1.4.6 Dividing by 9 Trick

From Section 1.4.2 it is explained how a remainder can be found when dividing by 9. However, you can continue this process of adding select digits to get the complete answer when dividing by 9. The following is the result when you divide a four digit number $abcd$ by 9 without carries. The details of the proof is omitted, only the result is shown:

$$abcd \div 9 =$$

• Fractional Part: $\frac{a + b + c + d}{9}$
• Ones: $a + b + c$
• Tens: $a + b$
• Hundreds: $a$

I think the gist of the trick is self explanatory, let’s look at a simple example:

3211 ÷ 9 =

• Fractional Part: $\frac{1 + 1 + 2 + 3}{9} = \frac{7}{9}$
• Ones: $1 + 2 + 3 = 6$
• Tens: $2 + 3 = 5$
• Hundreds: $3$

Answer: $356 \frac{7}{9}$

Here is a little bit more complicated of a problem involving a larger number being divided as well as incorporating carries:

32257 ÷ 9 =

• Fractional Part: $\frac{7 + 5 + 2 + 2 + 3}{9} = 2 \frac{1}{9}$ (Write 1/9, carry 2)
• Ones: $5 + 2 + 2 + 3 + 2 = 14$ (Write 4, carry 1)
• Tens: $2 + 2 + 3 + 1 = 8$
• Hundreds: $2 + 3 = 5$
• Thousands: $3$

Answer: $3584 \frac{1}{9}$

Problem Set 1.4.6