# Casting out 9s

When I published my book “Algebra Made Easy as Arithmetic”, Ms. Encarnacion Reyes, long time president of the San Pablo City chapter of the Mathematics Teachers Association of the Philippines commented: “I’m so glad that you included the casting out nines. I used to teach that several decades ago but I seldom find it in new books.”

By casting out nines and combining it with the Vedic Math sub-Sutra “The product of the Sums is the Sum of the Product”, this type of problem, which is often given in VM competitions, can be solved easily in less than 5 seconds.

If we will use the traditional long multiplication and division algorithms we would surely require more than 1 minute to give the answer. And of course, with long solutions, there is a greater chance of errors.

The procedure to solve this problem is fully explained in our “Algebra Made Easy as Arithmetic” online course which will start on March 19, 2022 and in our Vedic Math review series which will start on March 8, 2022. See https://www.math-inic.com/blog/revised-schedule-of-vedic-math-activities-in-the-philippines/

The procedure is really very simple.

- Start by casting out nines
- In the first factor, 981, 992, we can cast out or remove the 3 nines and the digits 8 and 1 which add up to 9. Only 2 remains. Note that you can add all the digits of the number, 9 + 8 + 1 + 9 + 9 + 2 to get 38. Add 3 and 8 to get 11 and finally add 1 + 1 to get 2. Thus casting out 9s is the easy way of determining the repeated digit sum of a number. And you can check that 2 is the remainder when 981, 992 is divided by 9.

- For the second factor, 799, 392, casting out the 3 nines and 7 and 2 will leave us only the digit 3. You can check that the digits of this number will add up to 39; 3+ 9 = 12 and 1+ 2 = 3. You can also verify that 3 is the remainder when the number is divided by 9.

- Now use the sub-Sutra” the Product of the (digit) Sums is the (digit) Sum of the Product (PSSP). 2 x 3 = 6. So 6 is the remainder when the product of of 981, 992 and 799, 392 is divided by 9.

You can use your calculator to verify that 981, 992 x 799,392 = 784,996, 548, 864 and when this product is divided by 9 the quotient is 87, 221, 838, 762 .67. The 0.67 means 2/3 or 6/9 which verifies that the remainder is 6

In addition to its use in multiplication, PSSP Sutra can be modified for use in Addition, subtraction and division. Its use can also be extended in algebraic multiplications.

Try finding the digit sums of the following by casting out 9s:

- 923 =
- 567 =
- 838 =
- 5,437 =
- 9,897 =
- 6, 789 =
- (123 x 345) =
- (4589 x 9281) =
- (123 + 456 + 789) =
- ( 23 x 592 + 785) =

Here are the answers to last week’s exercises:

- 12 ÷ 1.5 = 8
- 21 ÷ 1.5 = 14
- 28 ÷ 3.5 = 8
- 45.5 ÷ 3.5 = 13
- 27 ÷ 4.5 = 6
- 44 ÷ 5.5 = 8
- 12 ÷ 0.75 = 16
- 27 ÷ 0.5 = 54
- 23 ÷ 0.25 = 92
- 21 ÷ 0.125 = 168

We are are also inviting you to attend the 8^{th} Online Vedic Math Conference. The fee is $10/PhP500 for delegates and teachers but free for students 18 years or younger. You can register here: https://instavm.org/upcoming-events/