---
title: "Sharing and Division"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 17
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/17-sharing-and-division
---

# Chapter 17 — Sharing 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.](https://one-course.com/images/onecourse/chapters/math-1/g3-division/fig-be095898b2e9.svg)

*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.](https://one-course.com/images/onecourse/chapters/math-1/g3-division/fig-a041b2b0ae9c.svg)

*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](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) that fits under $58$ ($7 \times 9 = 63$ is too much). So

$$
58 = 7 \times 8 + 2 :
$$

[quotient](#def-g3-division-remainder) $8$, [remainder](#def-g3-division-remainder) $2$. Check: $56 + 2 = 58$, and $2 < 7$.

**Method 17.5 (Interpreting the remainder).**

After dividing, return to the story:

1. “how many each / how many full groups?” — the [quotient](#def-g3-division-remainder) ;
2. “how many left over?” — the [remainder](#def-g3-division-remainder) ;
3. “how many boxes to hold everything?” — the [quotient](#def-g3-division-remainder) *plus one* if the [remainder](#def-g3-division-remainder) is not [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-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](#def-g3-division-half) of $46$: since $46 = 40 + 6$, [half](#def-g3-division-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 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](#def-g3-division-remainder) and the [remainder](#def-g3-division-remainder): $17 \div 5$; $30 \div 4$; $50 \div 6$.

**Solution of Exercise 17.2.**

$17 = 5 \times 3 + 2$: [quotient](#def-g3-division-remainder) $3$, [remainder](#def-g3-division-remainder) $2$.

$30 = 4 \times 7 + 2$: [quotient](#def-g3-division-remainder) $7$, [remainder](#def-g3-division-remainder) $2$.

$50 = 6 \times 8 + 2$: [quotient](#def-g3-division-remainder) $8$, [remainder](#def-g3-division-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 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](#def-g3-division-half) of $68$; the third of $27$; the [quarter](#def-g3-division-half) of $48$; the [half](#def-g3-division-half) of $150$.

**Solution of Exercise 17.4.**

[Half](#def-g3-division-half) of $68$: $34$. Third of $27$: $9$. [Quarter](#def-g3-division-half) of $48$: $12$. [Half](#def-g3-division-half) of $150$: $75$.

**Exercise 17.5 ★.**

True or false? Check with a multiplication. “$45 \div 5 = 9$”; “$52 \div 6 = 8$ [remainder](#def-g3-division-remainder) $4$”; “$70 \div 8 = 9$ [remainder](#def-g3-division-remainder) $2$”.

**Solution of Exercise 17.5.**

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

“$52 \div 6 = 8$ [remainder](#def-g3-division-remainder) $4$”: true, $6 \times 8 + 4 = 52$ and $4 < 6$.

“$70 \div 8 = 9$ [remainder](#def-g3-division-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](https://one-course.com/books/math/1/en/chapter/11-shapes-and-solids#def-g2-shapes-solids) of division is this (sharing or grouping)?

**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 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 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](#def-g3-division-half) of $?$ is $35$.

**Solution of Exercise 17.9.**

$? = 7 \times 6 = 42$. $54 \div 9 = 6$, so the missing divisor is $9$. The number whose [half](#def-g3-division-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 of Exercise 17.10.**

$47 = 5 \times 9 + 2$: each friend gets $9$ stickers, and Amina keeps the [remainder](#def-g3-division-remainder), $2$ stickers.

**Exercise 17.11 ★★.**

When a number is divided by $6$, what are all the possible [remainders](#def-g3-division-remainder)? What is the biggest number whose division by $6$ has [quotient](#def-g3-division-remainder) $7$?

**Solution of Exercise 17.11.**

Dividing by $6$, the [remainder](#def-g3-division-remainder) is always smaller than $6$: it can be $0$, $1$, $2$, $3$, $4$ or $5$. The biggest number with [quotient](#def-g3-division-remainder) $7$ uses the biggest [remainder](#def-g3-division-remainder): $6 \times 7 + 5 = 47$.