# 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:

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

- 35
- 53
- 258
- 285
- 3478
- 3749
- 25688
- 52368
- 364869
- 754868

Answers to previous exercises:

- 542 ÷ 5 = 108.4
- 743 ÷ 50 = 14.86
- 867 ÷ 500 = 1.734
- 927 ÷ 0.5 = 1,854
- 1024 ÷ 5 = 204.8
- 7526 ÷ 50 = 150.52
- 13856 ÷ 500 = 27.712
- 23254 ÷ 0.5 = 46,508
- 78946 ÷ 5000 =15.7892
- 35795 ÷ 0.05 = 715,900