him after 2 days. How many more days will Anil now have to work to complete it? [ACCENTURE] (A) 2.75 days
(B) 3 days
(C) 4.66 days
(D) 5 days
5. A can have a piece of work done in 8 days, B can work three times faster than A, C can work five times faster than A. How many days will they take to do the work together? [TECH MAHINDRA] (A) 3 days
(B) 8/9 days
(C) 4 days
(D) Can't say
6. There are two pipes in a tank. Pipe A is for filling the tank and Pipe B is for emptying the tank. If A can fill the tank in 10 hours and B can empty the tank in 15 hours then find out how many hours it will take to completely fill a half empty tank? [ACCENTURE] (A) 30 hours
(B) 15 hours
(C) 20 hours
(D) 33.33 hours
7. A can do a work in 90 days, B in 40 days and C in 12 days. They worked for a day each in turn, i.e. A worked alone for the first day, B for the second day and C for the third day, then again A and so on. After finishing the work, they got Rs.240. Wages were divided in proportion to the
work done by them. A’s share is: [TCS] (A) Rs. 24
(B) Rs. 28
(C) Rs. 60
(D) Rs. 88
6
8.
It is often said, “Rome was not built in a day”. But if there are 2000 walls in Rome and each wall has 3000 bricks. It takes 2 seconds for one worker to fix a brick. How much time would have been taken to build Rome if there were 100 such workers? [ZENSAR] (A) less than a day
9.
(B) 1-2 days
(C) 2-3 days
(D) 4-5 days
Worker W produces ‘n’ units in 5 hours. Workers V and W, work independently but at the same time, produce ‘n’ units in 2 hours. How long would it take V alone to produce ‘n’ unit [HCL] (A) 1 hr 26 min
(B) 1 hr 53 min
(C) 2 hr 30 min
(D) 3 hr 20 min
10. 20 labourers can do a work in 20 days, if everybody works for 6 hours daily. Then 25 labourers can do the same work in 12 days by working daily for: [INFOSYS] (A) 8 hours
(B) 6 hours
(C) 4 hours
(D) 10 hours
ANSWER KEY LEVEL 1:
1D
2B
3A
4B
5A
6B
7 BA
8D
9B
10 C
6A
7C
8C
9C
10 B
7A
8B
9D
10 A
ANSWER KEY LEVEL 2:
1C
2D
3B
4C
5 B
11 D
12 D
13 D
14 C
15 B
ANSWER KEYCOMPANY SPECIFIC QUESTIONS:
1D
2D
3B
4C
5B
6B
CHAPTER 8 TIME, SPEED AND DISTANCE 8.1 Speed: 7
Speed = Distance/ Time or Distance = Speed x Time 1. Speed is directly proportional to distance 2. Distance is directly proportional to time 3. Speed and time are inversely proportional 8.2 Conversion Factors:
1 Km/Hr = 5/18 m/s 1 M/S = 18/5 Km/Hr Ex: A bowler has a run up of 150 m. if the speed of the bowler is 54 kmph, how much time would he take to complete the run up? Solution:
54 km/Hr = 54 x 5/18 m/s = 15 m/s Time = distance /speed =150/15 = 10 sec.
In travelling equal distance with speeds of X and Y , the average speed is given by (2XY)/X+Y Ex: Find the average speed of a car that covers 1 st 50% of distance at 40 Km/Hr and the 2 nd 50% of the distance at 60 Km/ Hr. Solution:
Average speed = 2 x X x Y / X+Y = 2 x 60 x 40/ 60 +40 =4800/100=48 km/hr 8.3 Relative Speed :
The speed of a body with the respect to another moving body is defined as relative speed. If two bodies are moving in same direction , the relative speed is given by the difference of speeds. X y 8
Relative speed = (X-Y) If two bodies are moving in the opposite direction, then the relative speed is given by the sum of the speeds. X Y Relative speed = (X+Y) Ex: If two trains of length 200m and 250m are having speed 18 km/hr and 36 km/ hr , find the time taken to cross each other
1. If they move in the same direction 2. If they move in opposite direction Solution:
Speed of first train = 18 x 5/18= 5m/s Speed of second train = 36 x 5/18= 10 m/s Distance they have to cross to cross each other is the sum of lengths of train = 200 + 250 =450m If they move in same direction, relative speed = 10-5 = 5 m/s Time taken = distance /Speed= 450/5 = 90 sec If they move in opposite direction, relative speed = 10 + 5=15 m/s Time taken = distance / speed = 450 /15 = 30 sec.
8.4 Boats and Streams:
If the speed of boat in still water is X km /hr and speed of stream is Y km/hr then Speed downstream =X + Y Speed Up Stream = X-Y Speed of boat in still water = (Speed up stream + speed Downstream)/2 Speed of Stream = (Speed downstream- speed upstream)/2
9
Ex: If the speed downstream is 12 Km/hr and speed upstream is 8 km/hr , find the speed of boat in still water and speed of stream. Solution:
Speed of boat in still water = (speed upstream + speed downstream)/2 = 12+8/2= 10 km/hr Speed of stream = (Speed downstream – speed upstream)/2 =12 -8/2= 2 km/hr
LEVEL 1 1. John covers half of a certain distance by bus at 40 Km/hrand the remaining by by train at 60 km/hr. John return to starting point riding a scooter at 52 Km/hr. Find his average speed for the whole Journey. a. 48 km
b. 49.52 km
c. 50.66 km
d.51 km
2. A train 120m in length passes a pole in 12sec and another train of length 100m travelling in opposite direction in 10sec. Find the speed of the second train in km per hour. a. 43.2 km/hr
b. 43 km/hr
c. 44 km/hr
d. 43.5 km/hr
3. Walking at the rate of 4kmph a man cover certain distance in 2hr 45 min. Running at a speed of 16.5 kmph the man will cover the same distance in. a. 12 min
b. 15 min
c. 25 min
d. 40 min
4. If 601 signal poles are arranged such that the distance between two consecutive signal poles is 0.1 km and a train 240 m long crosses them completely in 45 minutes. What is the speed of the train. a. 80.45 kmph
b. 80.40 kmph
c. 80.32 kmph
d. none
5. Rahul can row a certain distance downstream in 6 hour and return the same distance in 9 If the speed of Rahul in still water is 12 km/hr, Find the speed of the stream. a. 2 kmph
b. 2.4 kmph
c. 3 kmph
hour.
d. 1.5 kmph
10
6. Jay started cycling along the boundaries of a square field from corner point A. After half an hour he reached the corner point C, diagonally opposite to A. If his speed was 8km/hr, the area of the filed in square km is: a. 64
b.16
c. 9
d. 4
7. A man sitting in train travelling at the rate of 50 km/hr observes that it takes 9 sec for a goods train travelling in the opposite direction to pass him. If the goods train is 187.5m long. Find its speed. a. 40 kmph
b. 30 kmph
c. 24 kmph
d. 25 kmph
8. I travel the first part of my journey at 40 kmph and the second part at 60 kmph and cover the total distance of 240 km to my destination in 5 hours. How long did the first part of my journey last? a. 4 hours
b. 3 hours
c. 2 hours 24 minutes
d. 2 hours
9. A man can row 3 kmph in still water. When the river is running at 0.6 kmph, it takes him half an hour to row to a place and back. How far is the place? a. 0.7 km
b. 0.3 km
c. 280 mts
d. 720 mts
10. Sachin can cover a distance in 1hr 24min by covering 2/3 of the distance at 4 kmph and the rest at 5kmph. Then the total distance is? a. 5 km
b. 6 km
c. 7 km
d. 8 km
LEVEL 2 1. A man driving his bike at 24 kmph reaches his office 5 minutes late. Had he driven 25% faster on an average he would have reached 4 minutes earlier than the scheduled time. How far is his office? a. 24 km
b. 72 km
c. 18 km
d. Data Insufficient
2. Three athletes A, B and C run a race, B finished 24 meters ahead of C and 36 m ahead of A, while C finished 16 m ahead of A. If each athlete runs the entire distance at their respective constant speeds, what is the length of the race? a. 108 m
b. 90 m
c. 80 m
d. 96 m
3. Yana and Gupta leave points x and y towards y and x respectively simultaneously and travel in the same route. After meeting each other on the way, Yana takes 4 hours to reach her destination,
11
while Gupta takes 9 hours to reach his destination. If the speed of Yana is 48 km/hr, what is the speed of Gupta? a. 72 kmph
b. 32 kmph
c. 20 kmph
d. None of these
4. A man can row 50 km upstream and 72 km downstream in 9 hours. He can also row 70 km upstream and 90 km downstream in 12 hours. Find the rate of current. a. 3 km/hr
b. 5 km/hr
c. 8 km/hr
d. 4 km/hr
5. If a man cycles at 10 km/hr, then he arrives at a certain place at 1 p.m. If he cycles at 15 km/hr, he will arrive at the same place at 11 a.m. At what speed must he cycle to get there at noon? a. 11 km/hr
b. 12 km/hr
c. 13 km/hr
d. 14 km/hr
6. The Ghaziabad-Hapur-Meerut EMU and the Meerut-Hapur-Ghaziabad EMU start at the same time from Ghaziabad and Meerut and proceed towards each other at 16 km/hr and 21 km/hr respectively. When they meet, it is found that one train has travelled 60 km more than the other. The distance between two stations is: a. 445 km
b. 444 km
c. 440 km
d. 450 km
7. The speed of a motor boat itself is 20 km/h and the rate of flow of the river is 4 km/h. Moving with the stream the boat went 120 km. What distance will the boat cover during the same time going against the stream? a. 80 km
b. 180 km
c. 60 km
d. 100 km
8. Rajesh walks to and fro to a shopping mall. He spends 30 minutes shopping. If he walks at speed of 10 km an hour, he returns to home at 19.00 hours. If he walks at 15 km an hour, he returns to home at 18.30 hours. How fast must he walk in order to return at 18.15 hours? a. 17 km/hr
b. 18 km/hr
c. 19 km/hr
d. 20 km/hr
9. A man and a woman 81 miles apart from each other, start travelling towards each other at the same time. If the man covers 5 miles per hour to the women's 4 miles per hour, how far will the woman have travelled when they meet? a. 27
b. 36
c. 45
d. none of these
10. A bus without stopping travels at an average speed of 60 km/hr and with stoppages at an average speed of 40 km/hr. What is the total time taken by the bus for stoppages on a route of length 300km? a. 3 hr
b. 4 hr
c. 2.5 hr
d. 3.5 hr
12
11. A man travels from A to B at 4 mph over a certain journey and returns over the same route to A, at 5 mph. What is his average speed (in mph) for the journey?
(A) 4.5 (B) 17/4 (C) 19/4 (D) 40/9 12. The J& K Express from Delhi to Srinagar was delayed by snowfall for 16 minutes and made up for the delay on a section of 80 km travelling with a speed 10 km per hour higher than its normal speed. Find the original speed of the J& K Express (according to the schedule) (a) 60 km/ h
(b) 66.66 km/ h (c) 50 km/ h
(d) 40 km/ h
13. Ayrton Senna had to cover a distance of 60 km. However, he started 6 minutes later than his scheduled time and raced at a speed 1 km/ h higher than his originally planned speed and reached the finish at the time he would reach it if he began to race strictly at the appointed time and raced with the assumed speed. Find the speed at which he travelled during the journey described. (a) 25 km/ h
(b) 15 km/ h
(c) 10 km/ h
(d) 6 km/ h
14. Amitabh covered a distance of 96 km two hours faster than he had planned to. This he achieved by travelling travelling 1 km more every every hour than he intended intended to cover cover every 1 hour 15 minutes. minutes. What What was the speed at which Amitabh travelled during the journey? (a) 16 km/ h
(b) 26 km/ h
(c) 36 km/ h
(d) 30 km/ h
15. An urgent message had to be delivered from the house of the Peshwas in Pune to Shivaji who was camping in Bangalore. A horse rider travels on horseback from Pune to Bangalore at a constant speed. If the horse increased increased its speed by 6 km/ h, it would take the rider 4 hours less to cover that distance. And travelling with a speed 6 km/ h lower than the initial speed, it would take him 10 hours more than the time he would have taken had he travelled at a speed 6 kmph higher than the initial speed. Find the distance between Pune and Bangalore. (a) 120 km
(b) 600 km
(c) 720 km
(d) 750 km
COMPANY SPECIFIC QUESTIONS 1. A motor boat whose speed is 15 km/hr in still water goes 30km downstream and comes back in a total time of 4 hrs. 30 min., then the speed of stream (in km/hr) [CAPGEMINI] (A) 8 (B) 6 (C) 5 (D) None of these 2.
Two cars are running on a highway, such that the speed of one is thrice of other. Both of them started from same point and have same destination, but the faster one started after the slower one has covered some part of journey. What fraction of journey faster one has covered when slower one started? [ HP] 13
(A) 1/2 3.
(B) 1/3
(C) 2/3
(D) None of these
Two trains are 2 km apart. Speed of one train is 20m/s and the other train is running at 30 m/s .Lengths of the trains is 200 and 300m. In how much time (seconds) do the trains cross each other? [CAPGEMINI] (A) 50
(B) 60
(C) 30
(D) 120
4. Going at 12 steps per minute, a lion covers 240m. If the step size of lion is 5 times that of deer, how much distance deer covers in an hour if deer takes 10 steps in a minute? [ZENSAR] (A) 6 km
(B) 1.2 km
(C) 2.4 km
(D) None of these
5. A plane travelled K miles miles in the first 96 minutes of flight time. If it completed the remaining 300 miles of the trip in 1 minute, what was its average speed in miles per hour f or the entire trip? [HCL] (A) (K + 300)/ 97.
(B) 60 (K + 300)/ 97
(C) (K + 97) / 300
(D) (K + 96) /300
6. The charges of a hired car are Rs. 7 per km for the first 80 km, Rs. 5 per km for the next 60 km and Rs. 11 for every 5 km for further journey. If the charges for a journey is Rs. 1212, what is the distance travelled in the journey? [ACCENTURE] (A) 300 km
(B) 240 km
(C) 360 km
(D) 200 km
7. The boy goes to school reaches the railway station at his 1/3 of his journey & a mill at 1/4 of his journey. The The time taken for him to walk between between the railway station & the mill is 5 mins. mins. Also he reaches the railway station at 7.35am. So when does he start from his house? [INFOSYS] (A) 7:20 am (B) 7:15 am (C) 7:05 am (D) None of these 8. A train M leaves Meerut at 5 a.m. and reaches Delhi at 9 a.m. Another train leaves Delhi at 7 a.m. and reaches Meerut at 10.30 a.m. At what time do the two trains cross each other? [L&T] (A) 7:36am
(B) 7:56am
(C) 8am
(D) 8:26am
9. A boat takes 19 hours for travelling downstream from point A to point B and coming back to a point C midway midway between between A and B. If the velocity of the stream stream is 4 km/hr and the speed speed of the boat in still water is 14 km/hr, what is the distance distance between between A and B? B? [L&T] (A) 160 km
(B) 180 km
(C) 200 km
(D) 220 km
14
10. A lawyer goes for a daily walk and climbs up a hill at 2 km/hr. He comes back down the slope and reaches at the bottom of the hill. If he covers 4.8 km in 2 hours, how much his speed was when he came down the hill? [ZENSAR] (A) 2 km/hr
(B) 3 km/hr
(C) 4 km/hr (D) 2.7 km/hr
ANSWER KEY LEVEL 1:
1B
2A
3D
4C
5B
6D
7D
8B
9D
10 B
6B
7A
8D
9B
10 C
7B
8B
9B
10 B
ANSWER KEY LEVEL 2:
1C
2D
3C
4D
5 B
11 D
12 C
13 A
14 A
15 C
ANSWER KEYCOMPANY SPECIFIC QUESTIONS:
1C
2C
3A
4C
5B
6A
Chapter 9 Numbers 9.1 Types of Numbers: The number theory or number systems happens to be the backbone for CAT preparation. Number systems not only form the basis of most calculations and other systems in mathematics, but also form a major percentage of the CAT quantitative section. The reason for that is the ability of examiner to formulate tough conceptual questions and puzzles from this section. In number systems, there are hundreds of concepts and variations, along with various logics attached to them, which makes this seemingly easy looking topic most complex in preparation for the CAT examination. The students while going through these topics should be careful in capturing the concept correctly, as it’s not the speed but the concept that will solve the question here. The correct understanding of concept is the only way to solve complex questions based on 15
this
section.
9.1.1 Real numbers: The numbers that can represent physical quantities in a complete manner. All real numbers can be measured and can be represented on a number line. They are of two types: Rational numbers and Irrational numbers. 9.1.2 Rational numbers: A number that can be represented in the form p/q, where p and q are integers and q is not a zero. Example : 2/3, 1/10, 8/3, etc. They can be finite decimal numbers, whole numbers, and integers, fractions. 9.1.3 Irrational numbers: A number that cannot be represented in the form p/q, where p and q are integers and q is not a zero. An infinite non-recurring decimal is an irrational number. Example : √2, √5, √7 and Π (pie)=3.1416.
The rational numbers are classified into Integers and Fractions. 9.1.4 Integers: The set of numbers on the number line, with the natural numbers, zero and the negative numbers are called integers, I = {…..-3, -2, -1, 0, 1, 2, 3…….}
9.1.5 Fractions: A fraction denotes part or parts of an integer. For example, 1/6, which can represent 1/6th part of the whole, the type of fractions is:
16
1. Common fractions : The fractions where the denominator is not n ot 10 or a multiple of it. Example: 2/3, 4/5, etc.
2. Decimal fractions : The fractions where the denominator is 10 or a multiple of 10. Example:7/10, 9/100, etc.
3. Proper fractions : The fractions where the numerator is less than the denominator. Example: ¾, 2/5, etc. Its value is always alwa ys less than 1.
4. Improper fractions : The fractions where the numerator is greater than or equal to the denominator. Example: 4/3, 5/3, etc. Its value is always greater than or equal to 1.
5. Compound fraction : A fraction of a fraction is called a compound fraction Example: 3/5 of 7/9 = 3/5 x 7/9 = 21/45 6. Complex fractions : The combination of fractions is called a complex co mplex fraction. Example: (3/5) / (2/9)
7. Mixed fractions : A fraction, which consists of two parts, an integer and a fraction. Example: 3 ½, 6 ¾. Ex: Express 27/8 as a mixed fraction. Ans. Divide the numerator by the denominator; note the multiplier, whatever remainder is left divide it with the original denominator. For 2 7/8, 24/8 = 3, and remainder left is 3, therefore,
3 is the the mix mixed ed frac fracti tion on
Ex: Express 35
as an improper fraction.
m ultiply the denominator with the non-fraction part and add it to the Ans. Here, we need to multiply numerator and using the same denominator. For 35
=
The integers are classified into negative numbers and whole numbers.
17
9.1.6 Negative numbers:
All the negative numbers on the number line. For e.g.{…..-3, -2, -1}
9.1.7 Whole numbers: The set of all positive numbers and 0 are called whole numbers. For e.g. W = {0, 1, 2, 3, 4…….}.
The counting numbers 1, 2, 3, 4, 5……. are known as natural numbers, N = {1, 2, 3, 4, 5…..}. The natural numbers along with zero make the set of the whole numbers. 9.1.8 Natural numbers:
9.1.9 Even numbers: The numbers divisible by 2 are even numbers. For e.g. 2, 4, 6, 8,10, etc.
Even numbers can be expressed in the form 2n where ‘n’ is an integer other than 0.
9.1.10 Odd numbers: The numbers not divisible by 2 are odd numbers. For e.g. 1, 3, 5, 7, 9, etc. Odd numbers are expressible in the form (2n + 1) where ‘n’ is an integer other than 0. 9.1.11 Composite numbers: A composite number has other factors besides itself and unity. For e.g. 8,72, 39, etc. A real natural number that is not a prime number is a composite number. 9.1.12 Prime numbers: The numbers that has no other factors besides itself and unity is a prime number. Example : 2, 23,5,7,11,13, etc. Here are some properties of prime numbers:
• The only even prime number is 2 • 1 is neither a prime nor a composite number • 2,3,5,7,11,13,17,19,23,29 are first ten prime numbers (should be remembered). • Two numbers are supposed to be co-prime of their HCF is 1, e.g. 3 & 5, 14 & 29, etc. • A number is divisible by ‘ab’ only when that number is divisible by each one of a and b, where ‘a’ and ‘b’ are co prime. • To find a prime number, check the rough square root of the given number and divide the number by all the prime number numbe r lower than the estimated square root. i.e. n2> P. • All prime numbers can be expressed in the form 6n-1 or 6n+1, but all numbers that can be expressed in this form are not prime. Ex: If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for ‘a’ is:
(a) 1
(b) 2 (c) 3 (d) More than 3
(a) a, a + 2, a + 4 are prime numbers. The number fits is 1, 3, 5 and 3, 5, 7 but post this nothing will fit. Now 1, 3, 5 are not n ot prime numbers as 1 is not a prime number. So, only one possibility is there 3, 5, 7 for a = 3.
9.2 Prime Factors: The composite numbers express in factors, wherein all the factors are prime. To get prime factors, we divide the number by prime numbers till the remainder is a prime 18
number. All composite numbers can be expressed as prime factors, for example, prime factors of 150 are 2, 3, 5, 5. A composite number can be uniquely expressed as a product of the prime factors. Ex: 12 = 2 x 6 = 2 x 2 x 3 = 22 x 31 20 = 4 x 5 = 2 x 2 x 5 = 22 x 51, etc. Note: The number of divisors of a given number N (including one and the number itself ) where N = am x bn x c p……. Where a, b, c are prime numbers are = ( 1 + m ) ( 1 + n ) ( 1 + p )
…………..
Ex: 90 = 2 x 3 x 3 x 3 x 5 = 21 x 33 x 51
Hence here a = 2 b = 3 c = 5 m = 1 n = 3 p = 1, then the number of divisors are = ( 1 + m ) ( 1 + n ) ( 1 + p ) = 2 x 3 x 2 = 12 the factors of 90 = 1 , 2 , 3 , 5 , 6 , 9 , 10 , 15 , 18 , 30 , 45 , 90 = 1 Perfect number : If the sum of the divisor of N excluding N itself is equal to N, then N is called a perfect number. e.g. 6, 28, 496.
9.3 Absolute value of a number: The absolute value of a number ‘a’ is | a | and is always positive.
9.4 Divisibility Rules To determine whether a no. is divisible by a given number or not, we have the following rules: A number is divisible by
If
2
The unit digit is 0,2,4,6 or 8
3
The sum of the digits is divisible by 3
4
The last two digit of a number is divisible by 4 or last two digits are 0
5
The unit digit is 0 or 5
6
The number is divisible by 2 and 3 both
8
The last three digit of a number is divisible by 8
9
The sum of the digits is divisible by 9
11
The difference of the sum of alternate digits is 0 or a multiple of 11
12
The number is divisible by 4 and 3 both
19
9.5 HCF & LCM of Numbers
Highest Common Factor (HCF) The highest common factor (HCF) of two or more numbers is the greatest number that divides each one of them exactly. It is also called as Greatest Common Divisor (GCD). Ex. HCF of 24 and 36 is 12.
HCF can be found in two ways; Division Method and Factorization method. Factorization Method : We need to write each number in its prime factorization format and take the prime numbers common to all given numbers and their minimum power.
16=24, 24=23×3 Now, HCF of 16, 24 = 23 (we must not consider 3 because 16 does not contain the prime factor 3). Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. By the remainder last obtained till zero is obtained as remainder. The last divisor is the required H.C.F.
HCF of fractions
HCF of Fraction =
Ex.HCF of &
20
=
Least Common Multiple (LCM)
The least number, which is exactly divisible b y each one of the given numbers is called their L.C.M. Ex. LCM of 16 & 18 is 144.
There are two methods of finding the L.C.M. of a given set of numbers: Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of the highest powers of all the factors.
15 = 3 x 5 18 = 2×32 27 = 33 Now take all the primes number of the given numbers and write their maximum powers. So LCM of 15, 18, 27 = 2×33×5 = 270. Division Method: Arrange the given numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers. LCM of Fractions
LCM of Fraction =
Ex. LCM of &
21
=
If a, b, c are three numbers that divides a number ‘n’ to leave a remainder ‘r’, then the smallest value of n is n = (LCM of a, b, c) + r. If a, b, c are three numbers that divides a number ‘n’ to leave a different remainder p, q, r then say (a-p)=(b-q)=(c-r)=d, then the smallest value of ‘n’ is n=(LCM of a, b, c) -d. If a given number is divided successively by the different factors of the divisor leaving remainder r 1, r 2 and r 3, resp., then the true remainder (i.e. remainder when the number is divided by the divisor) can be obtained by using the formula. True Remainder = (first remainder) + (second remainder × first divisor) + (third remainder × first divisor × second divisor). (xn+yn) is divisible by (x+y) when n is an odd number. (xn-yn) is divisible by (x+y) when n is an even number. (xn+yn) is divisible by (x-y) when n is an odd or an even number.
9.6. Progression
A succession of numbers formed and arranged in a definite order according to certain definite rule is called a progression. 1. Arithmetic Progression (A.P.): If each term of a progression differs from its preceding term by a constant, then such a progression is called an arithmetical progression. This constant difference is called the common difference of the A.P.
An A.P. with the first term ‘a’ and common difference ‘d’ is given by a, (a + d), (a + 2d), (a + 3d),..... The nth term of this A.P. is given by Tn = a (n - 1) d. The sum of n terms of this A.P. is
Sn = n/2 [2a + (n - 1) d] = n/2 (first term + last term).
2. Geometrical Progression (G.P.): A progression of numbers in which every term bears a constant ratio with its preceding term is called a ge ometrical progression.
22
The constant ratio is called the common ratio of the G.P. A G.P. with first term ‘a’ and common ratio ‘r ‘is: a, ar, ar 2 In this, G.P. Tn = ar n-1
Sum of the ‘n’ terms, Sn= a(1-r n) (1-r)
9. 7 Important Results:
I.
Sum of first ‘n’ natural numbers =
=
II.
Sum of first ‘n’ odd numbers =
III.
Sum of first ‘n’ even numbers =
IV.
Sum of squares of first ‘n’ natural numbers =
V.
Sum of cubes of first ‘n’ natural numbers =[
= n2
= n(n+1)
]2
Level 1 1. For any natural number, a, b, and c, (a+b) (b+c) (A) always even
(B) always odd
(c+a) is ……………….
(C) sometime even
(D) sometimes Odd
2. Number of divisor of 37800, including 1 and itself is 23
(A) 96
(B) 70
(B) 81
(D) None of these
3. The rational number for recurring decimal 0.0593059305930... is: (A) 93 / 234
(B) 593 / 9999
(C) 1 / 593
(D) 100 / 593
4. The smallest number which when diminished by 3 is divisible by 21,28,36 and 45 is... (A) 869
(B) 859
(C) 4320
(D) 1263
5. If x and y are the two digits of the number 653xy such that this number is divisible by 80, Then x + y is equal to: (A) 2
(B) 3
(C) 4
(D) 6
6. Find the greatest of the four least common multiples of 3, 5 and 7. (A) 1 (B) 420 (C) 315 (D) 105 7. The largest fraction among the following is:
(A)
(B)
(C)
,
,
,
,
(D)
8. Three numbers, which are co-prime to one another are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is: (A ) 75
(B) 81
9. The simplification of 3. (A) 2.60
(C) 85
– 2.
(B) 2.
+ 1.
(D) 89 equals:
(C) 2.64
(D) 2.
10. Three numbers are in the ratio 3 : 4 : 5 and their L.C.M. is 2400. Their H.C.F. is: (A) 200
(B) 80
(C) 40
(D) 120
11. A number when successively divided by 5, 3, 2 gives remainder 0, 2, 1 respectively in that order. What will be the remainder when the same number is divided successively by 2, 3, 5 in that order. (A) 4, 3, 2
(B) 1, 0, 4
(C) 2, 1, 3
(D) 4, 1, 2
12. A school has 120, 192, and 144 students enrolled for its science, arts and commerce courses. All students have to be seated in rooms for an exam such that each room has students of only the 24
same course and also all rooms have equal number of students. What is the least number of rooms needed? (A) 21
(B) 19
(C) 24
(D) 20
13. The remainder when 1!+2!+3!...+50! divided by 5! will be (A) 21
(B) 14
(C) 9
(D) 33
14. Find the remainder when 2 31 is divided by 5? (A) 4
(B) 3
(C) 2
(D) None of these
15. Find the unit's digit in (5314)98 + (7454)151? (A) 0
(B) 2
(C) 1
(D) None of these
16. The LCM of a(a+b), a 2(a2-b2) and ab2(a+b)2 is: (A) Ab (a-b) (a+b)2
(B) a2 b2 (a+b) (a-b)2
(C)a2 b2 (a-b) (a+b)2 (D) ab (a-b) (a-b)2
17. The LCM of two numbers is 12 times their HCF. The sum of the HCF and LCM is 403. If one of the numbers is 93, then the other number is: (A) 124
(B) 128
(C) 134
(D) 138
18. What is the sum of the numbers between 300 and 400 such that when they are divided by 6, 9 and 12, they leave same remainders 4 in each case. (A) 482
(B) 692
(C) 512
(D) 342
19. Find the unit digit of 23 24×6413×8768 (A) 8
(B) 4
(C) 6
(D) 3
20. Find the least number which when divided by 5, 6 and 7 and leaves a remainder 3, but when divided by 9 leaves no remainder. (A) 288
(B) 213
(C) 207
(D) 423
LEVEL 2 1. The last digit of the number obtained by multiplying the numbers 81 x 82 x 83 x 84 x 85 x 86 x 87 x 88 x 89 will be 25
(a) 0
(b) 9
(c) 7
(d) 2
2. The sum of the digits of a two-digit number is 10, while when the digits are reversed, the number decreases by 54. Find the changed number. (a) 28
(b) 19
(c) 37
(d) 46
3. When we multiply a certain two-digit number by the sum of its digits, 405 is achieved. If you multiply the number written in reverse order of the same digits by the sum of the digits, we get 486. Find the number. (a) 81
(b) 45
(c) 36
(d) 54
4. The sum of two numbers is 15 and their geometric mean is 20% lower than their arithmetic mean. Find the numbers. (a) 11, 4
(b) 12, 3
(c) 13, 2
(d) 10, 5
5. The difference between two numbers is 48 and the difference between the arithmetic mean and the geometric mean is two more than half of 1/ 3 of 96. Find the numbers. (a) 49, 1
(b) 12, 60
(c) 50, 2
(d) 36, 84
6. If A381 is divisible by 11, find the value of the smallest natural number A. (a) 5
(b) 6
(c) 7
(d) 9
7. If 381A is divisible by 9, find the value of smallest natural number A. (a) 5
(b) 5
(c) 7
(d) 6
8. What will be the remainder obtained when (9 6 + 1) will be divided by 8? (a) 0
(b) 3
(c) 7
(d) 2
9. Find the ratio between the LCM and HCF of 5, 15 and 20. (a) 8 : 1
(b) 14 : 3
(c) 12 : 2
(d) 12 : 1
10. Find the LCM of 5/2, 8/9, 11/14. (a) 280
(b) 360
(c) 420
(d) None of these
11. If the number A is even, which of the following will be true? (a) 3A will always be divisible by 6 (c) (A 2 + 3) / 4 will be divisible by 7
(b) 3A + 5 will always be divisible by 11 (d) All of these 26
12. A five-digit number is taken. Sum of the first four digits (excluding the number at the units digit) equals sum of all the five digits. Which of the following will not divide this number necessarily? (a) 10
(b) 2
(c) 4
(d) 5
13. A number 15B is divisible by 6. Which of these will be true about the positive integer B? (a) B will be even (c) B will be divisible by 6
(b) B will be odd (d) Both (a) and (c)
14. Two numbers P = 2 3.310.5 and Q = 2 5.31.71 are given. Find the GCD of P and Q. (a) 2.3.5.7
(b) 3. 22
(c) 22.32
(d) 23.3
15. Find the units digit of the expression 25 6251 + 36528 + 7354. (a) 4
(b) 0
(c) 6
(d) 5
16. Find the units digit of the expression 55 725 + 735810 + 22853. (a) 4
(b) 0
(c) 6
(d) 5
17. Find the units digit of the expression 11 1 + 122 + 133 + 144 + 155 + 166. (a) 1
(b) 9
(c) 7
(d) 0
18. Find the units digit of the expression 11 1.122.133. 144 .155 .166 (a) 4
(b) 3
(c) 7
(d) 0
19. Find the number of zeroes at the end of 1090! (a) 270
(b) 268
(c) 269
(d) 271
20. If 146! is divisible by 5 n , then find the maximum value of n. (a) 34
(b) 35
(c) 36
(d) 37
21. Find the number of divisors of 1420. (a) 14
(b) 15
(c) 13
(d) 12
22. Find the HCF and LCM of the polynomials (x 2 – 5x + 6) and (x 2 – 7x + 10). (a) (x - 2), (x – 2) (x – 3) (x – 5) (c) (x – 3), (x – 2) (x – 3) (x – 5)
(b) (x – 2), (x – 2)( x – 3) (d) (x - 2), (x – 2) (x – 3) (x – 5)2
Directions for Questions 23 to 25: Given two different prime numbers P and Q, find the number of divisors of the following: 27
23. P.Q (a) 2
(b) 4
(c) 6
(d) 8
(b) 4
(c) 6
(d) 8
(b) 4
(c) 6
(d) 12
24. P2Q (a) 2 25. P3Q2 (a) 2
26. The sides of a pentagonal field (not regular) are 1737 meters, 2160 meters, 2358 meters, 1422 meters and 2214 meters respectively. Find the greatest length of the tape by which the five sides may be measured completely. (a) 7
(b) 13
(c) 11
(d) 9
27. There are 576 boys and 448 girls in a school that are to be divided into equal sections of either boys or girls alone. Find the minimum total number of sections thus formed. (a) 24
(b) 32
(c) 16
(d) 20
28. A milkman has three different qualities of milk. 403 gallons of 1st quality, 465 gallons of 2nd quality and 496 gallons of 3rd quality. Find the least possible number of bottles of equal size in which different milk of different qualities can be filled without mixing. (a) 34
(b) 46
(c) 26
(d) 44
29. What is the greatest number of 4 digits that when divided by any of the numbers 6, 9,12, 17 leaves a remainder of 1? (a) 9997
(b) 9793
(c) 9895
(d) 9487
30. Find the least number that when divided by 16, 18 and 20 leaves a remainder 4 in each case, but is completely divisible by 7. (a) 364
(b) 2254
(c) 2964
(d) 2884
COMPANY SPECIFIC QUESTIONS 1. Which of the following numbers are completely divisible by 11? (i) 3245682 (ii) 283712 (iii) 438416 (iv)36894 [ACCENTURE] (A) Only (i)
(B) Only (ii)
(C) Only (iii)
(D) All are divisible
28
2. The difference between a two-digit number and the number obtained by interchanging the digits is 36.What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1:2? [CAPGEMINI] (A) 4 (B) 8 (C) 16 (D) None of these 3. What is the largest integer that divides all three numbers 23400, 272304, 205248 without leaving a remainder? [TCS] (A) 48
(B) 24
(C) 96
(D) 72
4. Find the least number when divided by 7 gives the reminder 6, when divided by 6 gives reminder 5, when divided by 5 gives reminder 4 and so on.... (A) 419 5.
(B) 523
If 0.0ababab….= (A) 10
2 55
(C) 399
(D) None of these
, what is the value of a x b?
(B) 18
[WIPRO]
(C) 27
[TCS] (D) 36
6. Six bells commence tolling together and toll at intervals 2, 4, 6, 8, 10 and 12 seconds, respectively. In 30 minutes, how many times they toll together?[WIPRO] (A) 4
(B) 10
(C) 15
(D) 16
7. If a six-digit number 93p25q is divisible by 88, then the values of p and q are _______respectively. [INFOSYS] (A) 2 and 8
(B) 8 and 2
(C) 8 and 6
(D) 6 and 8
8. The LCM of two numbers is 80 and the product of the two numbers is 320. How many such possible pairs of numbers exist? [CYBAGE] (A) 1
(B) 3
(C) 2
9. Find the sum of prime factors of 561. (A) 31
(B) 12
(C) 40
(D) More than 3 [ACCENTURE]
(D) 41
10. 96 + 7 when divided by 8 would have a remainder of: (A) 0
(B) 6
(C) 5
[TCS]
(D) None of these 29
ANSWER LEVEL 1: 1.(A) 11.(B)
2.(A) 12.(B)
3.(B) 13.(D)
4.(D) 14.(B)
5.(D) 15.(A)
6.(B) 16.(C)
7.(C) 17.(A)
8.(C) 18.(B)
9.(D) 19.(B)
10.(C) 20.(D)
ANSWER LEVEL 2: 1.(A)
2.(A)
3.(B)
4.(B)
5.(A)
6.(C)
7.(D)
8.(D)
9.(D)
10.(D)
11.(A)
12.(C)
13.(D)
14.(D)
15.(B)
16.(C)
17.(B)
18.(D)
19.(A)
20.(B)
21.(D)
22.(A)
23.(B)
24.(C )
25.(D)
26.(D)
27.(C )
28.(D)
29.(B)
30.(D)
7.(C)
8.(B)
ANSWER COMPANY SPECIFIC QUESTIONS: 1.(D)
2.(B)
3.(B)
4.(A)
5.(B)
6.(D)
9.(A)
10.(A)
30
Chapter 10 PERMUTATIONS & COMBINATIONS 10.1 PERMUTATION : An arrangement that can be formed by taking some or all of a finite set of things (or objects) is called a Permutation . Order of the things is very important in case of permutation. A permutation is said to be a Linear Permutation if the objects are arranged in a line. A linear permutation is simply called as a permutation. A permutation is said to be a Circular Permutation, if the objects are arranged in the form of a circle. n Selection of objects from n objects and arranging them in r positions will be P r. n
Pr =
= n(n-1)(n-2)……………..(n-r +1)
10.2 COMBINATION :
A selection that can be formed by taking some or all of a finite set of things( or objects) is called a Combination.
Selection of objects from n objects will be nCr n
Cr =
10.3 Difference between Permutation and combination:
If order is important, then the problem relates to permutations and the number of possible samples is the required number of arrangement. If order is not important, then the problem relates to combinations and the number of possible samples is e the required number of combination. For e.g. Suppose we have to make a pair of numbers from three numbers, say, 2, 4, 7 If order is important, then first picking 2 then 4 is different from picking 4 then 2, that is, (2, 4) and (4, 2) are two different samples. Similarly (4, 7) and (7, 4) are different samples. Therefore, total permutation formed = 6 [i.e. (2,4),(4,2),(4,7),(7,4),(2,7),(7,2)]
If order is not important, then no matter which number is picked first, either 2 or 4, it doesn’t make any different sample. Therefore, total combination formed = 3 [i.e. (2,4),(4,7),(7,2)]
31
10.4 Fundamental Rule
If one operation can be performed in ‘m’ ways and second operation can be performed in ‘n’ ways then the no. of ways of performing both the operations will be m n. This is called Product rule.
So total route from Banglore to Delhi= 4×3=12.
If one operation can be performed in ‘m’ ways and second operation can be performed in ‘n’ ways then no. of ways of performing of either operation one or second will be m + n. This is called Sum rule. Therefore, total no. of routes to either go to Banglore or Delhi from Nagpur = 4+3=7. 10.5 Circular Permutation:
If there is a difference between clockwise and anti clockwise arrangement, the number of circular arrangement of n distinct items is (n-1)! If there is no difference between clockwise and an ti clockwise arrangement, the number of circular arrangement of n distinct items is (n-1)!/2
10.6 Useful Relations: n
I.
Cr = nCn-r
II.
n
Cr-1+nCr = n+1Cr
III.
n
Cr = nPr / r!
IV.
n
C0 = nCn = nP0=1
LEVEL-1 32
1. How many four letter words can be made from the word ‘INDEPENDENT’? (A) 98
(B) 110
(C) 212
(D) None of these
2. How many six digit odd numbers can be formed from the digits 0, 2, 3, 5, 6, 7, 8, 9 (Repetition not allowed)? (A) 9048
(B) 7224
(C) 8640
(D) None of these
3. If we have to make 9 boys sit with 9 girls around a round table, then the number of different relative arrangement of boys and girls that we can make so that there are no two boys nor any two girls sitting next to each other is: (A) 8!×9!
(B) 7!×9!
(C) 8!×9P8
(D) None of these
4. In how many ways can the letters of English alphabet can be arranged so that there are seven letters between the letter A and B? (A) 24C7
(B) 2!×24P7
(C) 24P7
(D) None of these
5. How many words can be formed by re-arranging the letters of the word AS CENT such that A and T occupy the first and last position respectively? (A) 5!
(B) 6!–2!
(C) 4!
(D) 6!
6. How many 6 digit numbers can be formed with 1, 2, 3, 4, 5 and 6 such that all the odd positions from the left end are odd digits and all even positions are even digits. Repetition of digits is not allowed. (A) 72
(B) 36
(C) 720
(D) 360
7. How many 6 digit numbers can be formed using the digits 1, 2, 3, 4 and 5 (repetitions allowed) so that the number reads the same from left to right as right to left? ( Eg:134431) (A) 6
(B) 12
(C) 24
(D) 36
8. How many 5-digit positive integers exist the sum of whose digits are odd? (A) 36000
(B) 38000
(C) 45000
(D) 90000
33
9. 36 identical chairs must be arranged in rows with the same number of chairs in each row. Each row must contain at least 3 chairs and there must be at least 3 rows. A row is parallel to the front of the room. How many different arrangements are possible? (A) 2
(B) 4
(C) 5
(D) 6
10. If nCr-1 = nCr+1=15 and nCr = 20, then the value of r C2 is: (A) 3
(B) 3!
(C) 4!
(D) 12
LEVEL-2
1. How many words of 4 consonants and 3 vowels can be made from 12 consonants and 4 vowels, if all the letters are different? (A) 16C7×7!
(B) 12C3×4C4
(C) 12C4×4C3×7!
(D) 12C4×4C3
2. In a group of 15 students, there are 6 singers. In how many wa ys can 12 students be selected so as to include at least 4 singers? (A) 435
(B) 425
(C) 415
(D) 405
3. In how many ways can 6 delegates sit in a round table conference so that all shall not have the same neighbours in any two arrangements? (A) 120
(B) 60
(C) 24
(D) None of these
4. Find the number of ways in which 5 identical rings can be worn on the four fingers of one hand if each finger can have any number of rings. (A) 5P4
(B) 54
(C) 5C4×5!
(D) 45
5. There are 10 points in a plane, of which 4 are collinear. Find the number of triangles formed. (A) 120
(B) 116
(C) 214
(D) None of these
34
6. From a total of six men and four ladies a committee of three is to be formed. If Mrs. X is not willing to join the committee in which Mr. Y is a member, whereas Mr.Y is willing to join the committee only if Mrs Z is included, how many such committee are possible? (A) 138
(C) 112
(B) 128
(D) 91
7. The letter of the word LABOUR are permuted in all possible ways and the words thus formed are arranged as in a dictionary. What is the rank of the word LABOUR? (A) 275
(B) 242
(C) 251
(D) 240
8. An auditorium with 20 rows of seats has 10 seats in the 1st row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for an exam is: (A) 150
(B) 180
(C) 100
(D) 200
9. There are 20 couples in a party. Every person greets every person except his or her spouse. People of the same sex shake hands and those of opposite sex greet each other with a namaskar (It means bringing one's own pal ms together and raising them to the chest level). What is the total number of handshakes and namaskar's in the party? (A) 760
(B) 1140
(C) 780
(D) 720
10. Six countries sent their Prime Ministers and Presidents for a UN Conclave. In a round of six discussions held at a conclave, each discussion was between a PM and President, both from different countries. Find the number of ways in which this conclave would have been conducted. (A) 984
(B) 1412
(C) 1628
(D) 265
11. There are 20 balls which are red, blue or green. If 7 balls are green and the sum of red balls and green balls is less than 13, at most how many red balls are there? (A) 4
(B) 5
(C) 6
(D) 7
12. In how many ways can we distribute 10 identical looking pencils to 4 students so that each student gets at least one pencil? 35
(A) 5040
(B) 210
(C) 84
(D) None of these
13. How many ways can 10 letters be posted in 5 post boxes, if each of the post boxes can take more than 10 letters? (A) 5^10
(B) 10^5
(C) 10P5
(D) 10C5
14. How many numbers of five digits can be formed with the digits 0, 1, 2, 4, 6 and 8? (A) 550
(B) 600
(C) 1200 (D) 500
15. There are 4 boxes colored red, yellow, green and blue. If 2 boxes are selected, how any combinations are there for at least one green box or one red box to be selected? (A) 1 (B) 6
(C) 9
(D) 5
LEVEL-3 (COMPANY SPECIFIC QUESTIONS)
1. There are two sections in a question paper each contain five questions. A student has to answer 6 questions. Maximum no. of questions that can be answered from any section is 4. How many ways he can attempt the paper? [WIPRO] (A) 50
(B) 100
(C) 120
(D) 200
2. In the 4 digits 1, 2, 3, 4, how many 4 digit numbers are possible which are divisible by 4? Repetitions are allowed. [L&T] (A) 64
(B) 72
(C) 48
(D) None of these
3. A student can select one of 6 different mathematics books, one of 3 different chemistry books and one of 4 different science books. In how many different ways can a student select a book of mathematics, a book of chemistry and a book of science? [TCS] (A) 14
(B)12
(C)72
(D) 74
4. A set of football matches is to be organized in a "round-robin" fashion, i.e., every participating team plays a match against every other team once and only once. If 21 matches are totally played, how many teams participated? [INFOSYS]
(A) 6
(B) 7
(C) 10
(D) 12 36
5. In a group of 8 semifinals, all but two will advance to the final round. If in the final round only top 3 will be awarded the medals, then how many groups of medal winners are possible? [ACCENTURE] (A) 120
(B) 560
(C) 720
(D) None of these
6. There are 3 sections with 5 questions each. If three questions are selected from each section, then in how many ways this selection can be done? [DELOITTE] (A) 1200
(B) 1000
(C) 1296
(D) None of these
In how many ways can 4 boys and 5 girls be seated in a row so that they are sitting alternate?
7.
[PERSISTANT] (A) 2400
(B) 120
(C) 2880
(D) None of these
8. On a triangle ABC, on the side AB, 5 points are marked, 6 points are marked on side BC, and 3 points are marked on the AC (none of the points being the vertex of the triangle). How many triangles can be made using these points.[TCS] (A) 333
(B) 328
(C) 273
(D) None of these
9. The letters of the word WOMAN are written in all possible orders and these words are written out as in a dictionary ,then the rank of the word 'WOMAN' is: [TECH-MAHINDRA] (A) 117
(B) 120
(C) 118
(D) 119
10. There are 12 points in a plane out of which 15 are collinear. Find the number of straight lines formed by joining them. [HP] (A) 57
(B) 45
(C) 48
(D) 55
Answers (Permutations & Combinations) LEVEL 1 1B
2C
3A
4B
5C
6B
7A
8C
9C
10 A
2A
3B
4D
5B
6D
7B
8D
9B
10 D
Level 2 1C
37
11 C
12 C
13 A
14 B
15 D
4B
5D
Company Specific Questions: 1D
2A
3C
6B
7C
8D
9A
10 A
CHAPTER-11 PROBABILITY 11.1 PROBABILITY: Probability of an event is the ratio of the number of observations of the event to the total numbers of the observations.
Probability of an event =
Probability of an event A is symbolized by P(A)
Probability of an event A lies between0 ≤ P(A) ≤ 1.
Probability is a chance of prediction. P(A) + P( ) = 1
If ‘a’ outcomes are favorable to A and ‘b’ outcomes are not favorable to A, then “Odds in favor of A” are a:b, and “Odds against of A” are b: a, hence, we can write P(A) =
and P( ) =
If P(A) = 0, then event A is impossible. If P(A) = 1, then event A is certain.
11.2 Random Experiment: An experiment in which all possible outcomes are known and the exact output cannot be predicted in advance is called a random experiment. EXAMPLE: 1) Rolling an unbiased dice. 2) Tossing a fair coin. 3) Drawing a card from a pack of well-shuffled cards. 4) Picking up a ball of certain color from a bag containing balls of different colors. Sample Space: The set of all possible outcomes of an experiment is called sample space(s). Mutually exclusive Events: Two events are mutually exclusive if the occurrence of the one excluded the 38
simultaneous occurrence of the other. For example, if a coin is thrown, getting a head and getting a tail are mutually exclusive. Independent Events: Two events are independent if the occurrence or non-occurrence of the one does not affect the other. For example, if two dice are thrown simultaneously, the number that one dice shows and the number that the other die shows are independent events. Addition of Probabilities: In two events A and B are mutually exclusive, then the probability of A or B happening is equal to the sum of the probability of A occurring and the probability of B occurring i.e., P(A or B) = P(A) + P(B). This holds good for three or more mutually exclusive events. If two events are not mutually exclusive, the probability of occurrence of A or B is P(A or B) = P(A) + P(B) – P(A and B). Multiplication of Probabilities: If two events are independent the probability that both will occur is the product of their individual probabilities i.e.,
P(A and B) = P(A) x P(B) This can be extended to three or more independent events A , B, C …… etc. Conditional Probability: If the occurrence of an event is related to the occurrence of another event, the occurrence of both events simultaneously is called a dependent compound event.
The notation P(A ∩ B) or simply P(AB) is used to denote the probability of the joint occurrence of the events A and B. If the events are independent P(AB) = P(A) x P(B) When events A and Bare not independent, P(AB) = P(A) = P(B) The probability of occurrence of the event B when it is known that event A has already occurred is called conditional Probability of B and is given by P
=
Level-1 1) In a simultaneous throw of dice, what is the probability of getting a sum of 7 ? (A) 1/2 (B)1/3 (C)1/6 (D)8/12 39
2) Two dice are thrown at a time. What is the probability that the sum of two numbers is 6 or 9? (A) 3/4 (B)1/4 (C)7/36 (D)1/2 3) A student has to select 3 subject out of 6 subject M,B,H,U,L and S. If he has chosen M, what is
the probability of ‘B’ being chosen? (A) 2/5
(B)3/5
(C)4/5
(D)1/5
4) A bag contains 8 red and 6 blue balls. If 5 balls are drawn at random, what is the probability that 3 are red and 2 are blue? (A) 50/143 (B)96/143 (C)14/143 (D)60/143 5) If 2 ball are drawn one after another from a bag containing 3 red and 5 black balls, what is the probability that the 1st ball is red and the 2 nd ball is black ? (A)7/56 (B)3/28 (C)15/56 (D)11/28 6) Two diced are tossed, what is the probability that the total score is a prime number? (A) 5/12 (B)7/36 (C)1/4 (D) None of these 7) An urn contains 6red, 5 blue and 2 green marbles. If 2 marbles are picked at random, what is the probability that both are red (A) 6/13 (B) 5/26 (C) 5/13 (D) 7/26 8) A bag contains 12 white and 18 black balls. Two balls are drawn in succession without replacement. What is the probability that first is white and second is black? (A) 18/145
(B) 18/29
(C) 36/135
(D) 36/145
9) A box contains 20 electric bulbs, out of which 4 are defective. Two bulbs are chosen at random from this box. The probability that at least one of these is defective is (A) 4/19
(B) 7/19
(C) 12/19
(D) 21/95
10) In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green? (A) 2/3
(B) 3/4
(C) 7/19
(D) 7/21
Level-2 40
1. If the letters of the word 'REGULATIONS' be arranged at random, what is the probability that there will be exactly 4 letters between R and E? (A) 6/55 (B) 6/44 (C) 5/23 (D) 4/18 2. A three-digit number is formed with the digits 1, 2, 3, 4, 5 at random. What is probability that number formed is divisible by 2 : (A) 2/5
(B) 3/5
(C) 5/2
(D) 7/2
3. The letters of the word CASTIGATION is arranged in different ways randomly. What is the chance that vowels occupy the even places ?
(A) 0.19
(B) 0.043
(C) 0.093
(D) 0.087
4. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5? (A) 9/20
(B) 8/15
(C) 2/5
(D) ½
5. I forgot the last digit of a 7-digit telephone number. If 1 randomly dial the final 3 digits after correctly dialing the first four, then what is the chance of dialing the correct number? (A) 1/1001
(B) 1/1000
(C) 1/999
(D) 1/990
6. In his wardrobe, Dexter has three trousers. One of them is black the second is blue, and the third brown. In his wardrobe, he also has four shirts. One of them is black and the other 3 are white. He opens his wardrobe in the dark and picks out one shirt and one trouser pair without examining the colour. What is the likelihood that neither the shirt nor the trouser is black? (A) 1/12
(B) 1/6
(C) 1/4
(D) 1/2
7. A man can hit a target once in 4 shots. If he fires 4 shots in succession, what is the probability that he will hit his target? (A) 1/256
(B) 81/256
(C) 1
(D) 175/256
8. Two brothers X and Y appeared for an exam. The probability of selection of X is 1/7 and that of B is 2/9. Find the probability that both of them are selected. (A) 1/63
(B) 2/35
(C) 2/63
(D) 9/14
9. he letters B,G,I,N and R are rearranged to form the word 'Bring'. Find its probability: 41
(A) 1/120
(B) 1/625
(C) 1/24
(D) 1/76
10. Dinesh speaks truth in 3/4 cases and Abhishek lies in 1/5 cases. What is the percentage of cases in which both Dinesh and Abhishek contradict each other in stating a fact? (A) 60%
(B) 35%
(C) 20%
11. A dice is rolled three times and the sum of the numbers appearing on The chance that the first roll was a four is: (A) 2/5
(B)1/5
(C) 1/6
(D) 15% the uppermost face is 15.
(D) None of these
12. There are two boxes namely A and B. A contains 20 green and 15 blue balls and B contains 7 green and 8 blue balls. You can move the balls between the two boxes. If you are allowed to choose a box at random then what will be the maximum probability of getting a green ball from the chosen box? (A) 55/58
(B) 40/58
(C) 41/58
(D) 39/58
13. There are three boxes namely A, B and C. A contains 20 – one rupee coins, B contains 12 – five rupee coins and C contains 31 – two rupee coin. You can move the coins between the three boxes. If you are allowed to choose a box at random then what will be the maximum probability of getting a two rupee coin from the chosen box? (A) 111/183 14.
(B) 169/183
(C) 151/183
(D) 171/183
In a game there are 70 people in which 40 are boys and 30 are girls, out of which 10 people are selected at random. One from the total group, thus selected is selected as a leader at random. What is the probability that the person, chosen as the leader is a boy?
(A) 4/7
(B) 4/9
(C) 5/7
(D) 7/9
15. An eight-digit telephone number consists of exactly two zeroes. One of the digits is repeated thrice. Remaining three digits are all distinct. If the first three digits(from left to right) are 987, then find the probability of having only one 9, one 8 and one 7 in the telephone number. (A) 1/18
(B) 1/20
(C) 5/47
(D) 1/10
COMPANY SPECIFIC QUESTIONS 1) A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is is:[CAPGEMINI] (A) 1/13
(B)2/13
(C)1/26
(D)1/52 42
2) Eric throws two dice, and his score is the sum of the values shown. Sandra throws one dice, and her score is the square of the value shown. What is the probability that Sandr a’s score will be
strictly higher than Eric’s score? [TCS] (A) 137/216
(B) 17/36
(C) 173/216
(D) 5/6
3) There are two boxes, one containing 10 red balls and the other containing 10 green balls. You are allowed to move the balls between the boxes so that when you choose a box at random and a ball at random from the chosen box, the probability of getting a red ball is maximized. This maximum probability is[TCS] (A) 1/2
(B) 3/7
(C) 37/38
(D) 14/19
4) Out of forty students, there are 14 who are taking Physics and 29 who are taking Calculus. What is the probability that a randomly chosen student from this group is taking only the Calculus class? (A) 0.5
[COGNIZANT] (B) 0.6
(C) 0.75
(D) None of these
5) If A speaks the truth 80% of the times, B speaks the truth 60% of the times. What is the probability that they tell the truth at the same time? [WIPRO] (A) 0.8 (B) 0.48 (C) 0.6 (D) 0.14
6) There are two boxes namely A and B. A contains 20 green and 15 blue balls and B contains 7 green and 8 blue balls. You can move the balls between the two boxes. If you are allowed to choose a box at random then what will be the maximum probability of getting a green ball from the chosen box? [ACCENTURE] (A) 55/58
(B) 40/58
(C) 41/58
(D) 39/58
7) There are three boxes namely A, B and C. A contains 20 – one rupee coins, B contains 12 – five rupee coins and C contains 31 – two rupee coin. You can move the coins between the three boxes. If you are allowed to choose a box at random then what will be the maximum probability of getting a two rupee coin from the chosen box?[ACCENTURE] (A)111/183
(B) 169/183
(C) 151/183
(D) 171/183
8) A group consists of equal number of men and women. Of them 10% of men and 45% of women are unemployed. If a person is randomly selected from the group. Find the probability for the selected person to be an employee. [ATOS] (A) 29/40
(B) 19/24
(C) 7/45
(D) None of these 43
9) A man has to decide among 10 lotteries to pick one. The probability that one of these have a prize is 1/3. If the number of lottery is of 3 digit, what are his chances that he will win the lottery? [CAPGEMINI] (A) 1/2
(B) 1/1000
(C) 1/10
(D) 1/30
10) A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is: (A) 1/13
(B)2/13
(C)1/26
(D)1/52
ANSWER SH EET L EVEL -1 :
1C
2B
3A
4D
5C
6A
7B
8D
9B
10 D
6D
7D
8C
9A
10 B
6A
7C
8A
9D
10 C
ANSWER SH EET L EVEL -2 :
1A
2A
3B
4A
5B
11 B
12 A
13 C
14 A
15D
: COM PANY SPECI FI C QUESTI ONS
1C
2A
3D
4B
5B
44
CHAPTER 12 SET THEORY
45
46
47
LEVEL 1
1. In a class, there are 150 people who speak English and 125 people who speak Hindi. 55 speak both languages and at least one student speaks one language. How many students speak at least one language? (A) 210
(B) 165
(C) 220
(D) None of these
2. In a class everyone will play at least one sport viz. table tennis, cricket and badminton. 73 students play table tennis, 79 play badminton, 75 play cricket and 25 students play both table tennis and badminton, 24 play both table tennis and cricket 30 play. Badminton and cricket and 16 students play all the sports. How many students are there in the class? (A) 161
(B)145
(C)130
(D) None of these
3. In a certain examination, 40% of the students failed in Mathematics, 30% failed in English and 10% failed in both. The percentage of students who passed in both the subjects is: (A) 20
(B) 40
4. If A and B are subsets of
(C) 30 ∪ such
(D) 10
that n( ∪ ) = 700,n(A) = 200, n(B) = 300, n(A ∩ B) =
100, then n (A ’ ∩ B’ ) is equal to: (A) 200
5.
(B) 300
(C) 400
(D) 600
In National Public school, out of every five pupils learning Marathi, two learn Hindi as well; and for every student learning at least one of these languages, there are three who learn neither. Nineteen percent of the school learns Hindi . What percent of the students learn neither of the languages?
(A) 75
(B) 71
(C) 66
(D) Indeterminate
6. A, B, and C are finite sets. A has twice as many elements as B. B has more elements than C. The number of subsets of B is 15 more than that of C. Then what is the number by which the number of subsets of A exceeds the number of subsets of B? 48
(B) 28
(A) 0
(D) 215
(C) 240
7. For any three sets A, B, C: A –(B∩C) is equal to: (A) (A-B) ∪ (A-C)
(B) (A∩B) −C
(C) (A−B) ∩ (A−C) (D) A ∪ (B−C)
8. Let A, B, C be three finite sets with k, l, m elements, respectively. If then the number of elements in the set A ×(B ∪C) is: (A) klm – n
(B) k + l + m – n
(C) k(l+m-n)
B∩C contains n elements,
(D) k(lm-n)
9. In a class of 100 students, the number of students passed in English is 46, in Maths is 46, in commerce is 58. The number who passes in English and Maths is 16, Maths and Commerce is 24 and English and commerce is 26, and the number who passed in all the subject is 7. Find the number of the students who failed in all the subjects. (A) 10
(B) 9
(C) 8
(D) 7
10. In a residential complex of 25 houses, each house has either a colour TV or a scooter, or both. The number of houses that have only colour TV is in excess of these with both colour TV and scooter by 3. The number of houses with scooter alone is less than the number with colour TV alone by 2. Then how many houses have both colour TV and scooter and what % of the houses have only scooter. (A) 10, 10%
(B) 7, 32%
(C) 6, 28%
(D) 9, 21%
11. At a party 30% of the people have only cocktail drinks and snacks, 20% have only food, and 70% have snacks. If nobody has all three and everybody has something, then how many have food? (A) 60%
(B) 70%
(C) 42%
(D) None of these
12. If A and B are two sets containing 4 and 7 distinct elements, respectively, find the minimum possible number of element (A U B). (A) 8 (B) 7 (C) 6 (D) None of these Combined question for Q.13 and Q.14
In a class of 75 students, 36 have opted for Mathematics, 24 have take Mathematics but not Biology. 13. Find the number of students who have taken mathematics and biology. (A) 18 (B) 4 (C) 12 (D) 16 14. Find those who have taken biology, but not mathematics.
49
(A) 48
(B) 39
(C) 40
(D) 4
15. If n(P) = 27, n(PUQ) = 48 and n(PᴨQ) = 2 then find n(P-Q) ? (A) 20
(B) 21
(C) 41
(D) 10
16. 65% of students in a class like cartoon movies, 70% like horror movies and 75% like war movies. What is the smallest % of students liking all the three type of movies? (A) 10%
(B) 5%
(C) 15%
(D) 20%
Combined question for Q.17-Q.20
In a survey of 2000 students, conducted in a college, it was found that 48% played Football (F), 54% played Cricket (C) and 64% played Badminton (B). Of the total 28% played F and C, 32% played C and B and 30% played F and B. Only 6% did not play any of the three games. Find the number of students playing. 17. All the three (A) 520 (B) 360 18. C and B, but not F (A) 280 (B) 512 19. C but not F (A) 480 (B) 520 20. Only F (A) 362 (B) 560
(C) 452
(D) None of these
(C) 340
(D) None of these
(C) 250
(D) 380
(C) 342
(D) None of these
COMPANY SPECIFIC QUESTIONS
1. 75 persons Major in Physics, 83 major in Chemistry, 10 not a major in these subjects. Find number of students majoring in both the subjects. [INFOSYS] (A) 60 (B) 68 (C) 63 (D) None of these 2. In a class 80% have passed English, 70% passed Hindi 10% did not passed either. If 144 students passed both, what is the total strength of the class? [WIPRO] (A) 240
(B) 255
(C) 235
(D) None of these
50
3. In town of 500 people, 285 read Hindu and 212 read Indian Express and 127 read Times of India, 20 read Hindu and Times of India and 29 read Hindu and Indian express and 35 read Times of India and Indian express. 50 read no newspaper. Then how many read only one paper? [COGNIZANT] (A) 60 4.
(B) 55
(C) 45
(D) None of these
In a group of persons traveling in a bus, 6 persons can speak Tamil, 15 can speak Hindi and 6 can speak Gujarati. In that group, none can speak any other language. If 2 persons in the group can speak two languages and one person can speak all the three languages, then how many persons are there in the group? [COGNIZANT] (A) 21
(B) 23
(C) 22
(D) 24
5. Out of a total of 120 musicians in a club, 5% can play all the three instruments- Guitar, violin and Flute. It so happens that the number of musicians who can play any two and only two of the above instruments is 30. The number of musicians who can play the guitar alone is 40. What is the total number of those who can play violin alone or flute alone? [COGNIZANT] (A) 30
(B) 38
(C) 44
(D) 45
6. In a town 65% people watched the news on television, 40% read a newspaper and 25% read a newspaper and watched the news on television also. What percent of the people neither watched the news on television nor read a newspaper ? [COGNIZANT] (A) 5 (B) 10 (C) 15 (D) 20 7. There are 76 persons. 53 can read the Hindu, 46 can read the Times, 39 can read the Deccan and 15 can read all. If 22 can read the Hindu and Deccan and 23 can read Deccan and Times then what is the number of persons who read only the Times and Hindu? [ACCENTURE] (A) 18
(B) 20 (C) 23
(D)16
8. A survey made among 280 college students highlighted the following facts: 1) 50 students are from the village, and they come by government bus and takes canteen lunch. 2) 110 students are from the village and takes canteen lunch. 51
3) 160 students are from the village. 4) 90 students come by government bus and takes canteen lunch. 5) 130 students come by government bus. 6) 30 students are from the village, and they come by government bus but do not take canteen lunch. 7) 50 students are not from the village, do not come by government bus and do not take canteen lunch. Find the number of students who take canteen lunch? [TCS] (A) 220
(B) 120
(C) 170
(D) None of these
9. In a school competition, 24 students participated in a dance, 11 participated in drama, 25 participated in vocal solo song, 7 participated in both dance and drama, 4 participated in both drama and vocal solo song, 12 participated in dance and vocal solo song and 3 students participated in all the three. If totally there were 50 students, then the number of students who participated in none of the three competition is: [TCS] (A) 10
(B) 20
(C) 15
(D) 25
10. A group of 50 students were required to clear 2 tasks, one in rock-climbing and the other in bridge crossing during an adventure sports expedition. 30 students cleared both the tasks. 37 cleared bridge crossing, 38 students cleared rock climbing. How many students could not clear any task? [ACCENTURE] (A) 0 (B) 3 (C) 5 (D) 9
LEVEL 1 1C
2A
3B
4C
5A
6C
7A
8C
9B
10 B
11 B
12 B
13 C
14 B
15 B
16 A
17 B
18 A
19 B
20 D
5C
6D
7A
8C
9A
10 C
COMPANY SPECIFIC QUESTIONS: 1B
2A
3C
4B
52