Most of us want to avoid the number **9 **in almost all calculations. But we can make calculations easier by thinking of **9** as **(10 – 1)**. This fact is particularly useful in division by **9**.

Every **10** contains a **9** and a remainder of **1**. So every multiple of ten that is less than 90 will have a quotient and remainder equal to its tens digit.

So **20/9 = 2 r 2**

** 40/9 = 4 r 4**

and **70/9 = 7 r 7.**

Extending this observation, we can readily obtain the quotient when small numbers are divided by 9.

Take the case of **34. **When divided by **9, **the quotient is equal to the tens digit, **3** and the remainder is equal to the sum of the tens and units digits, **3 + 4 **or **7.**

Similarly,

** 42/9 = 4 r (4+2) = 4 r 6**

** 71/9 = 7 r (7+1) = 7 r 8**

** 26/9 = 2 r (2+6) = 2 r 8**

** 69/9 = 6 r (6+9) = 6 r 15**

But wait! Since the remainder **15 **is greater than **9, **we can divide **15** by **9 **to get **1 r 6. **

So **69/9 = 6 r 15 = (6+1) r 6 = 7 r 6.**

At this point, we would like to stress that the following results are equivalent:

** 69/9 = 6 r 15 = 7 r 6 = 8 r -3 **but **7 r 6 **is the best form.

Example 11.1: **1321/9**

We can write the procedure as: **1 3 2 1 / 9**

Step 1. Bring down the first digit **(1) **to the answer row.

** 1 3 2 1 / 9**

** 1**

Step 2. Add the next digit of the dividend to this number to get the next digit of the quotient: **(1+3=4)**

** 1 3 2 1 / 9 **

** 1 4**

Step 3. Repeat the preceding procedure to get the next digit of the quotient: **(4+2=6)**

** 1 3 2 1 / 9**

** 1 4 6**

Step 4. The last sum is the remainder: **(6+1=7) ** ** 1 3 2 1 / 9**

** 1 4 6 r 7**

Example 11.2: **2023/9**

** 2 0 2 3 / 9**

** 2 2 4 r 7**

To check: the sum of the digits of the dividend should be equal to the remainder.

** 2 + 0 + 2 + 3 = 7**

Example 11.3: **4352/9**

** 4 3 5 2 / 9 **

** 4 7 12 r 14**

Here, we see that we have a 12 and a 14 in the quotient. The **1 **in the **12 **must be carried over to the **7 **to yield **482. ** There is also one **9** in the remainder **14.**

So the final answer is **483 r 5.**

We can modify our procedure to avoid double digits in the quotient.

** 4 3 5 2 / 9 **

** 4 **

Before writing down the **7 (4 + 3), **we see that the next addition **7 + 5 **will give a two digit result, **12. **So we anticipate the carry operation and write down **8** instead of **7**.

** 4 3 5 2 / 9 **

** 4 8**

We then proceed as before

** 4 3 5 2 / 9**

** 4 8**

**8 + 5 = 13. **But since we have performed the carry operation in the previous step, we will write down only the last digit **3.**

** 4 3 5 2 / 9**

** 4 8 3**

Finally we have the remainder: **3 + 2 = 5**

** 4 3 5 2 / 9**

** 4 8 3 r 5 **

**check: 4 + 3 + 5 + 2 = 14; 1 + 4 = 5**

The following are the decimal values of the remainder when dividing by 9.

1 – 1/9 = .1111… = 0.1

2 – 2/9 = .2222… = 0.2

3 – 3/9 = .3333… = 0.3

4 – 4/9 = .4444… = 0.4

5 – 5/9 = .5555… = 0.6

6 – 6/9 = .6666… = 0.7

7 – 7/9 = .7777… = 0.8

8 – 8/9 = .8888… = 0.9

Exercise 11: Divide the following numbers by 9

- ) 134 / 9 =
- ) 215 / 9 =
- ) 2231 / 9 =
- ) 4202 / 9 =
- ) 625 / 9 =
- ) 3030 / 9 =
- ) 7135 / 9 =
- ) 5672 / 9 =
- ) 3692 / 9 =
- ) 46893 / 9 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!