Primary & Middle School Mathematics · Grades 1–9

17Sharing and Division

Twenty marbles shared fairly among four children: how many each? Division answers all the sharing questions — and the grouping questions too — and it is multiplication’s twin: one undoes the other.

17.1 Two faces of division

Definition 17.1 (Division)

The division $20 \div 4$ answers two kinds of questions:

sharing: $20$ marbles shared among $4$ children — how many marbles each? ( $5$ each);
grouping: $20$ marbles put in bags of $4$ — how many bags? ( $5$ bags).

Both questions have the same answer, because both undo the same multiplication: $4 \times 5 = 20$ .

Sharing 20 into 4 equal groups. Grouping by 4 instead would give 5 groups — the same division. — Sharing $20$ into $4$ equal groups. Grouping by $4$ instead would give $5$ groups — the same division.

Method 17.2 (Dividing with the tables)

To compute $42 \div 6$ , ask: “ $6$ times what makes $42$ ?” The table of $6$ answers: $6 \times 7 = 42$ , so $42 \div 6 = 7$ . Every division question is a multiplication question read backwards.

17.2 When it does not come out even

Definition 17.3 (Remainder)

Sharing $23$ marbles among $4$ children gives $5$ marbles each — and $3$ marbles left over, not enough for another round. We write

23 = 4 \times 5 + 3 ,

and say: the quotient is $5$ , the remainder is $3$ . The remainder is always smaller than the number of children — otherwise another round of sharing would be possible.

Sharing 23 among 4: after 5 rounds, 3 marbles remain — too few for a 6th round. — Sharing $23$ among $4$ : after $5$ rounds, $3$ marbles remain — too few for a $6$ th round.

Example 17.4

Divide $58$ by $7$ . Scan the table of $7$ : $7 \times 8 = 56$ is the biggest product that fits under $58$ ( $7 \times 9 = 63$ is too much). So

58 = 7 \times 8 + 2 :

quotient $8$ , remainder $2$ . Check: $56 + 2 = 58$ , and $2 < 7$ .

Method 17.5 (Interpreting the remainder)

After dividing, return to the story:

“how many each / how many full groups?” — the quotient;
“how many left over?” — the remainder;
“how many boxes to hold everything?” — the quotient plus one if the remainder is not zero.

Example 17.6

$58$ students go canoeing, $7$ per canoe. $58 = 7 \times 8 + 2$ : eight canoes are full, and $2$ students remain — they need a canoe too. So $9$ canoes must be booked.

17.3 Halves, thirds, quarters

Definition 17.7 (Half, third, quarter)

The half of a number is the result of dividing it by $2$ ; the third, by $3$ ; the quarter, by $4$ . For instance the half of $18$ is $9$ , the third of $18$ is $6$ , the quarter of $36$ is $9$ .

Example 17.8 (Doubles and halves go together)

The half of $46$ : since $46 = 40 + 6$ , half of each part gives $20 + 3 = 23$ . Check with the double: $23 + 23 = 46$ . Knowing doubles by heart ( $25 \to 50$ , $45 \to 90$ , $150 \to 300$ ) makes halves instantaneous.

17.4 Exercises

Exercise 17.1 ★

Compute using the tables backwards: $36 \div 4$ ; $63 \div 9$ ; $40 \div 5$ ; $56 \div 8$ .

Solution

Solution of Exercise 17.1.

$36 \div 4 = 9$ (since $4 \times 9 = 36$ ); $63 \div 9 = 7$ ; $40 \div 5 = 8$ ; $56 \div 8 = 7$ .

Exercise 17.2 ★

For each division, write the equality $a = b \times q + r$ and give the quotient and the remainder: $17 \div 5$ ; $30 \div 4$ ; $50 \div 6$ .

Solution

Solution of Exercise 17.2.

$17 = 5 \times 3 + 2$ : quotient $3$ , remainder $2$ .

$30 = 4 \times 7 + 2$ : quotient $7$ , remainder $2$ .

$50 = 6 \times 8 + 2$ : quotient $8$ , remainder $2$ .

Exercise 17.3 ★

$28$ marbles are shared among $6$ children. How many marbles each, and how many left over? Draw the sharing if it helps.

Solution

Solution of Exercise 17.3.

$28 = 6 \times 4 + 4$ : each child gets $4$ marbles, and $4$ marbles are left over.

Exercise 17.4 ★

Compute: the half of $68$ ; the third of $27$ ; the quarter of $48$ ; the half of $150$ .

Solution

Solution of Exercise 17.4.

Half of $68$ : $34$ . Third of $27$ : $9$ . Quarter of $48$ : $12$ . Half of $150$ : $75$ .

Exercise 17.5 ★

True or false? Check with a multiplication. “ $45 \div 5 = 9$ ”; “ $52 \div 6 = 8$ remainder $4$ ”; “ $70 \div 8 = 9$ remainder $2$ ”.

Solution

Solution of Exercise 17.5.

“ $45 \div 5 = 9$ ”: true, $5 \times 9 = 45$ .

“ $52 \div 6 = 8$ remainder $4$ ”: true, $6 \times 8 + 4 = 52$ and $4 < 6$ .

“ $70 \div 8 = 9$ remainder $2$ ”: false — $8 \times 9 + 2 = 74$ , not $70$ . Correct: $70 = 8 \times 8 + 6$ .

Exercise 17.6 ★

$32$ photos are glued into an album, $4$ per page. How many pages are used? Which face of division is this (sharing or grouping)?

Solution

Solution of Exercise 17.6.

$32 \div 4 = 8$ pages. This is grouping: the photos are put in groups of $4$ , and we count the groups.

Exercise 17.7 ★

A rope of $75$ cm is cut into pieces of $9$ cm. How many full pieces, and how long is the leftover bit?

Solution

Solution of Exercise 17.7.

$75 = 9 \times 8 + 3$ : eight full pieces, and a leftover bit of $3$ cm.

Exercise 17.8 ★

$50$ children go on a trip in cars of $4$ seats. How many cars are needed? (Careful: everyone must ride!)

Solution

Solution of Exercise 17.8.

$50 = 4 \times 12 + 2$ : twelve cars carry $48$ children, and $2$ children still need a ride. $13$ cars are needed.

Exercise 17.9 ★

Find the missing numbers: $? \div 7 = 6$ ; $54 \div ? = 6$ ; the half of $?$ is $35$ .

Solution

Solution of Exercise 17.9.

$? = 7 \times 6 = 42$ . $54 \div 9 = 6$ , so the missing divisor is $9$ . The number whose half is $35$ is $70$ .

Exercise 17.10 ★★

Amina shares her $47$ stickers among her $5$ friends, as fairly as possible, and keeps the leftover for herself. How many stickers does each friend get, and how many does Amina keep?

Solution

Solution of Exercise 17.10.

$47 = 5 \times 9 + 2$ : each friend gets $9$ stickers, and Amina keeps the remainder, $2$ stickers.

Exercise 17.11 ★★

When a number is divided by $6$ , what are all the possible remainders? What is the biggest number whose division by $6$ has quotient $7$ ?

Solution

Solution of Exercise 17.11.

Dividing by $6$ , the remainder is always smaller than $6$ : it can be $0$ , $1$ , $2$ , $3$ , $4$ or $5$ . The biggest number with quotient $7$ uses the biggest remainder: $6 \times 7 + 5 = 47$ .