Check and Solve Division Word Problems with Remainders | Grade 5 Math | LevelKite

How do you know a division answer is right? And which number really answers the question? Start with a small story.

You have $100$ stickers and you put $12$ on each page. You fill $8$ pages, and $4$ stickers are left over. As a division, that is $100 \div 12 = 8$ R $4$ . The R stands for remainder, the amount left over.

Each number has a name:

The dividend is the total you start with: $100$ stickers.
The divisor is the number you divide by: $12$ stickers on a page.
The quotient is how many times the divisor fits into the dividend: $8$ times, for $8$ full pages.
The remainder is what is left after that: $4$ stickers, too few to fill another page.

All four are whole numbers. Whole numbers are $0$ , $1$ , $2$ , $3$ and so on.

Multiplication puts the total back together. $8$ pages of $12$ stickers are $96$ stickers. Add the $4$ left over and you are back at $100$ . Here is that as one number sentence, with each number's name under it:

\underbrace{8}_{\text{quotient}} \times \underbrace{12}_{\text{divisor}} + \underbrace{4}_{\text{remainder}} = \underbrace{100}_{\text{dividend}}

Key concept

The multiplication check

Division splits a total into equal groups and a leftover. Multiplication builds the groups again, so it can test your answer.

When you divide whole numbers, a quotient and a remainder are right when both of these are true, and wrong when either one fails:

Quotient times divisor, plus remainder, gives back the dividend.
The remainder is smaller than the divisor. If it is not, at least one more group still fits.

When nothing is left over, the remainder is $0$ , and quotient times divisor equals the dividend.

\text{quotient} \times \text{divisor} + \text{remainder} = \text{dividend}

Worked example

Check a division with multiplication

Jo writes $620 \div 17 = 36$ R $8$ . Use multiplication to check whether she is right.

Multiply the quotient by the divisor. This builds the 36 full groups of 17 again. Split 17 into 10 and 7 to make the multiplying easier: 36 × 10 = 360 and 36 × 7 = 252.
$36 \times 17 = 360 + 252 = 612$
Add the remainder, because the leftover is part of the total too. The result must equal the dividend, 620. It does.
$612 + 8 = 620$
Compare the remainder with the divisor. The remainder 8 is smaller than 17, so no more groups of 17 fit.
$8 < 17$

Answer

Both tests pass, so Jo is right: $620 \div 17 = 36$ R $8$ .

The check can also show that a quotient is wrong, even when you have no remainder to look at. Multiply the quotient by the divisor. Then compare the result with the dividend.

If the result is more than the dividend, the quotient is too big. The groups would need more than the total you have.
If it is not, subtract the result from the dividend to find the leftover. If the leftover is as big as the divisor, or bigger, the quotient is too small. One more group still fits.
If the leftover is smaller than the divisor, the quotient is right. The leftover is the remainder, and it may be $0$ .

Three students each give a different quotient for $850 \div 32$ . Only one can be right.

Three quotients for 850 ÷ 32: which can be right?

Quotient given	Quotient × 32	Leftover from 850	Can it be right?
$25$	$25 \times 32 = 800$	$850 - 800 = 50$	No. $50$ is not smaller than $32$ . One more group fits, so $25$ is too small.
$26$	$26 \times 32 = 832$	$850 - 832 = 18$	Yes. $18$ is smaller than $32$ , so $850 \div 32 = 26$ R $18$ .
$27$	$27 \times 32 = 864$	You cannot subtract: $864$ is more than $850$ .	No. $27$ groups need more than the total, so $27$ is too big.

Now put division into a story. A story calls for division when it splits a total into equal groups and you know either the size of each group or the number of groups. Before you divide, name the total and ask what the divisor stands for. That tells you what the quotient will count.

Here is $850 \div 32 = 26$ R $18$ in two stories:

Packing. You pack $850$ pencils $32$ to a box. The divisor is the size of each group, so the quotient counts groups: $26$ full boxes.
Sharing. You share $850$ pencils equally among $32$ classes. The divisor is the number of groups, so the quotient counts how many are in each group: $26$ pencils for each class.

In both stories the remainder counts the same kind of thing as the dividend: $18$ pencils left over. The check is the same for both: $26 \times 32 + 18 = 832 + 18 = 850$ .

Worked example

Say what the quotient counts

A teacher has $1{,}300$ beads to share equally among $28$ students. Each student gets as many beads as possible. How many beads does each student get?

Name the total and the groups. The total is 1,300 beads. The teacher shares them into 28 equal groups, one for each student, so you divide. The divisor 28 is the number of groups, so the quotient will count the beads for each student.
$1{,}300 \div 28$
Divide the way you already know. Giving each student 40 beads uses 40 × 28 = 1,120 beads and leaves 180. Giving each student 6 more uses 6 × 28 = 168 beads and leaves 12. Twelve beads cannot give all 28 students one more, so stop. Each student has 40 + 6 = 46 beads.
$1{,}300 \div 28 = 46 \text{ R } 12$
Check with multiplication. 46 beads for each of 28 students make 1,120 + 168 = 1,288 beads. Add the 12 left over to get back to the dividend, 1,300. The remainder 12 is smaller than the divisor 28. Both tests pass.
$46 \times 28 + 12 = 1{,}288 + 12 = 1{,}300$
Say what each number counts. The quotient 46 counts the beads for each student. The remainder 12 counts beads left over. The question asks for each student's equal share, so the answer is the quotient.

Answer

Each student gets $46$ beads. The $12$ beads left over are too few to give every student one more.

A division gives you two numbers: the quotient and the remainder. The question tells you how to use them. After you divide and check, read the question again and ask, What happens to the leftover?

You set the leftover aside. The question asks how many full groups you can make, or how many each group gets in an equal share. The answer is the quotient.
The leftover needs a group too. The question asks how many groups you need to hold everything. If the remainder is not $0$ , the answer is the next whole number, which is the quotient plus $1$ . If the remainder is $0$ , the quotient is already enough.
The question asks about the leftover. The answer is the remainder.

Try it with one division and three questions. A baker has $530$ muffins and boxes that hold $24$ muffins each. Dividing gives $530 \div 24 = 22$ R $2$ . The check agrees: $22 \times 24 + 2 = 528 + 2 = 530$ , and $2$ is smaller than $24$ .

One division, three questions: 530 muffins, boxes of 24

Question	What happens to the 2 leftover muffins	Answer
How many boxes can the baker fill completely?	You set them aside. They do not fill a box. Use the quotient.	$22$ boxes
How many boxes does the baker need to pack every muffin?	They need a box too. Use the next whole number.	$22 + 1 = 23$ boxes
How many muffins are left after the baker fills as many boxes as possible?	They are what the question asks for. Use the remainder.	$2$ muffins

Worked example

When the leftover needs one more group

A ferry can carry up to $42$ passengers on each trip. On a busy day, $1{,}250$ people have tickets. What is the smallest number of trips the ferry must make so that everyone gets a ride?

Name the total and the groups. The total is 1,250 people. To make as few trips as possible, the ferry fills each trip with 42 people. You need to know how many groups of 42 are in 1,250, so you divide. The divisor is the size of each group, so the quotient will count full trips.
$1{,}250 \div 42$
Divide the way you already know. First, 20 full trips carry 20 × 42 = 840 people and leave 410. Then 9 more full trips carry 9 × 42 = 378 people and leave 32. That makes 20 + 9 = 29 full trips, with 32 people left.
$1{,}250 \div 42 = 29 \text{ R } 32$
Check with multiplication. 29 trips of 42 people carry 840 + 378 = 1,218 people. Add the 32 people left to get back to the dividend, 1,250. The remainder 32 is smaller than the divisor 42. Both tests pass.
$29 \times 42 + 32 = 1{,}218 + 32 = 1{,}250$
Read the question again. It asks how many trips will carry everyone. The 32 people left over do not fill a trip, but they need a trip too. The remainder is not 0, so the answer is the next whole number after the quotient.
$29 + 1 = 30$
Test the answer against the story. 30 trips have room for 30 × 42 = 1,260 people, which is enough for 1,250. With only 29 trips, 32 people would be left behind.
$30 \times 42 = 1{,}260$

Answer

The smallest number of trips is $30$ : $29$ full trips and one more trip for the last $32$ people.

Here is the idea to keep. A division word problem is not finished when you find a quotient. Finish it in three moves:

Solve. Divide, and say what the quotient and the remainder count.
Check. Quotient × divisor + remainder must give back the dividend, and the remainder must be smaller than the divisor.
Choose. Read the question again. Use the quotient when it asks for full groups or an equal share. Use the next whole number when it asks how many groups will hold everything and the remainder is not $0$ . Use the remainder when it asks about the leftover.

Next, you will use these moves in multi-step word problems that mix multiplication and division. You will also find and fix mistakes in solved problems.