Remainders and Whole-Number Word Problems | Grade 6 Math | LevelKite

When you divide, you split an amount into equal groups. Sometimes the split comes out even. Often it does not, and something is left over. You already know how to carry out the division. In this lesson you will decide what the leftover means in a real problem.

Start with $50 \div 12$ . These words name its parts:

The dividend is the amount you start with. Here it is $50$ .
The divisor is the number you divide by. Here it is $12$ . Every divisor in this lesson is a whole number greater than zero, because you cannot divide by zero.
A quotient is the result of a division. The whole quotient counts how many full times the divisor fits into the dividend. Here it is $4$ , because $12 \times 4 = 48$ fits inside $50$ but $12 \times 5 = 60$ does not.
The remainder is the amount of the dividend that is left over. Here it is $50 - 48 = 2$ .

One equation holds all four numbers together:

50 = 12 \times 4 + 2

Read the equation as a story. The division split $50$ into $4$ full groups of $12$ , with $2$ left over. Multiplying joins the full groups again: $12 \times 4 = 48$ . Adding the remainder puts back the part that was in no group: $48 + 2 = 50$ .

This works because multiplying by a nonzero number and dividing by that same number are inverse operations, which means each one undoes the other. Joining equal groups is the opposite of splitting into equal groups. Here the multiplication rebuilds only the full groups, so you add the remainder to get back everything you started with.

Two more facts hold for whole numbers whenever the divisor is not zero:

The remainder is always less than the divisor. If $12$ or more were left, you could make another full group.
When the split comes out even, there is no remainder, so there is nothing to add. For example, $48 \div 12 = 4$ exactly, and the check is simply $4 \times 12 = 48$ .

The numbers $4$ and $2$ are facts about the division. They are not yet the answer to a problem. A word problem asks a question, and the question tells you what to do with the remainder. Deciding what the remainder means in the situation is called interpreting the remainder.

The table shows four questions. Each one uses $50 \div 12$ , yet the four answers are all different.

One division, four questions: 50 divided by 12 makes 4 full groups with 2 left over

Question	What you do with the remainder	Answer
$50$ eggs go into cartons of $12$ . How many cartons can you fill?	Drop it. The $2$ extra eggs cannot fill a carton.	$4$ cartons
$50$ eggs go into cartons of $12$ . How many eggs are left over?	Report it. The remainder is the answer.	$2$ eggs
$50$ students ride in vans with $12$ seats each. How many vans are needed?	Round the quotient up. The $2$ extra students need a van too.	$5$ vans
$50$ feet of rope is cut into $12$ equal pieces. How long is each piece?	Share it. Rope can be cut, so the $2$ extra feet are split among the $12$ pieces.	$4\frac{2}{12} = 4\frac{1}{6}$ feet

Look again at the last row. Rope, juice, time and distance can all be split into smaller parts. For quantities like these, a leftover does not have to stay left over. You can share it equally, so each group gets a fraction of a unit.

Keep two cases apart:

No remainder. $48 \div 12 = 4$ exactly. Nothing is left, so there is nothing to share.
A fractional share. $50 \div 12$ leaves $2$ . When those $2$ units can be split among the $12$ groups, each group gets $\frac{2}{12}$ more. The quotient becomes $4\frac{2}{12}$ . Two twelfths make one sixth, so this equals $4\frac{1}{6}$ .

Each of these is an exact quotient: the full result of the division, with nothing left out. With a nonzero divisor, an exact quotient times the divisor equals the dividend: $4 \times 12 = 48$ , and $4\frac{1}{6} \times 12 = 4 \times 12 + \frac{1}{6} \times 12 = 48 + 2 = 50$ . The two cases are still different. In the first, there was never a leftover. In the second, there was a leftover, and the situation let you split it. Never write a fraction for things that cannot be split, such as students or vans.

Key concept

The question decides what the remainder means

When you divide a whole number by a divisor that is not zero, you get a whole quotient and sometimes a remainder, which is always less than the divisor. The question then tells you what to do with a remainder: drop it, round the quotient up, report it as the answer, or share it as a fraction.

Match your check to your answer:

Whole quotient with leftovers (things that cannot be split): divisor times whole quotient, plus remainder, equals the dividend.
Exact quotient (no remainder, or a fractional share included): quotient times divisor equals the dividend.
Rounded quotient (a quotient moved to a nearby whole number): quotient times divisor only lands near the dividend, so the check is approximate.

\text{dividend} = \text{divisor} \times \text{whole quotient} + \text{remainder} \qquad (\text{divisor} \neq 0)

Worked example

Every leftover needs a seat

A school is taking $437$ students on a field trip. Each bus has seats for $52$ students. How many buses does the school need so that every student gets a seat?

Name each quantity. The dividend is 437 students. The divisor is 52, the number of students each bus can seat, and it is not zero. You are splitting the students into groups of 52, so you divide.
$437 \div 52$
Find the whole quotient, which is the number of full buses. Eight buses seat 416 students. Nine buses would seat 468 students, which is more than 437. So exactly 8 buses are full.
$52 \times 8 = 416, \qquad 52 \times 9 = 468$
Find the remainder by subtracting the students on full buses from all the students. The result is 21, which is less than the divisor 52, as a remainder must be.
$437 - 416 = 21$
Check the division. Students cannot be split, so use the check for a whole quotient with leftovers: divisor times whole quotient, plus remainder, must equal the dividend.
$52 \times 8 + 21 = 416 + 21 = 437$
Interpret the remainder. The 21 leftover students still need seats, so the school needs one more bus than the whole quotient. This rounds the quotient up.
$8 + 1 = 9$
Check the rounded answer. Because 9 is a rounded quotient, 9 times 52 lands near 437 but does not equal it. This approximate check still shows the answer is sensible: 9 buses have enough seats, and 8 buses do not.
$52 \times 9 = 468, \qquad 416 < 437 < 468$

Answer

The school needs $9$ buses. Eight buses are full, and the ninth bus carries the other $21$ students.

Now use remainders inside longer word problems. Many problems need more than one step, so decide which operation fits each step. Think about equal groups:

Multiply when you know the number of equal groups and the size of each group, and you want the total.
Divide when you know the total, and you want the number of groups or the size of each group. The number you divide by must not be zero.

Write a unit next to every number. A unit with the word per, such as beads per bracelet, means for each one. It tells you the size of every group. Units show what each number counts, so they also show what your answer counts.

Worked example

Divide, then multiply

A craft club has $950$ beads. Each bracelet uses $24$ beads. The club makes as many bracelets as it can and sells each one for $16$ dollars. How much money does the club collect, and how many beads are left over?

Find how many bracelets the beads make. The total, 950 beads, is split into groups of 24 beads per bracelet. The divisor 24 is not zero, so you divide. Start with 30 bracelets. They use 720 beads and leave 230.
$24 \times 30 = 720, \qquad 950 - 720 = 230$
Make more bracelets from the 230 beads. Nine bracelets use 216 beads. A tenth would raise that to 240, which is more than 230, so stop at 9. The whole quotient is 30 plus 9, which is 39. The remainder is 14, which is less than the divisor 24.
$24 \times 9 = 216, \qquad 230 - 216 = 14, \qquad 30 + 9 = 39$
Check the division. Beads are not split, so use the check for a whole quotient with leftovers: divisor times whole quotient, plus remainder, must equal the dividend.
$24 \times 39 + 14 = 936 + 14 = 950$
Interpret the remainder. Only complete bracelets can be sold, so the 14 extra beads do not make a bracelet, even though 14 is more than half of 24. The club makes 39 bracelets. The beads left over are the remainder itself.
Find the money collected. You know the number of equal groups, 39 bracelets, and the size of each group, 16 dollars per bracelet, so you multiply. Split 16 into 10 and 6: 39 groups of 16 are 39 groups of 10 together with 39 groups of 6.
$39 \times 16 = 39 \times 10 + 39 \times 6 = 390 + 234 = 624$

Answer

The club collects $624$ dollars from $39$ bracelets, and $14$ beads are left over.

Worked example

Multiply, then divide and share

A camp cook makes $18$ batches of lemonade. Each batch is $15$ liters. All of the lemonade is poured equally into $40$ coolers. How many liters of lemonade go into each cooler?

Multiply to find the total amount: 18 equal batches of 15 liters per batch. Split 15 into 10 and 5, as in the last example.
$18 \times 15 = 18 \times 10 + 18 \times 5 = 180 + 90 = 270$
Divide the 270 liters among 40 coolers. The divisor 40 is not zero. Six liters per cooler use 240 liters, and 7 liters per cooler would need 280 liters. So the whole quotient is 6, and 30 liters remain.
$40 \times 6 = 240, \qquad 270 - 240 = 30$
Interpret the remainder. Lemonade can be split, so share the 30 leftover liters among the 40 coolers. Each cooler gets 30 fortieths of a liter more. Ten fortieths make one fourth, so 30 fortieths equal 3 fourths.
$\frac{30}{40} = \frac{3}{4}$
Each cooler gets the 6 whole liters and the fractional share. Add them to get the exact quotient.
$6 + \frac{3}{4} = 6\frac{3}{4}$
Check. The quotient is exact, so quotient times divisor must equal the dividend. Multiply the whole part and the fraction part by 40, then add.
$40 \times 6 + 40 \times \frac{3}{4} = 240 + 30 = 270$

Answer

Each cooler gets $6\frac{3}{4}$ liters of lemonade, and none is left over.

Takeaway. A division with a leftover is not finished until you read the question again. When the divisor is not zero, divisor times whole quotient, plus remainder, equals the dividend. The question then tells you what the remainder means:

Drop it when only full groups count.
Round the quotient up when every leftover needs a place.
Report it when the leftover is what the question asks for.
Share it as a fraction when the quantity can be split. This is not the same as having no remainder.

In any word problem, label your units and use equal groups to choose between multiplying and dividing. Finish with the check that matches your answer.