Division is an interesting operation. Both these signs tell you to divide:

$8\div4$              $\require{enclose}4\enclose{longdiv}{8}$

Division is the opposite of multiplication.

Multiplication takes equal-sized groups and puts the groups together to find the total.

3 × 4 = 12

Division takes the total and separates that amount into equal groups. You can find the number of equal groups or the size of each group.

12 ÷ 4 = 3

Learn this vocabulary for division:

• Dividend – The number or quantity to be divided; the amount altogether.
• Divisor – The number we divide by. The divisor tells us the number of groups or the quantity in each group that the dividend is to be separated into.
• Quotient – (“kw ō shent”) The answer to a division question.

$\begin{array}{r}\text{quotient}\\\text{divisor} \enclose{longdiv}{\text{dividend}}\end{array}$             $\text{dividend}\div\text{divisor}=\text{quotient}$

To use the times table chart for division facts:

• Find the divisor in the column on the left of the times table chart.
• Run your finger in a straight line across the divisor row until you come to the dividend.
• Go straight up that column and the quotient should be the top number.

Try a few while you are doing Exercise 1.

Exercise 1

Complete this chart to help yourself understand the connection between multiplication and division. The first one is done for you. Check your work using the answer key at the end of the exercise.

Answers to Exercise 1

Exercise 2

Check your division facts by quickly doing this exercise. Check your work using the answer key at the end of the exercise.

1. 72 ÷ 6 =
2. 12 ÷ 2 =
3. 3 ÷ 1 =
4. 80 ÷ 10 =
5. 18 ÷ 6 =
6. 40 ÷  4 =
7. 21 ÷ 7 =
8. 50 ÷ 5 =
9. 54 ÷ 9 =
10. 8 ÷  2 =
11. 22 ÷ 11 =
12. 45 ÷ 9 =
13. 4 ÷ 4 =
14. 24 ÷  6
15. 81 ÷ 9 =
16. 88 ÷ 8 =
17. 30 ÷ 3 =
18. 12 ÷ 4 =
19. 33 ÷ 3 =
20. 66 ÷ 11 =
21. 20 ÷ 5 =

Answers to Exercise 2

1. 12
2. 6
3. 3
4. 8
5. 3
6. 10
7. 3
8. 10
9. 6
10. 4
11. 2
12. 5
13. 1
14. 4
15. 9
16. 11
17. 10
18. 3
19. 11
20. 6
21. 4

Exercise 3

Check your division facts by quickly doing this exercise. Check your work using the answer key at the end of the exercise.

1. 90 ÷ 10 =
2. 70 ÷ 7 =
3. 28 ÷ 7 =
4. 32 ÷ 8 =
5. 24 ÷ 3 =
6. 36 ÷ 12 =
7. 84 ÷ 7 =
8. 10 ÷ 2 =
9. 64 ÷ 8 =
10. 6 ÷ 6 =
11. 60 ÷ 12 =
12. 48 ÷ 4 =
13. 72 ÷ 9 =
14. 20 ÷ 10 =
15. 49 ÷ 7 =
16. 48 ÷ 6 =
17. 36 ÷ 9 =
18. 21 ÷ 3 =

Answers to Exercise 3

1. 9
2. 10
3. 4
4. 4
5. 8
6. 3
7. 12
8. 5
9. 8
10. 1
11. 5
12. 12
13. 8
14. 2
15. 7
16. 8
17. 4
18. 7

Exercise 4

Check your division facts by quickly doing this exercise. Check your work using the answer key at the end of the exercise.

1. 12 ÷ 6 =
2. 27 ÷ 9 =
3. 56 ÷ 7 =
4. 3 ÷ 1 =
5. 20 ÷ 2 =
6. 9 ÷ 3 =
7. 55 ÷ 5 =
8. 14 ÷ 7 =
9. 42 ÷ 6 =
10. 18 ÷ 3 =
11. 88 ÷ 11 =
12. 63 ÷ 9 =
13. 28 ÷ 4 =
14. 6 ÷ 1 =
15. 30 ÷ 5 =
16. 4 ÷ 2 =
17. 7 ÷ 7 =
18. 48 ÷ 12 =

Answers to Exercise 4

1. 2
2. 3
3. 8
4. 3
5. 10
6. 3
7. 11
8. 2
9. 7
10. 6
11. 8
12. 7
13. 7
14. 6
15. 6
16. 2
17. 1
18. 4

Remainders

You have been practicing the division facts that always work out evenly – nothing is left over.

Well, in the real world things are not usually so perfect!

You have 7 candies to share among your 3 children.

We call the left-over the remainder. For now, put R and the left over number after your quotient.

$7\div3=2\text{ R}1$                            $\begin{array}{r}2\text{ R}1\\3 \enclose{longdiv}{7}\hspace{1.3em}\end{array}$

The remainder must not be the same size or bigger than the divisor. If it is bigger, it means another group could be made.

Exercise 5

Draw pictures to find the quotient and the remainder. Write each question using the other division sign as well. Have your instructor check your work.

1. $23\div10=2\text{ R}3$         $\begin{array}{r}2\text{ R}3\\10 \enclose{longdiv}{23}\hspace{1.3em}\end{array}$

2. 24 ÷ 7 =
3. 19 ÷ 3 =
4. 39 ÷ 12 =

You cannot always draw pictures, so how should you find the quotients?

Exercise 6

Find the quotients and remainders (divide, multiply, subtract, compare).

1. $5\enclose{longdiv}{28}$
2. $4\enclose{longdiv}{15}$
3. $6\enclose{longdiv}{47}$
4. $9\enclose{longdiv}{37}$
5. $2\enclose{longdiv}{13}$
6. $6\enclose{longdiv}{25}$
7. $8\enclose{longdiv}{75}$
8. $3\enclose{longdiv}{19}$
9. $7\enclose{longdiv}{32}$
10. $4\enclose{longdiv}{9}$
11. $9\enclose{longdiv}{55}$
12. $10\enclose{longdiv}{98}$
13. $3\enclose{longdiv}{26}$
14. $8\enclose{longdiv}{47}$
15. $9\enclose{longdiv}{46}$
16. $6\enclose{longdiv}{43}$
17. $5\enclose{longdiv}{49}$
18. $4\enclose{longdiv}{38}$
19. $2\enclose{longdiv}{19}$
20. $7\enclose{longdiv}{61}$
21. $3\enclose{longdiv}{23}$
22. $8\enclose{longdiv}{78}$
23. $9\enclose{longdiv}{67}$
24. $6\enclose{longdiv}{45}$

Answers to Exercise 6

1. 5 R3
2. 3 R3
3. 7 R5
4. 4 R1
5. 6 R1
6. 4 R1
7. 9 R3
8. 6 R1
9. 4 R4
10. 2 R1
11. 6 R1
12. 9 R8
13. 8 R2
14. 5 R7
15. 5 R1
16. 7 R1
17. 9 R4
18. 9 R2
19. 9 R1
20. 8 R5
21. 7 R2
22. 9 R6
23. 7 R4
24. 7 R3

Exercise 7

Find the quotients and remainders (divide, multiply, subtract, compare.) Check your work using the answer key at the end of the exercise.

Example

59 ÷ 7 =

Rewrite: $7\enclose{longdiv}{59}$

Then solve: $\begin{array}{r}8\\7 \enclose{longdiv}{59}\\-56\\\hline3\end{array}$

Answer: 8 R3

1. 27 ÷ 5 =
2. 13 ÷ 2 =
3. 46 ÷ 9 =
4. 38 ÷ 6 =
5. 61 ÷ 7 =
6. 14 ÷ 5 =
7. 49 ÷ 8 =
8. 28 ÷ 3 =
9. 78 ÷ 8 =
10. 37 ÷ 4 =
11. 67 ÷ 9 =
12. 52 ÷ 6 =
13. 45 ÷ 8 =
14. 25 ÷ 7 =
15. 11 ÷ 3 =
16. 53 ÷ 9 =
17. 19 ÷ 4 =
18. 77 ÷ 8 =
19. 20 ÷ 3 =
20. 11 ÷ 2 =
21. 23 ÷ 5 =
22. 54 ÷ 7 =
23. 87 ÷ 9 =
24. 9 ÷ 4 =

Answers for Exercise 7

1. 5 R2
2. 6 R1
3. 5 R1
4. 6 R2
5. 8 R5
6. 2 R4
7. 6 R1
8. 9 R1
9. 9 R6
10. 9 R1
11. 7 R4
12. 8 R4
13. 5 R5
14. 3 R4
15. 3 R2
16. 5 R8
17. 4 R3
18. 9 R5
19. 6 R2
20. 5 R1
21. 4 R3
22. 7 R5
23. 9 R6
24. 2 R1

Topic A: Self-Test

Mark /12          Aim 9/12

A. Give the answer. (6 marks)

1. 63 ÷ 9 =
2. 21 ÷ 7 =
3. 72 ÷ 8 =
4. 65 ÷ 4 =
5. 86 ÷ 4 =
6. 75 ÷ 6 =

B. Find the quotient. (6 marks)

1. $6\enclose{longdiv}{59}$
2. $9\enclose{longdiv}{87}$
3. $7\enclose{longdiv}{51}$
4. $8\enclose{longdiv}{76}$
5. $5\enclose{longdiv}{49}$
6. $3\enclose{longdiv}{26}$

Answers to Topic A Self-Test

A. Give the answer.

1. 7
2. 3
3. 9
4. 9
5. 8
6. 8

B. Find the quotient.

1. 9 R5
2. 9 R6
3. 7 R2
4. 9 R4
5. 9 R4
6. 8 R2