Lesson 8 of 10 · Grades 3 to 4
Division with remainders
Some sharing problems do not come out even. Fifteen biscuits shared among four friends leaves three biscuits over. Division still works; the leftover amount is called the remainder, and it has one important rule.
What is left over
Some piles do not split evenly. Fifteen biscuits shared among four friends gives three biscuits each, and three biscuits are still in the tin. That leftover amount is the remainder. We write the answer as 3 remainder 3, or as 15 = 4 × 3 + 3.
A remainder is not a mistake. It is information about the sharing: the groups came out as full as possible, and the leftover part was not big enough to make another full group.
The remainder rule
The remainder must always be smaller than the divisor. If the leftover part were the same size as the divisor or bigger, it could still make another group, and the division would not be finished. When you divide by 4, the remainder can be 0, 1, 2 or 3. It can never be 4 or more.
The rule gives you a quick way to spot an unfinished answer. If someone writes 30 ÷ 4 = 6 remainder 6, the remainder 6 is bigger than the divisor 4, so a group is hiding in the leftover. Taking 4 out of the 6 makes a seventh group, and the answer becomes 7 remainder 2.
Finding the remainder
To divide with a remainder, find the largest multiple of the divisor that is not more than the dividend. For 29 ÷ 7, the multiples of 7 are 7, 14, 21, 28, 35 and so on. The largest that fits inside 29 is 28. Since 28 ÷ 7 = 4, the quotient is 4, and the remainder is 29 - 28 = 1.
Check the answer by multiplying and adding: 7 × 4 + 1 = 29. If that sentence is true, the quotient and the remainder are both correct.
Watch out for this
Read word problems closely, because the remainder sometimes changes the answer. If 26 eggs are packed into boxes of 4, six boxes are full and 2 eggs are left over. But if the question asks how many boxes are needed to hold every egg, the answer is 7, because the last 2 eggs still need a box. The remainder tells you whether to round up or report the leftover.
Worked example
29 stickers are shared equally among 7 friends. How many stickers does each friend get, and how many are left over?
Step 1: find the largest multiple of 7 that is not more than 29. That is 28.
Step 2: divide: 28 ÷ 7 = 4, so each friend gets 4 stickers.
Step 3: find the remainder: 29 - 28 = 1, so 1 sticker is left over.
Step 4: check the rule. The remainder 1 is smaller than the divisor 7, so the answer is complete: 29 = 7 × 4 + 1.
Answer: 4 stickers each, with 1 left over (4 remainder 1).
Remember
- The remainder is what is left after making full groups.
- The remainder is always smaller than the divisor.
- Check with divisor × quotient + remainder = dividend.
- In a word problem, decide whether the remainder needs its own group.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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What is 13 ÷ 4?
Show answer
3 remainder 1. Three groups of 4 make 12, and 1 is left over.
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What is 17 ÷ 5?
Show answer
3 remainder 2.
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What is the remainder when 20 is divided by 6?
Show answer
2. 6 × 3 = 18, and 20 - 18 = 2.
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What is 25 ÷ 7?
Show answer
3 remainder 4.
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What is 29 ÷ 6?
Show answer
4 remainder 5.
Puzzle · Biscuit tin
Puzzle time
Fifty biscuits are shared equally among 7 hikers. Each hiker takes the same whole number of biscuits, and the leftover biscuits stay in the tin. How many biscuits stay in the tin?
Show a hint
Find the largest multiple of 7 that is not more than 50. How far is it from 50?
Show the answer
Answer: 1. 7 × 7 = 49, so each hiker takes 7 biscuits and 50 - 49 = 1 biscuit stays in the tin. The answer 6 would be almost a whole extra share.