B. Number · Operations: Numbers to 100 000, Decimals, Fractions, Ratios and Rates

Remainders: What Should We Do with the Leftovers?

B2.7 A1About 40 minutes

Learning goals

I can divide a 3-digit number by a 2-digit number and find the remainder.
I can show a remainder in an area model and in long division.
I can decide whether to round up, drop the remainder, use it as the answer, or share it as a fraction or decimal.

Words to know

remainder
What is left over when a number can't be divided into equal groups exactly. It must be smaller than the divisor.

interpret
Decide what an answer means in a story.

mixed number
A whole number and a fraction, like 11 3/12.

Remember from Grade 4

In Grade 4 you found remainders when dividing by 1-digit numbers, like 35 ÷ 4 = 8 R3. You learned to round up, drop the remainder, or share it as a fraction. In Grade 5 you'll do the same with 2-digit divisors, and see where the remainder shows up in each method.

Let's start with a story

250 people are going to a hockey game. Each bus holds 48 people. How many buses do they need?

250 ÷ 48: 48 × 5 = 240. 250 − 240 = 10 left over. So 250 ÷ 48 = 5 R10.

Is the answer 5 buses? No! Ten people would be left at the curb. They need 6 buses.

The math gives 5 R10, but the story decides what that means.

Five buses are full with 48 people each (240 people). The remainder, 10 people, still needs a bus, so the answer to the story is 6 buses.

Where is the remainder?

In the area model, the remainder is the piece that is too small to make another full column.
In long division, the remainder is the number left after the last subtraction.

Either way, the remainder must be smaller than the divisor. If it isn't, you can make another group!

An area model for 500 ÷ 24. A rectangle with height 24 and width 20 has area 480. The leftover piece of 20 (dashed) is too small to make another full column of 24, so it is the remainder.

Worked example

Example 1: Find the remainder

Solve 500 ÷ 24.

24 × 20 = 480. 500 − 480 = 20 left.
Can you take away another 24? No, 20 is smaller than 24.
So the quotient is 20 and the remainder is 20.
Check: 24 × 20 + 20 = 480 + 20 = 500. ✓

500 ÷ 24 = 20 R20

Five things a remainder can mean

Round up: everyone or everything needs a place. 250 people, buses of 48 → 6 buses.
Drop it: only full groups count. 200 eggs, cartons of 12 → 200 ÷ 12 = 16 R8 → 16 full cartons.
The remainder is the answer: the question asks what's left. 175 cookies in bags of 12 → 14 R7 → 7 cookies left over.
Share it as a fraction: the leftover can be cut up. 135 L of water shared equally in 12 jugs → 11 R3 → 11 3/12 L in each jug.
Share it as a decimal: money can be split into cents. $250 shared by 20 people → 12 R10 → the $10 left is $0.50 each → $12.50 each.

Worked example

Example 2: Round up or drop?

A school has 400 maple candies to put in boxes of 36. (a) How many full boxes can they fill? (b) How many boxes are needed to hold all the candies?

400 ÷ 36: 36 × 10 = 360, 40 left. 36 × 1 = 36, 4 left. So 400 ÷ 36 = 11 R4.
(a) Only full boxes count, so drop the remainder: 11 full boxes.
(b) The 4 extra candies need a box too, so round up: 12 boxes.

(a) 11 full boxes (b) 12 boxes

Worked example

Example 3: Share the remainder as a decimal

$186 is shared equally by 12 friends. How much does each friend get?

186 ÷ 12: 12 × 10 = 120, 66 left. 12 × 5 = 60, 6 left. So 186 ÷ 12 = 15 R6.
Each friend gets $15, and $6 is left.
$6 is 600 cents. 600 ÷ 12 = 50 cents, or $0.50 each.
$15 + $0.50 = $15.50. Check: 12 × 15.50 means 12 × $15 = $180 plus 12 × 50¢ = $6, total $186. ✓

$15.50 each

What does the question ask?

Before you write your final answer, re-read the question and ask:

Does everything need a place? → round up
Do only full groups count? → drop the remainder
Is the question asking what's left? → the remainder is the answer
Can the leftover be cut or split? → fraction or decimal

Just for fun

Riddle: What did the remainder shout at the bus driver?

"Wait! Don't leave without the last 10 of us!"

Hands-on activity

Remainder stories

You need: paper, a pencil and a grown-up.

Your grown-up picks a division with a remainder, like 350 ÷ 24 (= 14 R14).
Make up three stories for it: one where you round up (15), one where you drop the remainder (14), and one where the remainder is the answer (14).
Bonus: make up a money story for $360 ÷ 25 where each person gets dollars and cents. (360 ÷ 25 = 14 R10. The $10 left is 1000 cents, and 1000 ÷ 25 = 40 cents, so each person gets $14.40.)

Math talk: "How did the question tell you what to do with the remainder?"

Your turn

200 ÷ 15 = ?

350 ÷ 24 = ?

487 ÷ 32 = ?

150 students go on a trip in vans that hold 12. How many vans are needed?

You have $100. Books cost $15 each. How many books can you buy?

300 hockey cards go in a binder with 18 cards on each page. How many pages are needed to hold all the cards?

175 cookies are packed in bags of 12. How many cookies are left over?

More practice: warm-up

25 ÷ 12 = ?

50 ÷ 24 = ?

100 ÷ 30 = ?

61 ÷ 20 = ?

130 ÷ 11 = ?

More practice: on your own

625 ÷ 14 = ?

230 guests sit at tables for 12. How many tables are needed?

135 L of water is shared equally among 12 jugs. How much is in each jug? Write the answer as a mixed number.

$250 is shared equally by 20 people. How much does each person get?

Mo says 300 ÷ 24 = 11 R36. What is wrong? Fix it.

Challenge

Find a number between 200 and 300 that leaves a remainder of 5 when divided by 25.

500 ÷ 24 = 20 R20. Write one story where the answer is 21 and one where the answer is 20.

412 students form teams of 15 for a field day. How many full teams can they make? How many more students would they need to make one more full team?

Check your understanding

Quiz

250 people ride buses that hold 48. How many buses are needed?

200 eggs go in cartons of 12. How many full cartons?

What is 350 ÷ 24?

A remainder must always be ...

$250 is shared equally by 20 people. How much does each get?

135 L of water is shared equally among 12 jugs. How much is in each jug?

For parents and teachers

Big idea: B2.7 asks students to divide three-digit numbers by two-digit numbers "while expressing any remainder appropriately." That means deciding, from the story, whether to round up, drop the remainder, use the remainder as the answer, or share it as a fraction or decimal.

Common difficulties:

Writing "5 R10 buses". Always ask "What does the remainder mean here?"
A remainder bigger than the divisor. Teach your child to check this after every division.
Sharing money. Changing leftover dollars to cents makes it easier (6 dollars = 600 cents).

How to help: Real situations are perfect: "We have 150 party favours and 12 guests. How many each? How many left?" Ask your child to explain why they rounded up or down; this reasoning is part of the A1 process expectations.