One division
26 ÷ 4 = 6 remainder 2. Four goes into 26 six times, because 6 × 4 = 24, and 26 − 24 = 2 are left over.
That is the arithmetic, and it is the same every time. The answer to a question can still be 7, 6 or 2, because the question decides what to do with the remainder.
Six full groups of 4, and a last group with 2 in it and 2 empty places.
26 ÷ 4 = 6 remainder 2: 6 tickets: round down
Find the question whose answer is 7
The same 26 ÷ 4 = 6 remainder 2 under three questions. The cars count every group, the short one too; the tickets count only the full groups; the eggs left out are the remainder.
Round up: everyone needs a place
26 children are going on a trip, and each car seats 4. Six cars carry 6 × 4 = 24 children, and 2 children are still waiting. They have to travel too, so they need a seventh car, even though it will not be full.
The answer is 7 cars. When the remainder is people or things that still need a place, round the answer up to the next whole number.
Round down: only whole ones count
You have $26, and a ticket costs $4. Six tickets cost 6 × $4 = $24, and $2 is left. $2 is not enough for another ticket, so you can buy 6 tickets.
The answer is 6 tickets. When the question asks how many whole things can be made or bought, the remainder is too small to make another one, so leave it out.
The remainder is the answer
26 eggs are packed into boxes of 4. How many eggs are left out of the boxes? Six boxes hold 24 eggs, and 26 − 24 = 2 eggs are left out. This question asks for what is left over, so the answer is the remainder: 2 eggs.
Read the question before the answer
Answering 6 cars leaves 2 children behind. Answering 7 tickets spends 7 × $4 = $28, which is more than the $26 there is. And the 2 eggs are not a number of boxes: the question about eggs left over is answered by the remainder alone.
So after dividing, ask what the question counts. Places that everyone needs: round up. Whole things that can be made or bought: round down. What is left over: the remainder.
Worked example: Coaches and Minibuses for a School Trip
Question A school is taking 468 students and teachers on a trip to the science center. A coach seats 44 people. (a) How many coaches must the school hire so that everyone has a seat? (b) Instead, the school hires only as many coaches as it can fill completely, and everyone left over travels in minibuses that seat 12 people each. How many minibuses does it need?
1.Draw the 468 people as one bar and cut it into coach loads of 44. Long division tells us how many loads there are.
The 468 people, cut into coach loads of 44. 2.Since 4 is less than 44, start with 46: 46 ÷ 44 = 1 remainder 2. Bring down the 8 to make 28. Since 28 is less than 44, the next digit of the quotient is 0, so 468 ÷ 44 = 10 remainder 28.
Long division: 46 ÷ 44 = 1 remainder 2; bring down the 8 to make 28, which is less than 44. So 468 ÷ 44 = 10 remainder 28. 3.(a) Ten coaches carry 10 × 44 = 440 people, and the 28 left over still need seats. Round up: the school must hire 10 + 1 = 11 coaches.
(a) The 28 left over still need seats, so round up: 10 + 1 = 11 coaches. 4.Part (b) reads the same division another way. Only full coaches go, so the school hires 10 coaches, and the remainder, 28 people, travels in minibuses.
Part (b) keeps only the 10 full coaches. The remainder, 28 people, goes by minibus. 5.28 ÷ 12 = 2 remainder 4. Two minibuses carry 24 people, and the last 4 still need seats, so round up again.
28 ÷ 12 = 2 remainder 4. The last 4 people still need seats. 6.(b) The school needs 2 + 1 = 3 minibuses. Check: 10 × 44 + 28 = 440 + 28 = 468, and 3 minibuses have 3 × 12 = 36 seats for the 28 people.
(b) Round up again: 2 + 1 = 3 minibuses.
Answer: (a) 11 coaches; (b) 3 minibuses
Common mistakes
- Answering 10 coaches for part (a) because the quotient is 10. The 28 people in the remainder still have to travel, so one more coach is needed.
- Starting part (b) from the 11 coaches of part (a), which leaves nobody for the minibuses. Part (b) hires full coaches only, so it keeps the quotient, 10, and the minibuses carry the remainder, 28.