Day of the Date Here are some shortcuts/tips to find out the day of the week from the given date. You can play these trick as instructed, with your parents or friends and prove your talent to them. Day of the Week: January has 31 days. It means that every date in February will be 3 days later than the same date in January(28 is 4 weeks exactly). The below table is calculated in such a way. Remember this table which will help you to calculate. January 0 February 3 March 3 April 6 May 1 June 4 July 6 August 2 Septemb 5 er October 0 Novembe 3 r Decembe 5 r Step1: Ask for the Date. Ex: 23rd June 1986 Step2: Number of the month on the list, June is 4. Step3: Take the date of the month, that is 23 Step4: Take the last 2 digits of the year, that is 86. Step5: Find out the number of leap years. Divide the last 2 digits of the year by 4, 86 divide by 4 is 21. Step6: Now add all the 4 numbers: 4 + 23 + 86 + 21 = 134. Step7: Divide 134 by 7 = 19 remainder 1. The reminder tells you the day. Sunday 0 Monday 1 Tuesday 2 Wednesda 3 y Thursday 4 Friday 5 Saturday 6 1
Answer: Monday
Beauty of Mathematics Sequential Inputs of numbers with 8 1x8+1=9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432 123456789 x 8 + 9 = 987654321
Sequential 1's with 9 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 1111 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111 123456789 x 9 + 10 = 1111111111
Sequential 8's with 9 2
9 x 9 + 7 = 88 98 x 9 + 6 = 888 987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 + 1 = 88888888 98765432 x 9 + 0 = 888888888
Numeric Palindrome with 1's 1x1=1 11 x 11 = 121 111 x 111 = 12321 1111 x 1111 = 1234321 11111 x 11111 = 123454321 111111 x 111111 = 12345654321 1111111 x 1111111 = 1234567654321 11111111 x 11111111 = 123456787654321 111111111 x 111111111 = 12345678987654321
Without 8 12345679 x 9 = 111111111 12345679 x 18 = 222222222 12345679 x 27 = 333333333 12345679 x 36 = 444444444 12345679 x 45 = 555555555 3
12345679 x 54 = 666666666 12345679 x 63 = 777777777 12345679 x 72 = 888888888 12345679 x 81 = 999999999
Sequential Inputs of 9 9 x 9 = 81 99 x 99 = 9801 999 x 999 = 998001 9999 x 9999 = 99980001 99999 x 99999 = 9999800001 999999 x 999999 = 999998000001 9999999 x 9999999 = 99999980000001 99999999 x 99999999 = 9999999800000001 999999999 x 999999999 = 999999998000000001 ......................................
Sequential Inputs of 6 6 x 7 = 42 66 x 67 = 4422 666 x 667 = 444222 6666 x 6667 = 44442222 66666 x 66667 = 4444422222 666666 x 666667 = 444444222222 6666666 x 6666667 = 44444442222222 66666666 x 66666667 = 4444444422222222 4
666666666 x 666666667 = 444444444222222222 ......................................
Amazing Prime Numbers Here are few amazing prime numbers, these prime numbers were proved by the XVIIIth century. 31 331 3331 33331 333331 3333331 33333331 The next number 333333331 is not a prime number. Whereas it is multiplied by 17 x 19607843 = 333333331.
2519 Mod n means the reminder of 2519/n, here / is the integer division. 2519 Mod n 2519 Mod 2 = 1 2519 Mod 3 = 2 2519 Mod 4 = 3 2519 Mod 5 = 4 2519 Mod 6 = 5 2519 Mod 7 = 6 2519 Mod 8 = 7 2519 Mod 9 = 8 2519 Mod 10 = 9 5
Sequential Numbers with 2519 1259 x 2 + 1 = 2519 839 x 3 + 2 = 2519 629 x 4 + 3 = 2519 503 x 5 + 4 = 2519 419 x 6 + 5 = 2519 359 x 7 + 6 = 2519 314 x 8 + 7 = 2519 279 x 9 + 8 = 2519 251 x 10 + 9 = 2519
Names for Powers of 10 Values 100 101 102 103 104 106 109 1012 1015 1018 1021 1024 1027 1030 1033 1036 1039 1042 1045 1048 1051 1054 1057 1060 1063 1066 1069
Zero's 0 1 2 3 4 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69
Names One Ten Hundred Thousand Myriad Million Billion Trillion Quadrillion Quintillion Sextillion Septillion Octillion Nonillion Decillion Undecillion Duodecillion Tredecillion Quattuordecillion Quindecillion Sexdecillion Septdecillion / Septendecillion Octodecillion Nondecillion / Novemdecillion Vigintillion Unvigintillion Duovigintillion 6
1072 1075 1078 1081 1084 1087 1090 1093 1096 1099 10100 10102 10120 10123 10138 10153 10180 10183 10213 10240 10243 10261 10273 10300 10303 10309 10312 10351 10366 10402 10603 10624 10903 102421 103003 103000003
72 75 78 81 84 87 90 93 96 99 100 102 120 123 138 153 180 183 213 240 243 261 273 300 303 309 312 351 366 402 603 624 903 2421 3003 3000003
Trevigintillion Quattuorvigintillion Quinvigintillion Sexvigintillion Septenvigintillion Octovigintillion Novemvigintillionn Trigintillion Untrigintillion Duotrigintillion Googol Trestrigintillion Novemtrigintillion Quadragintillion Quinto-Quadragintillion Quinquagintillion Novemquinquagintillion Sexagintillion Septuagintillion Novemseptuagintillion Octogintillion Sexoctogintillion Nonagintillion Novemnonagintillion Centillion Duocentillion Trescentillion Centumsedecillion Primo-Vigesimo-Centillion Trestrigintacentillion Ducentillion Septenducentillion Trecentillion Sexoctingentillion Millillion Milli-Millillion
Age Calculation Here you can find the age using some tricks. You can play these trick as instructed, with your parents or friends and prove your talent to them. 7
Age Calculation Age Calculation Tricks: Step1: Multiply the first number of the age by 5. (If <10, ex 5, consider it as 05. If it is >100, ex: 102, then take 10 as the first digit, 2 as the second one.) Step2: Add 3 to the result. Step3: Double the answer. Step4: Add the second digit of the number with the result. Step5: Subtract 6 from it.
Answer: That is your age.
Math Magic/Tricks Here we have mentioned few math trick play. You can play these tricks as instructed, with your parents or friends and prove your talent to them.
Math Magic/Tricks Trick 1: Number below 10 Step1: Think of a number below 10. Step2: Double the number you have thought. Step3: Add 6 with the getting result. Step4: Half the answer, that is divide it by 2. Step5: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought.
Answer: 3 Trick 2: Any Number Step1: Think of any number. Step2: Subtract the number you have thought with 1. Step3: Multiply the result with 3. Step4: Add 12 with the result. Step5: Divide the answer by 3. Step6: Add 5 with the answer. Step7: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought.
Answer: 8 Trick 3: Any Number Step1: Think of any number. Step2: Multiply the number you have thought with 3. Step3: Add 45 with the result. 8
Step4: Double the result. Step5: Divide the answer by 6. Step6: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought.
Answer: 15 Trick 4: Same 3 Digit Number Step1: Think of any 3 digit number, but each of the digits must be the same as. Ex: 333, 666. Step2: Add up the digits. Step3: Divide the 3 digit number with the digits added up.
Answer: 37 Trick 5: 2 Single Digit Numbers Step1: Think of 2 single digit numbers. Step2: Take any one of the number among them and double it. Step3: Add 5 with the result. Step4: Multiply the result with 5. Step5: Add the second number to the answer. Step6: Subtract the answer with 4. Step7: Subtract the answer again with 21. Answer: 2 Single Digit Numbers. Trick 6: 1, 2, 4, 5, 7, 8 Step1: Choose a number from 1 to 6. Step2: Multiply the number with 9. Step3: Multiply the result with 111. Step4: Multiply the result by 1001. Step5: Divide the answer by 7. Answer: All the above numbers will be present. Trick 7: 1089 Step1: Think of a 3 digit number. Step2: Arrange the number in descending order. Step3: Reverse the number and subtract it with the result. Step4: Remember it and reverse the answer mentally. Step5: Add it with the result, you have got.
Answer: 1089 Trick 8: x7x11x13 9
Step1: Think of a 3 digit number. Step2: Multiply it with x7x11x13. Ex: Number: 456, Answer: 456456 Trick 9: x3x7x13x37 Step1: Think of a 2 digit number. Step2: Multiply it with x3x7x13x37. Ex: Number: 45, Answer: 454545 Trick 10: 9091 Step1:Think of a 5 digit number. Step2:Multiply it with 11. Step3:Multiply it with 9091. Ex: Number: 12345,Answer:1234512345
Math Magic / Number fun / Maths Tricks This ia a simple math magic. Here involved few maths tricks to play. You can play these number tricks as instructed, with your parents or friends and prove your talent to them. Have fun with maths.
Math Magic / Number fun / Maths Tricks Trick 1: 2's trick Step1: Think of a number . Step2: Multiply it by 3. Step3: Add 6 with the getting result. Step4: divide it by 3. Step5: Subtract it from the first number used.
Answer:2 Trick 2: Any Number Step1: Think of any number. Step2: Double the number. Step3: Add 9 with result. Step4: sub 3 with the result. Step5: Divide the result by 2. 10
Step6: Subtract the number with the number with first number started with.
Answer: 3 Trick 3: Any three digit Number Step1:Add 7 to it. Step2:Multiply the number with 2. Step3:subtract 4 with the result. Step4: Divide the result by 2. Step5:Subtract it from the number started with.
Answer: 5
Math Magic / Phone Number / Missing Digit / Funny Tricks Trick 1: Phone Number trick Step1: Grab a calculator (You wont be able to do this one in your head) . Step2: Key in the first three digits of your phone number (NOT the area code-if your number is 01-123-4567, the 1st 3 digits are 123). Step3: Multiply by 80. Step4: Add 1. Step5: Multiply by 250. Step6: Add the last 3 digits of your phone number with a 0 at the end as one number step7: Repeat step 6 step8: Subtract 250 step9: Divide number by 20 Answer: The 3 digits of your phone number Funny Trick That results on your exact phone number!!
Trick 2: Missing digit Trick Step1: Step2: Step3: Step4: Step5:
Choose a large number of six or seven digits. Take the sum of digits. Subtract sum of digits from any number chosen. Mix up the digits of resulting number. Add 25 to it. 11
Step6: Cross out any one digit except zero. step7: Tell the sum of the digits. Subtract the sum of the digits from 25. Answer: Inorder to find out the missing digit, subtract the sum of digits from 25. The difference is the missing digit.
Math Magic Tricks - Birthday Date Calcualtions Trick 1: Birthday magic Step1: Add 18 to your birth month. Step2: Multiply by 25. Step3: Subtract 333. Step4: Multiply by 8. step5: Subtract 554. step6: Divide by 2. step7: Add your birth date. step8: Multiply by 5. step9: Add 692. step10: Multiply by 20. step11: Add only the last two digits of your birth year. step12: Subtract 32940 to get your birthday!. Example: If the answer is 123199 means that you were born on December 31, 1999. If the answer is not right, you followed the directions incorrectly or lied about your birthday.
Math Magic / Phone Number / Missing Digit / Funny Tricks Trick 1: 2's trick Step1: Think of a number . Step2: Multiply it by 3. Step3: Add 6 with the getting result. Step4: divide it by 3. Step5: Subtract it from the first number used. Answer:2
Trick 2: Any Number Step1: Think of any number. Step2: Double the number. Step3: Add 9 with result. Step4: sub 3 with the result. Step5: Divide the result by 2. 12
Step6: Subtract the number with the number with first number started with. Answer: 3
Trick 3: Any three digit Number Step1: Add 7 to it. Step2: Multiply the number with 2. Step3: subtract 4 with the result. Step4: Divide the result by 2. Step5: Subtract it from the number started with. Answer: 5
Maths Tricks with Numbers Trick 1:Quick Square Step1: If you need to square a 2 digit number ending in 5, you can do so very easily with this trick. Step2: Mulitply the first digit by itself + 1, and put 25 on the end. Step3:That is all! Step4: Ex:25 ie(2*(2+1))&25 Answer:625 Trick 2: Multiply by 5 Step1: Take any number, then divide it by 2 Step2:If the result is whole, add a 0 at the end. Step3: If it is not, ignore the remainder and add a 5 at the end. Step4: 2682 x 5 = (2682 / 2) & 5 or 0 Step5:2682 / 2 = 1341 (whole number so add 0) Step6:Ans:13410 Step7:Let’s try another:5887 x 5,2943.5 (fractional numbers (ignore remainder, add 5) Answer: 29435 13
Trick 3: Tough Multiplication Step1: If you have a large number to multiply and one of the numbers is even, you can easily subdivide to get to the answer Step2:Ex:32 x 125, is the same as: 16 x 250 is the same as: 8 x 500 is the same as: 4 x 1000 = 4,000 Trick 4: Subtracting from 1,000 Step1:To subtract a large number from 1,000 you can use this basic rule: subtract all but the last number from 9, then subtract the last number from 10: Step2:Ex:1000-648 Step3: subtract 6 from 9 = 3 Step4: subtract 4 from 9 = 5 step5: subtract 8 from 10 = 2 Answer: 352
Finger Multiplication of 6,7,8,9,10 Time Tables | Math Tricks Place your fingers as in the below image and consider the value of fingers in each hand to be 6, 7, 8, 9 and 10 - in the order from small finger to thumb.
Finger Multiplication of Ninth (9th) Time Table | Math Tricks Place your fingers as in the below image and number the fingers from 1 to 10.
Example Consider the multiplication of 4 × 9. Step 1: Here 4 is to be multiplied by 9. 14
So, fold the 4th finger and count the ones before the folded finger. Number of fingers before the folded finger = 3 fingers Multiply the number by 10, = 3 × 10 = 30 -----> (1) (as shown in the below figure)
Step 2: Count the fingers after the folded finger = 6 -----> (2) (as shown in the below figure)
15
Step 3: Add (1) and (2), = 30 + 6 = 36 Therefore the solution to our problem 4 × 9 = 36 is easily found using the above trick. Note: If there is no finger below/above the considered (folded) finger, then consider the value as zero (0). Play the same for the below multiplications 1×9=? 2×9=? 3×9=? 4×9=? 5×9=? 6×9=? 7×9=? 8×9=? 9×9=? 10 × 9 = ?
16
Example Consider the multiplication of 7 × 8. Make the finger numbered 7 in the left hand to touch the finger numbered 8 in the right hand.
Step 1: Now in the left hand, count the finger which is touching (7) and the ones below that = 2 fingers Similarly in the right hand, count the finger which is touching (8) and the ones below that = 3 fingers Add the above counted fingers = 2 + 3 = 5 fingers Multiply the number by 10 = 5 × 10 = 50 -----> (1) Step 2: 17
In the left hand, count the fingers above the touching finger = 3 fingers Similarly in the right hand, count the fingers above the touching finger = 2 fingers Multiply both = 3 × 2 = 6 -----> (2) Step 3: Add (1) and (2), = 50 + 6 = 56 So, the answer for 7 × 8 = 56 which is easily found through the above trick. Note: If there is no finger above the considered (touched) finger, then consider the value as zero (0). Play the same for the below multiplications 6×6 =?
6×7 =?
6×8 =?
6×9 =?
6 × 10 =?
7×6 =?
7×7 =?
7×8 =?
7×9 =?
7 × 10 =?
8×6 =?
8×7 =?
8×8 =?
8×9 =?
8 × 10 =?
9×6 =?
9×7 =?
9×8 =?
9×9 =?
9 × 10 =?
10 × 6 =?
10 × 7 =?
10 × 8 =?
10 × 9 =?
10 × 10 =?
Adding Two Digit Numbers Trick Let us begin with the basic method of adding two numbers Take the example of 71 + 69
This is the usual method which we do with the help of pen and paper Are you interested to do these calculations mentally? 18
If your answer is yes,then read below to learn the trick.
Trick: In addition, adding 10 to a two digit number is easy as we just need to increase the digit in the tenth place by 1. 12 + 10 is 22,34 + 10 is 44 and so on. We use the same rule while adding any two numbers.To be clear, see how 71 and 69 is added
Method 1: Step 1: 71 + 69 can be written as 71 + 70 -1 In this step, 69 is split as 70 and 1
Step 2: Addition of 71 + 70 is easy which gives 141 ie 71 + 70 = 141 Now subtract 1 from 141 ie 141 - 1 = 140.
This gives the answer i.e. 71 + 69 = 140
Method 2: Alternatively, the below method can also be used.The choice of method depends on the numbers that are added
Step 1: 71 + 69 can be written as 71 + 60 + 9 In this step, 69 is split as 60 and 9
19
Step 2: Addition of 71 + 60 is easy which gives 131 ie 71 + 60 = 131 Now add 9 to 131 ie 131 + 9 = 140.
This gives the answer i.e. 71 + 69 = 140 When you practice this, you can increase the speed and improve your addition skills.Try some practice questions
Practice Questions 1.19+26=? 2.31+39=? 3.14+45=? 4.11+79=?
Multiplying Two-digit Numbers By 11 Example 1: Let us take the example of 12 * 11
Step 1: In the first step, let us focus on 12. Split the number 12 like this,1-------2
Step 2: Now add the two digits of 12 That is1+2=3
Step 3: 20
Place 3 in the middle of the two digits of 12 and the result is 132 Now, we have arrived at the answer for 12*11=132.
Example 2: Now let us take another example.65 * 11
Step 1: As mentioned above in the first step, focus on 65. Split the number 65 like this,6-------5
Step 2: Now add the two digits of 65 That is6+5=11
Step 3: However, we cannot place 11 in the middle of the two digits of 65.Write 1 in between 6 and 5. Carry over the remaining 1 to the previous number 6 as given below.
Add the 1 with 6 to get the digit in the hundredth place. We have arrived at the answer for 65*11=715.
Angel Number 421 | Math Tricks Rules 21
Step 1 Step 2 Step 3
Select any whole number. If it is an even number, divide by 2; if it is odd number multiply by 3 and add 1. Repeat the process mentioned in step 2 until you get the loop value 4, 2, 1 in repetition.
Example
Whole number is 15. 15 is an odd no; so (15 × 3) + 1 = 46 46 is an even no; so 46 / 2 = 23 23 is an odd no; so (23 × 3) + 1 = 70 70 is an even no; so 70 / 2 = 35 35 is an odd no; so (35 × 3) + 1 = 106 106 is an even no; so 106 / 2 = 53 53 is an odd no; so (53 × 3) + 1 = 160 160 is an even no; so 160 / 2 = 80 80 is an even no; so 80 / 2 = 40 40 is an even no; so 40 / 2 = 20 20 is an even no; so 20 / 2 = 10 10 is an even no; so 10 / 2 = 5 5 is an odd no; so (5 × 3) + 1 = 16 16 is an even no; so 16 / 2 = 8 8 is an even no; so 8 / 2 = 4 4 is an even no; so 4 / 2 = 2 2 is an even no; so 2 / 2 = 1 1 is an odd no; so (1 × 3) + 1 = 4 4 is an even no; so 4 / 2 = 2 2 is an even no; so 2 / 2 = 1 So the loop 4..2..1 goes on and on.
Tip : The angel number 421 is the smallest prime formed by the two powers in logical order from right to left.
Kaprekar Number 6174 Theory Rules Step 1 Step 2 Step 3 Step 4 Step 5
Select any four digit number, but do not select the ones in which all the four digits are the same like 1111, 2222, ... Arrange the digits in decreasing order. Arrange the digits in increasing order. Subtract the smaller number from the larger number. Repeat the steps 2, 3 and 4 until you get 6174 in repetition.
Example
22
Step : Step : Step : Step :
1 ==> 4 digit number is 5620. 2 ==> Arrange decreasing order : 6520 3 ==> Arrange increasing order : 0256 4 ==> Subtract the smaller number from the larger number : 6520 - 0256 = 6264 Step : 5 ==> Take the final result and repeat the steps 2, 3, and 4 until you get the repetition of 6174. Like,
6264 = 6642 - 2466 = 4176 4176 = 7641 - 1467 = 6174 6174 = 7641 - 1467 = 6174
Have fun with this math trick and discover why all the 4 digit numbers on subtracting using the kaprekar operation reach to the mysterious number 6174.
Long Multiplication reduced to one-line shortcut The Sanskrit sutra "Urdhva Tiryagbhyam", meaning "Vertically and Crosswise", gives a general technique for reducing long multiplication to a single-line shortcut. It works like this. To multiply two numbers (of two or more digits), split each number into two parts. If the first number is a1 + b1 and the second number is a2 + b2, then the product of the two numbers is: (a1 x a2) + (a1 x b2 + b1 x a2) + (b1 x b2)
The solution comprises three parts (as shown by the boxes and arrows above): the head, the middle, and the tail. 1. The digits on the right are multiplied vertically to get the tail part: b1 x b2 (excess carried over) 23
2. All digits are multipled crosswise and added together to get the middle part: a1 x b2 + b1 x a2 (excess carried over) 3. The digits on the left are multiplied vertically to get the head part: a1 x a2
Here is a simple example to illustrate this technique. 23 x 41 = 943
The steps are: 1. 3 x 1 = 3 2. 2 x 1 + 3 x 4 = 14, put down 4 and carry over 1 3. 2 x 4 = 8, plus the 1 carried over, is 9
The speed gain using this technique (over the conventional method of multiline long multiplication) becomes more apparent when handling larger numbers. Here is another example involving excess carryover at each stage. 108 x 64 = 6912
24
The steps are: 1. 8 x 4 = 32, put down 2 and carry over 3 2. 10 x 4 + 8 x 6 = 88, plus the 3 carried over, is 91; put down 1 and carry over 9 3. 10 x 6 = 60, plus the 9 carried over, is 69
This powerful technique can be expanded upon to cover all cases of multiplication, not just two or three-digit numbers.
Long Division reduced to one-line shortcut The Sanskrit sutras offer several techniques for division. Some of them work spectacularly to reduce to a single line certain cases of division that may take dozens of lines of calculation in the conventional method of long division taught in schools. While there are special techniques reserved for special cases, we highlight here a general-purpose long division technique that will work for both small and large numbers. It is basically applying in reverse the Urdhva Tiryagbhyam (Vertically and Crosswise) principle used to simplify multiplication. Such is the power and elegance of this technique that we can easily take a large number and make short work of it by any divisor of two or more digits. Take for example, 716769 ÷ 54. Yes, you too CAN work it out manually -- and in one line -- without having to reach for the calculator! You will instead rely on the calculator inside your head and the Dhvajanka sutra, meaning "on top of the flag". The trick is to reduce the divisor to a mentally manageable value by putting its other digits "on top of the flag". In this example, the divisor will be reduced to 5 (instead of 54) by pushing the 4 up the flagpost, as shown below. Corresponding to the number of digits flagged on top (in this case, one), the rightmost part of the number to be divided is split to mark the placeholder of the decimal point or the remainder portion. 25
Now observe carefully as we walk through the steps of this example: 716769 ÷ 54 = 13273.5
1. 7 ÷ 5 = 1 remainder 2. Put the quotient 1, the first digit of the solution, in the first box of the bottom row and carry over the remainder 2 2. The product of the flagged number (4) and the previous quotient (1) must be subtracted from the next number (21) before the division can proceed. 21 - 4 x 1 = 17 17 ÷ 5 = 3 remainder 2. Put down the 3 and carry over the 2 3. Again subtract the product of the flagged number (4) and the previous quotient (3), 26 - 4 x 3 = 14 14 ÷ 5 = 2 remainder 4. Put down the 2 and carry over the 4 4. 47 - 4 x 2 = 39 39 ÷ 5 = 7 remainder 4. Put down the 7 and carry over the 4 5. 46 - 4 x 7 = 18 18 ÷ 5 = 3 remainder 3. Put down the 3 and carry over the 3 6. 39 - 4 x 3 = 27. Since the decimal point is reached here, 27 is the raw remainder. If decimal places are required, the division can proceed as before, filling the original number with zeros after the decimal point 27 ÷ 5 = 5 remainder 2. Put down the 5 (after the decimal point) and carry over the 2 7. 20 - 4 x 5 = 0. There is nothing left to divide, so this cleanly completes the division
While this example is easily solved, there are finer details in the application of technique that will be highlighted in subsequent examples later.
Squaring First, a nifty shortcut! The square of a number ending in 5 is almost a nobrainer. 26
If n is the number formed by the preceding digit/s (before the 5), get the product of n and n+1. Then just append 25 (i.e. 5 x 5) to this product. For example, 752: 7 x 8 = 56; therefore solution is 5625. Another example, 1152: 11 x 12 = 132; therefore solution is 13225 For other cases of squaring, the same shortcut techniques used in multiplication may be utilised. Especially the general-purpose Urdhva Tiryagbhyam (Vertically and Crosswise) formula. To get the square of a number (of two or more digits), simplify by splitting it into at least two parts, a and b. Thus (a + b)2 = a2 + 2ab + b2
The solution comprises three parts, neatly fitting the three boxes shown above. Just adjust for excess carry over. 1. the head: a2 2. the middle: crosswise multiplication and doubling a x b x 2 3. the tail: b2
Here is a simple example to illustrate this technique. 232 = 529
27
The steps are: 1. tail: 32 = 9, put it down in the rightmost box 2. middle: 2 x 3 x 2 = 12, put down the 2 in the middle box and carry over the 1 3. head: 22 = 4, plus the 1 carried over, is 5 in the left box
Another example. 1082 = 11664
The steps are: 1. tail: 82 = 64, put down the 4 and carry over the 6 2. middle: 10 x 8 x 2 = 160, plus the 6 carried over, is 166; put down the 6 and carry over the 16 3. head: 10 x 10 = 100, plus the 16 carried over, is 116
The same technique can be expanded upon to handle the squaring of bigger numbers too.
Cubing Extrapolating on the principles used for squaring, to get the cube of a number (of two or more digits), simplify by splitting the number into two parts, a and b. Thus (a + b)3 = a3 + 3a2b + 3ab2 + b3
The solution comprises four parts, neatly fitting the four boxes shown above. Just adjust for excess carry over. 1. 2. 3. 4.
a3 a2 x b x 3 b2 x a x 3 b3 28
A simple example to illustrate this technique. 233 = 12167
The steps are: 1. 33 = 27, put down the 7 in the rightmost box and carry over the 2 2. 32 x 2 x 3 = 54, plus the 2 carried over is 56, put down the 6 and carry over the 5 3. 22 x 3 x 3 = 36, plus the 5 carried over is 41, put down the 1 and carry over the 4 4. 23 = 8, plus the 4 carried over, is 12 in the leftmost box
Another example. 1083 = 1259712
The steps are: 1. 83 = 512, put down the 2 in the rightmost box and carry over the 51 2. 82 x 10 x 3 = 1920, plus the 51 carried over is 1971, put down the 1 and carry over the 197 3. 102 x 8 x 3 = 2400, plus the 197 carried over is 2597, put down the 7 and carry over the 259 4. 103 = 1000, plus the 259 carried over, is 1259 in the leftmost box
That's a million-dollar figure worked out manually!
To the fourth power Extrapolating on the principles used for cubing, to get the fourth power of a number (of two or more digits), simplify by splitting the number into two parts, a and b. 29
Thus (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
The solution comprises five parts, neatly fitting the five boxes shown above. Just adjust for excess carry over. 1. 2. 3. 4. 5.
a4 a3 x b x 4 a2 x b2 x 6 b3 x a x 4 b4
A simple example to illustrate this technique. 234 = 279841
The steps are: 1. 34 = 81, put down the 1 in the rightmost box and carry over the 8 2. 33 x 2 x 4 = 216, plus the 8 carried over is 224, put down the 4 and carry over the 22 3. 22 x 32 x 6 = 216, plus the 22 carried over is 238, put down the 8 and carry over the 23 4. 23 x 3 x 4 = 96, plus the 23 carried over is 119, put down the 9 and carry over the 11 5. 24 = 16, plus the 11 carried over, is 27 in the leftmost box
Another example. 1084 = 136048896
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The steps are: 1. 84 = 4096, put down the 6 in the rightmost box and carry over the 409 2. 83 x 10 x 4 = 20480, plus the 409 carried over is 20889, put down the 9 and carry over the 2088 3. 102 x 82 x 6 = 38400, plus the 2088 carried over is 40488, put down the 8 and carry over the 4048 4. 103 x 8 x 4 = 32000, plus the 4048 carried over is 36048, put down the 8 and carry over the 3604 5. 104 = 10000, plus the 3604 carried over, is 13604 in the leftmost box
That's a hundred million-dollar figure worked out manually! . The 11 Times Trick We all know the trick when multiplying by ten – add 0 to the end of the number, but did you know there is an equally easy trick for multiplying a two digit number by 11? This is it: Take the original number and imagine a space between the two digits (in this example we will use 52: 5_2 Now add the two numbers together and put them in the middle: 5_(5+2)_2 That is it – you have the answer: 572. If the numbers in the middle add up to a 2 digit number, just insert the second number and add 1 to the first: 9_(9+9)_9 (9+1)_8_9 10_8_9 1089 – It works every time. 2. Quick Square If you need to square a 2 digit number ending in 5, you can do so very easily with this trick. Mulitply the first digit by itself + 1, and put 25 on the end. That is all! 31
252 = (2x(2+1)) & 25 2x3=6 625 3. Multiply by 5 Most people memorize the 5 times tables very easily, but when you get in to larger numbers it gets more complex – or does it? This trick is super easy. Take any number, then divide it by 2 (in other words, halve the number). If the result is whole, add a 0 at the end. If it is not, ignore the remainder and add a 5 at the end. It works everytime: 2682 x 5 = (2682 / 2) & 5 or 0 2682 / 2 = 1341 (whole number so add 0) 13410 Let’s try another: 5887 x 5 2943.5 (fractional number (ignore remainder, add 5) 29435
4. Multiply by 9 32
This one is simple – to multiple any number between 1 and 9 by 9 hold both hands in front of your face – drop the finger that corresponds to the number you are multiplying (for example 9×3 – drop your third finger) – count the fingers before the dropped finger (in the case of 9×3 it is 2) then count the numbers after (in this case 7) – the answer is 27. 5. Multiply by 4 This is a very simple trick which may appear obvious to some, but to others it is not. The trick is to simply multiply by two, then multiply by two again: 58 x 4 = (58 x 2) + (58 x 2) = (116) + (116) = 232
There are many useful Math techniques, tricks, and secrets that can be valuable in your day to day work or study. Knowing these shortcuts are key to computation speed and accuracy in your Mental Mathematical Aptitude. This can be useful in applications such as IQ tests, aptitude tests, job application tests, military tests, college entrance tests and so many more uses in the office, workplace, or in your own home.
MULTIPLICATION: Multiplication using multiples 12 x 15 = 12 x 5 x 3 = 60 x 3 = 180 Multiplication by distribution 12 x 17 = (12 x 10) + (12 x 7) ---> 12 is multiplied to both 10 & 7 = 120 + 84 = 204 Multiplication by "giving and taking" 12 x 47 = 12 x (50 - 3) = (12 x 50) - (12 x 3) = 600 - 36 = 564 33
Multiplication by 5 --> take the half(0.5) then multiply by 10 428 x 5 = (428 x 1/2) x 10 = 428 x 0.5 x 10 = 214 x 10 = 2140 Multiplication by 10 ---> just move the decimal point one place to the right 14 x 10 = 140 ---> added one zero Multiplication by 50 ---> take the half(0.5) then multiply by 100 18 x 50 = (18/2) x 100 = 18 x 0.5 x 100 = 9 x 100 = 900 Multiplication by 100 ---> move the decimal point two places to the right 42 x 100 = 4200 ---> added two zeroes Multiplication by 500 ---> take the half(0.5) then multiply by 1000 21 x 500 = 21/2 x 1000 = 10.5 x 1000 = 10500 Multiplication by 25 ---> use the analogy $1 = 4 x 25 cents 25 x 14 = (25 x 10) + (25 x 4) ---> 250 + 100 ---> $2.50 + $1 = 350 Multiplication by 25 ---> divide by 4 then multiply by 100 36 x 25 = (36/4) x 100 = 9 x 100 = 900 Multiplication by 11 if sum of digits is less than 10 72 x 11 = 7_2 ---> the middle term = 7 + 2 = 9 = place the middle term 9 between 7 & 2 = 792 34
Multiplication by 11 if sum of digits is greater than 10 87 x 11 = 8_7 ---> the middle term = 8 + 7 = 15 because the middle term is greater than 10, use 5 then add 1 to the first term 8, which leads to the answer of = 957 Multiplication of 37 by the 3, 6, 9 until 27 series of numbers --> the "triple effect" solve 37 x 3 multiply 7 by 3 = 21, knowing the last digit (1), just combine two more 1's giving the triple digit answer 111 solve 37 x 9 multiply 7 by 9 = 63, knowing the last digit (3), just combine two more 3's giving the triple digit answer 333 solve 37 x 21 multiply 7 by 21 = 147, knowing the last digit (7), just combine two more 7's giving the triple digit answer 777 Multiplication of the "dozen teens" group of numbers -(i.e. 12, 13, 14, 15, 16, 17, 18, 19) by ANY of the numbers within the group: solve 14 x 17 4 x 7 = 28; remember 8, carry 2 14 + 7 = 21 add 21 to whats is carried (2) giving the result 23 form the answer by combinig 23 to what is remembered (8) giving the answer 238 Multiplication of numbers ending in 5 with difference of 10 45 x 35 first term = [(4 + 1) x 3] = 15; (4 is the first digit of 45 and 3 is the first digit of 35 --> add 1 to the higher first digit which is 4 in this case, then multiply the result by 3, giving 15) last term = 75 combining the first term and last term, = 1575 75 x 85 first term = (8 + 1) x 7 = 63 last term = 75 combining first and last terms, = 6375 35
15 x 25 = 375 Multiplication of numbers ending in 5 with the same first terms (square of a number) 25 x 25 first term = (2 + 1) x 2 = 6 last term = 25 answer = 625 ---> square of 25 75 x 75 first term = (7 + 1) x 7 = 56 last term = 25 answer = 5625 ---> 75 squared
DIVISION: Division by parts ---> imagine dividing $874 between two persons 874/2 = 800/2 + 74/2 = 400 + 37 = 437 Division using the factors of the divisor: "double division" 70/14 = (70/7)/2 ---> 7 and 2 are the factors of 14 = 10/2 =5 Division by using fractions: 132/2 = (100/2 + 32/2) ---> break down into two fractions = (50 + 16) = 66 Division by 5 ---> divide by 100 then multiply by 20 1400/5 = (1400/100) x 20 = 14 x 20 = 280 Division by 10 ---> move the decimal point one place to the left 36
0.5/10 = 0.05 ---> 5% is 50% divided by ten Division by 50 ---> divide by 100 then multiply by 2 2100/50 = (2100/100) x 2 = 21 x 2 = 42 700/50 = (700/100) x 2 =7x2 = 14 Division by 100 ---> move the decimal point two places to the left 25/100 = 0.25 Division by 500 ---> divide by 100 then multiply by 0.2 17/500 = (17/100) x 0.2 = 0.17 x 0.2 = 0.034 Division by 25 ---> divide by 100 then multiply by 4 500/25 = (500/100) x 4 =5x4 = 20 750/25 = (750/100) x 4 = 7.5 x 2 x 2 = 30
ADDITION: Addition of numbers close to multiples of ten (e.g. 19, 29, 89, 99 etc.) 116 + 39 = 116 + (40 - 1) = 116 + 40 - 1 ---> add 40 then subtract 1 = 156 - 1 = 155 37
116 + 97 = 116 + (100 - 3) = 116 + 100 - 3 ---> add 100 then subtract 3 = 216 - 3 = 213 Addition of decimals 12.5 + 6.25 = (12 + 0.5) + (6 + 0.25) = 12 + 6 + 0.5 + 0.25 ---> add the integers then the decimals = 18 + 0.5 + 0.25 = 18.75
SUBTRACTION: Subtraction by numbers close to 100, 200, 300, 400, etc. 250 - 96 = 250 - (100 - 4) = 250 - 100 + 4 ---> subtract 100 then add 4 = 150 + 4 = 154 250 - 196 = 250 - (200 - 4) = 250 - 200 + 4 ---> subtract 200 then add 4 = 50 + 4 = 54 216 - 61 = 216 - (100 - 39) = 216 - 100 + 39 = 116 + (40 - 1) ---> now the operation is addition, which is much easier = 156 - 1 = 155 Subtraction of decimals 47 - 9.9 = 47 - (9 + 0.9) ---> "double subtraction" = 47 - 9 - 0.9 ---> subtract the integer first then the decimal = 38 - 0.9 = 37.1 18.3 - 0.8 38
= 18 + 0.3 - 0.8 = (18 - 0.8) + 0.3 ---> subtract 0.8 from 18 first = 17.2 + 0.3 = 17.5
WORKING ON PERCENTAGES:
30% of 120 = 10% x 3 x 120 ---> it is much easier working with tens (10%) = 10% x 120 x 3 = 12 x 3 = 36 five percent of a number: 5% 360 x 5% = 360 x 10%/2 ---> take the 10% and divide by 2 = 36/2 = 18 360 x 5% = 360 x 50%/10 ---> take the half(0.5) and divide by 10 = (360/2)/10 = 180/10 = 18 ninety percent of a number: 90% 90% of 700 = (100% - 10%) x 700 = (100% x 700) - (10% x 700) ---> 100% minus 10% of the number = 700 - 70 = 630 What is 18 as a percentage of 50? = 18/50 = (18/100) x 2 ---> method: division by 50 (explained above) = 0.18 x 2 = 0.36 = 36% What is 132 as a percentage of 200? = 132/200 = (132/2)/100 39
= [100/2 + 32/2]/100 ---> solution by "double division" = (50 + 16)/100 = 66/100 = 0.66 = 66% What is 270 as a percentage of 300? = 270/300 = [(270/3)/100] ---> "double division" (using the factors of 300) = 90/100 = 90% What is 17 as percentage of 500? = 17/500 = (17/50)/10 = (17/100) x 2/10 ---> solution using the procedure: division by 50 = (0.17 x 2)/10 = 0.34/10 = 0.034 = 3.4 % percentages close to 100: 95% of 700 = (100% - 5%) x 700 = (100% x 700) - (5% x 700) = 700 - (10% x 700/2) -------> 5% is 10%/2 = 700 - 70/2 = 700 - 35 = 665 percentages less than 10 percent: 3% of 70 = (3/100) x 70 = (70/100) x 3 ---> divide by 100 then multiply the percent value = 0.7 x 3 = 2.1
DECIMALS: To convert or express percentages as decimals, divide by 100 or simply just move the decimal point by two places to the left of the given number, thus: 1% = 1/100 = 0.01 2% = 2/100 = 0.02 = 1/50 40
3% = 3/100 = 0.03 4% = 4/100 = 0.04 = 1/25 5% = 5/100 = 0.05 = 1/20 6.25% = 6.25/100 = 0.0625 = 1/16 7% = 7/100 = 0.07 7.5% = 7.5/100 = 0.075 10% = 10/100 = 0.1 = 1/10 12.5% = 12.5/100 = 0.125 = 1/8 20% = 0.2 = 1/5 21% = 0.21 25% = 0.25 = 1/4 30% = 0.3 = 3/10 33.33% = 33.33/100 = 0.3333 = 1/3 37.5% = 0.375 = 3/8 40% = 0.4 = 2/5 50% = 0.5 = 1/2 60% = 0.6 = 3/5 62.5% = 0.625 = 5/8 66.66% = 66.66/100 = 2/3 75% = 0.75 = 3/4 80% = 0.8 = 4/5 87.5% = 0.875 = 7/8 100% = 1 125% = 1.25 = 1 1/4 150% = 1.5 = 1 1/2 200% = 2 FRACTIONS: What is three quarters of 80? = 3/4 x 80 = (80/4) x 3 ---> divide by 4 then multiply by 3 = 20 x 3 = 60 How many quarters in two and a half? 2.5/.25 = 10 ---> there are 10 quarters in $2.50 Improper fractions: 3/2 = 1 1/2 = 1.5 = 150% 4/3 = 1 1/3 = 1.3333 = 133.33% ---> useful number for volume of sphere, etc. 41
9/5 = 1 4/5 = 1.8 = 180% ---> conversion factor for Celsius/Fahrenheit temperatures V = 4/3 pi * r^3 where: V = volume of sphere r = radius of sphere F = (1.8 C) + 32 where: F = temperature in Fahrenheit C = temperature in Celsius
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