Divide by One-Digit Numbers and Interpret Remainders | Grade 5 Math | LevelKite

Some numbers do not split into equal groups with nothing left over. Say you share 17 stickers equally among 5 friends. Each friend can get 3 stickers, because $5 \times 3 = 15$ . Giving 4 each would take 20 stickers, and you have only 17. So 15 stickers are shared and 2 are left.

The amount left over is called the remainder. In this lesson you will divide bigger numbers one place at a time, check your answers, and decide what the remainder means in a word problem.

Here is the sticker sharing written as a number sentence. The R stands for remainder.

17 \div 5 = 3 \text{ R } 2

The four numbers in the sticker division

Number	Its name	What it is	In the story
$17$	dividend	the number you divide	17 stickers to share
$5$	divisor	the number you divide by	5 friends
$3$	quotient	the whole-number result	3 stickers for each friend
$2$	remainder	the amount left over	2 stickers left

Key concept

Equal groups plus leftovers make the whole

Division puts a number into equal groups. The quotient is the whole-number result. The remainder is whatever does not fit into the equal groups. The remainder is always less than the divisor. If it is not, you can still divide some more.

This gives you a way to check any division. Multiply the quotient by the divisor to rebuild the equal groups. Add the remainder to put the leftovers back. You must get the dividend, the number you started with.

For the stickers: $3 \times 5 + 2 = 15 + 2 = 17$ .

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

Bigger numbers work the same way. You just do not share them all at once. You share them place by place.

Think of $437$ as 4 hundreds, 3 tens and 7 ones. To divide it by $3$ , picture 3 equal groups. Start with the biggest place, so that anything left can move down to the next place.

Share as many as you can in that place. Your multiplication and division facts tell you how many.
Trade each one that is left for 10 of the next smaller place. The amount stays the same, because 1 hundred equals 10 tens and 1 ten equals 10 ones.
Add what was already in that smaller place, then share again.

What is left after you share the ones is the remainder. A number with more digits works the same way. It just starts at a bigger place.

Worked example

Divide place by place

Find $437 \div 3$ . Write the quotient and the remainder, then check your answer.

Start with the hundreds. Share 4 hundreds among 3 equal groups. Each group gets 1 hundred. It cannot get 2, because that would take 6 hundreds. Giving out 3 hundreds leaves 1 hundred.
$4 \text{ hundreds} \div 3 = 1 \text{ hundred, } 1 \text{ hundred left}$
Trade the leftover hundred for 10 tens, because 1 hundred equals 10 tens. Add the 3 tens that were already there to get 13 tens. Share them: each group gets 4 tens, since 3 times 4 is 12. That leaves 1 ten.
$13 \text{ tens} \div 3 = 4 \text{ tens, } 1 \text{ ten left}$
Trade the leftover ten for 10 ones. Add the 7 ones that were already there to get 17 ones. Share them: each group gets 5 ones, since 3 times 5 is 15. That leaves 2 ones. There is no smaller place to trade into, so 2 is the remainder.
$17 \text{ ones} \div 3 = 5 \text{ ones, } 2 \text{ ones left}$
Put the shares together. Each group has 1 hundred, 4 tens and 5 ones, which is 145. That is the quotient. The remainder is 2. It is less than the divisor, 3, so you cannot share any more.
$437 \div 3 = 145 \text{ R } 2$
Check. Multiply the quotient by the divisor to rebuild the 3 equal groups. Then add the remainder to put the leftovers back. You land on 437, the number you started with, so the division is correct.
$145 \times 3 + 2 = 435 + 2 = 437$

Answer

When you divide $437$ by $3$ , the quotient is $145$ and the remainder is $2$ , so $437 \div 3 = 145 \text{ R } 2$ .

Dividing 437 by 3: each place gives one digit of the quotient

Place	You share	Each group gets	Quotient digit	Left over
Hundreds	4 hundreds	1 hundred	$1$	1 hundred, traded for 10 tens
Tens	$10 + 3 = 13$ tens	4 tens	$4$	1 ten, traded for 10 ones
Ones	$10 + 7 = 17$ ones	5 ones	$5$	2 ones, the remainder

In a word problem, dividing is only part of the job. Here is why. 26 children line up for a boat ride, and each boat holds 4 children. The division is $26 \div 4 = 6 \text{ R } 2$ . In the story, that means 6 full boats, with 2 children still waiting.

So what is the answer? It depends on the question. The question tells you what to do with the remainder:

Drop the remainder when the question asks only about full groups.
Round the quotient up when the leftovers need a group too. The answer is then 1 more than the quotient.
Use the remainder as the answer when the question asks what is left over.

If the remainder is 0, nothing is left over, so there is nothing to decide.

The table shows all three choices for the boat ride.

The boat ride: one division, three different answers

The question	What to do	Why	Answer
How many boats leave completely full?	Drop the remainder.	The 2 waiting children cannot fill a boat.	6 boats
How many boats are needed so that every child gets a ride?	Round the quotient up.	The 2 waiting children need one more boat: $6 + 1 = 7$ .	7 boats
How many children are still waiting after the full boats leave?	Use the remainder as the answer.	The remainder counts the children who are left.	2 children

Worked example

Full bags and leftover rolls

A baker has $347$ rolls. She packs them in bags of $8$ . How many full bags does she make, and how many rolls are left over?

Write the division. The rolls go into equal groups of 8, so you divide 347 by 8. The quotient will count bags. The remainder will count rolls.
$347 \div 8$
Start with the hundreds. 3 hundreds divided by 8 gives 0 hundreds, because 3 is less than 8. A number does not start with 0, so write nothing yet. Trade the 3 hundreds for 30 tens and add the 4 tens already there to get 34 tens. 8 times 4 is 32, which is as close to 34 as you can get without going over. The quotient gets 4 tens, and 2 tens are left.
$34 \text{ tens} \div 8 = 4 \text{ tens, } 2 \text{ tens left}$
Trade the 2 leftover tens for 20 ones. Add the 7 ones already there to get 27 ones. 8 times 3 is 24, so the quotient gets 3 ones, and 3 ones are left.
$27 \text{ ones} \div 8 = 3 \text{ ones, } 3 \text{ ones left}$
The quotient is 4 tens and 3 ones, which is 43. The remainder is 3. Check by multiplying the quotient by the divisor and adding the remainder. You get back 347, so the division is correct.
$43 \times 8 + 3 = 344 + 3 = 347$
Go back to the question. It asks two things. For full bags, use the quotient and drop the remainder, because 3 rolls cannot fill a bag of 8. For rolls left over, the remainder is the answer. Name the unit, the thing each number counts: 43 counts bags and 3 counts rolls.
$347 \div 8 = 43 \text{ R } 3$

Answer

The baker makes $43$ full bags, and $3$ rolls are left over.

Worked example

Enough cartons for every egg

A farm collects $1{,}225$ eggs. Each carton holds $6$ eggs. How many cartons does the farm need so that every egg is packed?

The eggs go into equal groups of 6, so divide 1,225 by 6. Start with the thousands. 1 thousand divided by 6 gives 0 thousands, because 1 is less than 6. Trade it for 10 hundreds. Add the 2 hundreds already there to get 12 hundreds. 6 times 2 is 12, so the quotient gets 2 hundreds and none are left.
$12 \text{ hundreds} \div 6 = 2 \text{ hundreds, } 0 \text{ left}$
Now the tens. You have only 2 tens. 2 tens divided by 6 gives 0 tens, because 2 is less than 6. The quotient has already started, so write that 0. It holds the tens place, so the 2 you wrote before still means 2 hundreds. The 2 tens are left over.
$2 \text{ tens} \div 6 = 0 \text{ tens, } 2 \text{ tens left}$
Trade the 2 leftover tens for 20 ones. Add the 5 ones already there to get 25 ones. 6 times 4 is 24, so the quotient gets 4 ones and 1 one is left. The quotient is 2 hundreds, 0 tens and 4 ones, which is 204. The remainder is 1.
$1{,}225 \div 6 = 204 \text{ R } 1$
Check. Multiply the quotient by the divisor and add the remainder. You get back the 1,225 eggs you started with, so the division is correct.
$204 \times 6 + 1 = 1{,}224 + 1 = 1{,}225$
Go back to the question. It asks for enough cartons to pack every egg. 204 full cartons hold 1,224 eggs. The 1 egg that is left still needs a carton, so round the quotient up by 1.
$204 + 1 = 205$

Answer

The farm needs $205$ cartons. $204$ of them are full, and the last one holds $1$ egg.

Here is the idea to keep. Dividing by a one-digit number is sharing place by place. Start at the biggest place. Share what you can, then trade each leftover for 10 of the next smaller place. Once the quotient has started, every place gets a digit, even when that digit is 0. What is left at the end is the remainder, and it is always less than the divisor.

Check every division: quotient times divisor, plus remainder, gives back the number you started with.

In a word problem, read the question again after you divide. Drop the remainder for full groups. Round the quotient up when the leftovers need a group too. Use the remainder itself when the question asks what is left over. Then name the unit, the thing your answer counts, such as bags, cartons or children.