# Multiplication by 9

Many young school children find it hard to multiply by 9, using the conventional right to left method, whether or not they mastered the 9 times table.

But as we have discussed in our books 25 Math Short Cuts (pp. 25 – 27) and Algebra Made Easy as Arithmetic (pp. 14 – 15), multiplying by 9 is easy if we consider it as (10 -1).

Using this method we could easily get 4 x 9 = 40 – 4 = 36 and 7 x 9 = 70 -7 = 63.

Also, we have,  13 x 9 = 130 – 13 = 117 and  257 x 9 = 2570 – 257 = 2313.

But it would not be easy for the following: 74 x 9 = 740 – 74 or 864 x 9 = 8640 – 864.

However, we can develop a simple way to multiply by 9 and mastering this would be very valuable in higher mathematics.

Let us consider one of our examples, 257 x 9. If we perform the subtraction using the left to right method,

2 5 7 0

•  2 5 7

2 3 2 (7)

We can see that after bringing down 2, we subtracted 2 from 5, which is the next digit to it in the multiplicand to get 3. Then we also deducted 5 from 7 which again is the next digit to it  to get 2 and finally we are subtracted the last digit 7 from 0 to get negative or “bar” 7 which we represented as (7).

We can then evaluate 232(7) as 2320 – 7 or 2313. (Please see https://www.math-inic.com/blog/msc-4-using-bar-numbers/ for a discussion on bar numbers.)

We can now multiply long numbers by 9 using our technique.

1 4 5 8 x 9 = 1313(8) = 13,122

Here we used the following steps:

1. Copy the first digit of the multiplicand as the 1st digit of the answer – 1
2. Subtract the first from the next digit in the multiplicand – (4 – 1 = 3)
3. Repeat step 2 starting from the second digit up to the penultimate digit of the multiplicand (5 – 4 = 1) and (8 – 5 = 3)
4. Write down the last digit as a negative number which is the same as subtracting it from 0.
5. Express the answer in the conventional format.

Let have some more examples:

789 x 9 = 711(9) = 7101

3584 x 9 = 323(4)(4) = 32,256

35174 x 9 = 32(4)6(3)(4) = 316,566

Try multiplying these numbers by 9 using our technique.

1. 35
2. 53
3. 258
4. 285
5. 3478
6. 3749
7. 25688
8. 52368
9. 364869
10. 754868