Casting Out Nines

                          Casting Out Nines

When you solve a sum whether it is addition, subtraction, multiplication or division, a doubt constantly haunts you. To avoid any mistake you begin to check every step and thus lose your precious time. Don’t you think that there should be a single method which is equally beneficial in all the basic mathematical operations? The answer to your question is YES, there is a method to check all the four fundamental mathematical operation i.e. addition, subtraction, multiplication or division and that too in few seconds. This wonder method is called Casting out Nines or Nines- Remainder Method.

How does this method work?

This is a big question. But you will be surprised to know that this method is so simple that even a primary school going kid can understand and check his/her calculation.

 Casting out Nines literally means to throw nines. Now let us focus on its working.

v  Add the digits of a number across, dropping out 9, to get a single figure. If it is not a single figure, add the digits obtained so as to get a single figure between  0 to 8.

v  9 is not taken into account in this process, as a digit sum of 9 is the same as a digit sum of zero.

Before coming to the actual problem, shouldn’t we understand the basic meaning of above two points? Let us take some examples to understand what meaning the above line has.

Example: Find the digit sum of 54653

Verification: Digit sum of 54653= 5 + 4 + 6 + 5 +3 = 23

                Since 23 is a double figure number so to get a single figure, we have to sum it again.

                 Digit sum of 23 = 2 + 3 = 5

The digit sum of 54653 can be done in other way very easily. As discussed above, we need not take 9 into account

Digit sum of 54653 = 5   +   4    +    6 +      5   +      3

The two group of number 5+ 4 and 6+ 3 can easily be left out while finding the digit sum of 54653, as their sum is equal to 9.

Example: Find the digit sum of 438219

Verification: Add all the digits

                        4 + 3 + 8 + 2 + 1 + 9 = 27

                 Digit sum of 27 = 2 + 7 = 9 

Check for Addition

Whatever we do to the numbers, we also do to their digit sum; then the result that we have from the digit sum of the numbers, must be equal to the digit sum of answer.

Example: Verify 47385 + 69384 + 58769+ 38173 + 29464 = 243144

Verification:                                                   Digit sum of number

                        47385                                                    0             

                        69384                                                    3

                        58769                                                    8

                        38173                                                    4

               +       29464                                                    3

                       243144                                               ?

LH.S. = Digit sum of 243144 = 2 + 4 + 3 + 1 + 4 + 4 = 0

R.H.S. = Sum of the Digit sum of number = 0 + 3 + 8 + 4 + 3 = 0

Since LHS = RHS

Result Verified

Example: Verify   87643 + 84397+ 38549 + 29765 = 240354

Verification                                                                    Digit sum of number

                        87643                                                                    1

                        84397                                                                    4

                        38549                                                                    2

                        29765                                                                    2

                        240354                                                                  ?

Check for subtraction

The process of subtraction is same as applied in addition. Remember that the value of the digit sum of minuend should be greater than that of subtrahend.

Example: Verify the result, 8934 – 6758 = 2176

Verification:                                                                Digit sum of number

                        8934                                                       6

                 -    6758                                                        8

                      2176                                                         ?

Since the digit sum of Minuend is less than the digit sum of Subtrahend, hence we need to replace the digit sum of 8934, in such a way that the final digit sum remains the same.

Here the value of the digit sum of 8934 = 6

This digit sum 6 can also be written in so many ways as the digit sum of 15, 24, 33, 42, 51 and 60 also gives the same value 6. Therefore, the need of the hour is to replace the digit sum of 8934 with any of the given value 15, 24, 33,42,51,60.

Digit sum of number

                        8934                                       15 or 24 or 33 or 42 or 51 or 60

                 -    6758                                                        8

                      2176                                                         ?

LHS = Digit sum of 2176 = 2 + 1 + 7+ 6 = 7

RHS = Digit sum of (15 – 8) = 7

          Or, Digit sum of (24 – 8 = 16) = 1 + 6 = 7

          Or, Digit sum of (33 – 8 = 25) = 2 + 5 = 7

          Or, Digit sum of (42 – 8 = 34) = 3 + 4 = 7

          Or, Digit sum of (51 – 8 = 43) = 4 + 3 = 7

          Or, Digit sum of (60 – 8 = 52) = 5+ 2 = 7

In all the above cases

LHS = RHS

LH.S. = Digit sum of 240354 = 2 + 4 + 0 + 3 + 5 + 4 = 0

R.H.S. = Sum of the Digit sum of number = 1 + 4 + 2 + 2 = 0

Check for Multiplication

Multiplication is the most error prone fundamental operation in mathematics. Students always have doubts about the accuracy of their result and waste time in re-checking every operation again and again. This method will prove a panacea for all those who are not very much sure about their result. Let us take few examples to understand the modus operandi of this method.

We know,

Multiplicand X Multiplier = Product

 Example: Verify 12876 x 43853 = 564651228

Verification:-          

 Digit sum of Multiplicand = 1 + 2 + 8 + 7 + 6 = 6

 Digit sum of Multipliers =4 + 3 + 8 + 5 + 3 = 5                              

LHS = Digit sum of Multiplicand and Multiplier taken together (6 x 5= 30) = 3 RHS= Digit sum of Product =5 + 6 + 4 + 6 + 5+ 1 + 2 + 2 + 8 = 3

Since LHS = RHS

Result Verified

Example: Verify 5972 X 4853 = 29882116

Verification:-          

 Digit sum of Multiplicand = 5 + 9 + 7 + 2 = 5

 Digit sum of Multipliers = 4 + 8+ 5 + 3 = 2                     

LHS = Digit sum of Multiplicand and Multiplier taken together (5 X 2= 10)= 1 RHS=  Digit sum of Product = 2 + 9 + 8 + 8 + 2 + 1 + 1 + 6 = 1

Since LHS = RHS

Result Verified

Example: Verify 12589476 x 43256853 = 54058115267908

Verification:-          

 Digit sum of Multiplicand =1 + 2 + 5 + 8 + 9 + 4+ 7 + 6 = 6

 Digit sum of Multipliers =4 + 3 +2 + 5+ 6+ 8 + 5 + 3 = 5                             

LHS = Digit sum of Multiplicand and Multiplier taken together (6 x 5= 30) = 3 RHS= Digit sum of Product = 5+ 4+ 0+ 5+ 8+1+1+ 5+ 2+ 6+ 7+ 9+ 0+ 8 = 7

Since LHS ≠ RHS

Result Incorrect

Check for Division

We Know

Dividend = Divisor x Quotient + Remainder

The casting out nines method described in the very beginning will suffice to check the division operation effectively. You have only one thing to do-

Find the digit sum of Dividend, Divisor, Quotient and Remainder and put the value of digits sum in LHS and RHS. If the same digit sum is obtained in both the side, it ultimately tells you that you have performed the right operation. Let us take some examples to understand how effectively this method work for Division.

Example: Verify 876543 ÷ 123, Q = 7126 and R = 45

Verification:-

Here Dividend = 876543

Digit sum of Dividend = 8 + 7 + 6 + 5 + 4 + 3 = 6

Divisor = 123

Digit sum of Divisor = 1 + 2 + 3 = 6

Quotient = 7126

Digit sum of quotient = 7 + 1 + 2 + 6= 7

Remainder = 45

Digit sum of Remainder = 4 + 5 = 0

Putting the digit sum value in the given formulae, we get,

L.H.S = Digit sum of Dividend = 6

R.H.S = Divisor x Quotient + Remainder

          = 6 X 7 + 0 = 42

Digit sum of 42 = 6

Hence LHS = RHS

Result Verified

Example: Verify 8765 ÷ 243, Q = 36 and R = 23

Finding percentage in your head

Finding Percentage in your head

Percentage is merely a two place decimal without the decimal point shown. Percent has its origin from the Latin word per centum meaning per hundred. It can best be defined as: - A fraction whose denominator is 100 is called a percentage and the numerator of the fraction is called the rate percent.

The importance of percentage can be estimated with the fact that whether you go to Bank for taking loan or go to shopping mall for buying something you always feel the importance of calculating percentage. Let’s take some example to understand the importance of percent in daily life.

a) If the bank offers you the interest rate of home loan @10% per annum it simply means that you have to pay ` 10 per ` 100.

b) Suppose while reading newspaper early in the morning you get an ad claiming that you will get 50% + 50% off on buying a pair of jeans and in the evening you rush to the shop in a hope you will get 100% discount on jeans then you are really thinking it other way because you have miscalculated the percentage discount the company or shop is offering you.

c) We all know that in order to get 1^st Division in School/ College examination you need to acquire 60% marks of the total.  

It is not all, in every competitive examination you appear; you encounter some problem on percentage directly or indirectly.

This altogether shows that it is an important tool for our life. Here I shall not deal with the typical problem asked in examination involved percentage which comes in the form of Profit and Loss, Simple or Compound Interest etc. but I shall simply show you the way how you can calculate the simple percentage.

This clearly shows the importance of percentage. 1 % of a number is 1/100 of the number; 15% of a number is 15/100 and so on. This reminds me a question which is crucial in primary level and is used to some extent in senior or competitive level – what percent one number is of other. Let me explain it before I move to the main discussion.

What percent one number is of other?

Follow these simple steps and get the answer in your mind.

·         Put the number that follows the word what percent is in the numerator of the fraction

·         Place the other number to the denominator of the fraction.

·         Reduce the fraction in simplest part if possible and finally multiply the result by 100.

Example; - What percent is 16 of 80?

Solution: - Place 16 in numerator and 80 at denominator and multiply the fraction by 100.

Example: What percent is 36 of 4?

Solution: - Place 36 at the numerator and 4 at the denominator and multiply the result by 100

Let’s learn some special technique to find a certain percentage of a number. These method will give you an edge to do the calculation fast.

Finding 2 ½ % of a number

Suppose you are asked to find 2 ½ % of a number you will simply convert it into a fraction and multiply the number by 5/2 and divide it by 100 but this will take undoubtedly a minute from you. Let’s learn some simple trick and do it in your mind.

1.      Divide the number whose 2 ½ % you are going to calculate by 4

2.      Move decimal point one place to left.

Hey, did you get the answer? Of course, YES. Isn’t it super simple?

Let me take some examples to put focus on its working.

Example: Find 2 ½ % of 86

Solution: -  

Divide 86 by 4; 86 ÷ 4 = 21.5                                                                                                             Move decimal point one place to left = 2.15

Hence, 86 × 2 ½ % = 2.15

Example: Find 2 ½ % of 648

Solution: -  

Divide 648 by 4; 648 ÷ 4 = 162                                                                                                          Move decimal point one place to left = 16.2

Hence, 648 × 2 ½ % = 16.2

Finding 5% of a number

Let me ask you a simple question: - Are you comfortable dividing a number by 2? Your answer is a big YES. Finding 5% of a number is as simple as dividing a number by 2. Let’s see how it works.

1.      Divide the number by 2

2.      Move the decimal point one place to left

Example: - Find 5% of 850

Solution: - Divide 850 by 2; 850 ÷ 2 = 425

                  Move decimal point one place to left = 42.5

Hence, 850 × 5% = 42.5

Example: - Find 5% of 326

Solution: - Divide 326 by 2; 326 ÷ 2 = 163

                  Move decimal point one place to left = 16.3

Hence, 326 × 5% = 16.3

Finding 10% of a number

Finding 10% of a number is a child’s play. You follow the simple steps and get the answer in second.

1.       Simply Move the decimal point one place to left

Example: - Find 10% of 729

Solution: - Move decimal point one place to left = 72.9

Hence, 729 ×10% = 72.9

Example: - Find 10% of 2549

Solution: - Move decimal point one place to left = 254.9

Hence, 2549 ×10% = 254.9

Finding 15% of a number

If I ask you to multiply a number by 2 then you will comfortably do it. If I again ask you to divide a number by 2 then also you will get your answer correct. That all you have to do in order to find 15% of a number. Follow the steps.

1.      Divide the number by 2

2.      Multiply the result obtained by 3

3.      Move the decimal point one place to left

Example: - Find 15% of 43

Solution: - Divide 43 by 2; 43 ÷ 2 = 21.5

                  Multiply it by 3 = 21.5 × 3 = 64.5

                  Move decimal point one place to left = 64.5

Hence, 43 × 15% = 64.5

Example: - Find 15% of 438

Solution: - Divide 483 by 2; 438 ÷ 2 = 219

                  Multiply it by 3 = 219 × 3 = 657

                  Move decimal point one place to left = 65.7

Hence, 438 × 15% = 65.7

Finding 20% of a number

 Follow the steps.

1.      Divide the number by 5

Example: - Find 20% of 43

Solution: - Divide 43 by 5; 43 ÷ 5 = 8.6

Hence, 43 × 20% = 8.6

Example: - Find 20% of 348

Solution: - Divide 348 by 5; 348 ÷ 5 = 69.6

Hence, 348 × 20% = 69.6

Finding 25% of a number

Follow the steps.

1.      Divide the number by 4

Example: - Find 25% of 86

Solution: - Divide 86 by 4; 86 ÷ 4 = 21.5

Hence, 86 × 25% = 21.5

Example: - Find 25% of 484

Solution: - Divide 484 by 4; 484 ÷ 5 = 121

Hence, 484 × 25% =121

Finding 33 ¹/₃ % of a number

Follow the steps.

1.      Divide the number by 3

Example: - Find 33 ¹/₃% of 69

Solution: - Divide 69 by 3; 69 ÷ 3 = 23

Hence, 69 × 33 ¹/₃ % = 23

Example: - Find 33 ¹/₃ % of 921

Solution: - Divide 921 by 3; 921 ÷ 3 = 307

Hence, 921 × 33 ¹/₃ % = 307

Finding 40 % of a number

Follow the steps.

1.      Multiply the number by 4

2.      Move decimal one point left

Example: - Find 40% of 24

Solution: - Multiply the number by 4:   24×4 = 96

                  Move decimal one point left = 9.6

Hence, 24 × 40 % = 9.6

Example: - Find 40% of 49

Solution: - Multiply the number by 4:   49×4 = 196

                  Move decimal one point left = 19.6

Hence, 49 × 40 % = 19.6

Finding 45% of a number

Multiplying a number by 9 is very easy. Leave the unit digit apart and subtract 1 more than the remaining digits from the original digit and place it in the LHS, in the RHS place the complement of unit digit.

Example: - 112 x 9 =?

Solution: - 112 – (11+ 1) / Complement of unit digit 2 = 1008

In order to find 45% of a number follow the steps.

1.      Divide the number by 2

2.      Multiply the result obtained by 9

3.      Move the decimal point one place to left

Example: - Find 45% of 36

Solution: - Divide 36 by 2; 36 ÷ 2 = 18

                  Multiply it by 9 = 18 ×9 = 162

                  Move decimal point one place to left = 16.2

Hence, 36 × 45% = 16.2

Example: - Find 45% of 640

Solution: - Divide 640 by 2; 640 ÷ 2 = 320

                  Multiply it by 9 = 320 ×9 = 2880

                  Move decimal point one place to left = 288

Hence, 460 × 45% = 288

Finding 50% of a number

It is as simple as asking a Grade 7 student to read the table of 2. Simply divide the number whose 50% you are intend to find by 2 and you get the answer.

Example: - Find 50% of 630

Solution: - Divide 630 by 2: 630÷2 = 315

Example: - Find 50% of 6850

Solution: - Divide 6850 by 2: 6850÷2 = 3425

Finding 55% of a number

Multiplying a number by 11 is very easy. In Multiplication chapter I have described a simple rule to multiply any number by 11. Simply put two zeros (along both side one each) with the number and keep adding from right to left to get the answer.

Example: - 112 ×11 =?

Solution: - 0(112)0 = 0+1 / 1+1 / 1+2 / 2+0 = 1232

In order to find 50% of a number follow the steps.

1.      Divide the number by 2

2.      Multiply the result obtained by 11

3.      Move the decimal point one place to left

Example: - Find 55% of 36

Solution: - Divide 36 by 2; 36 ÷ 2 = 18

                  Multiply it by 11 = 18 ×11 = 198

                  Move decimal point one place to left = 19.8

Hence, 36 × 55% = 19.8

Example: - Find 55% of 580

Solution: - Divide 640 by 2; 580 ÷ 2 = 290

                  Multiply it by 11 = 290 × 11 = 3190

                  Move decimal point one place to left = 319

Hence, 460 × 55% = 319

Read more on percentage in my book MATHS MADE EASY published by Rupa Publication

http://www.rupapublications.co.in/books/maths-made-easy

Send your comments at

Rajesh Kumar Thakur

rkthakur1974@gmail.com