Round Decimal Quotients and Solve Word Problems | Grade 5 Math | LevelKite

You already know how to divide decimals. But some divisions do not come out neatly.

Take $7.4 \div 3$ . The dividend is $7.4$ , the number you divide. The divisor is $3$ , the number you divide by. The answer is the quotient.

When you work out $7.4 \div 3$ , the division never ends. However far you go, there is always a little left over. Other divisions do end, but they give more digits than you need.

In both cases you can round the quotient. To round a number means to replace it with the nearest number that ends at a chosen place, such as the tenths place. In this lesson you will round quotients, solve word problems, and check your answers.

Say you want $7.4 \div 3$ to the nearest tenth. If you divide as far as the tenths place, you get $2.4$ with a little left over. So the quotient lies between $2.4$ and $2.5$ . Which one is it nearer to? Halfway between them is $2.45$ . To see which side of halfway the quotient is on, you need one more digit: the hundredths digit.

That gives you a plan for rounding a quotient:

Divide one place further to the right than the place you need. This gives you one extra digit.
Look at the extra digit. If it is 5 or more, the quotient is at the halfway point or past it, so round up. If it is 4 or less, the quotient is below halfway, so round down.
Drop the extra digit.

For $7.4 \div 3$ , rounding up gives $2.5$ and rounding down gives $2.4$ . You can stop dividing once you have the extra digit. The digits after it are too small to change which way you round.

Worked example

Round a quotient to the nearest tenth

Find $7.4 \div 3$ . Round the quotient to the nearest tenth.

You need tenths, so plan to divide one place further, as far as the hundredths place. The hundredths digit will tell you which way to round.
Divide the ones and the tenths. 7 ones divided by 3 is 2 ones, with 1 one left over. That 1 one is 10 tenths. With the 4 tenths in 7.4, you have 14 tenths. 14 tenths divided by 3 is 4 tenths, with 2 tenths left over. The dots show that more digits follow.
$7.4 \div 3 = 2.4\ldots$
Now find the extra digit. The 2 tenths left over are the same as 20 hundredths. 20 hundredths divided by 3 is 6 hundredths, with 2 hundredths left over. You have reached the hundredths place, so stop dividing.
$7.4 \div 3 = 2.46\ldots$
Round to the nearest tenth. The extra digit is 6, which is 5 or more. So the quotient is past 2.45, the halfway point between 2.4 and 2.5. Round up to 2.5 and drop the extra digit.
$2.46\ldots \text{ rounds to } 2.5$

Answer

To the nearest tenth, $7.4 \div 3$ is $2.5$ .

The same plan works for other places

Round to the nearest	Divide as far as the	Division so far	Extra digit	Rounded quotient
whole number	tenths place	$8.9 \div 3 = 2.9\ldots$	9, so round up	$3$
tenth	hundredths place	$7.4 \div 3 = 2.46\ldots$	6, so round up	$2.5$
hundredth	thousandths place	$4.1 \div 6 = 0.683\ldots$	3, so round down	$0.68$

Now put division to work in word problems. In the problems here, you split a total into equal parts. Call each equal part a share. A problem can ask you for one of two things:

The size of a share. You know the total and the number of shares. Divide the total by the number of shares to find how big one share is.
The number of shares. You know the total and the size of one share. Divide the total by the size of one share to find how many shares there are.

Either way, the total is the dividend. So before you divide, ask yourself: Do I know how many shares there are, or how big each share is?

One rope, two different questions

The problem	What you know	What you find	Number sentence
You cut a 7.5-meter rope into 5 equal pieces. How long is each piece?	Total: 7.5 meters. Number of shares: 5.	The size of one share	$7.5 \div 5 = 1.5$ , so each piece is 1.5 meters long.
You cut a 7.5-meter rope into pieces that are 1.5 meters long. How many pieces do you get?	Total: 7.5 meters. Size of one share: 1.5 meters.	The number of shares	$7.5 \div 1.5 = 5$ , so you get 5 pieces.

Worked example

Find the number of shares

A baker has $7.2$ kilograms of flour. Each batch of bread uses $0.6$ kilograms of flour. How many batches can the baker make?

Decide what the problem asks for. You know the total: 7.2 kilograms of flour. You know the size of one share: 0.6 kilograms for each batch. You need the number of shares, so divide the total by the size of one share.
$7.2 \div 0.6$
Make the divisor a whole number, as you have done before. Multiply both numbers by 10. Both become 10 times as large, so the quotient stays the same.
$7.2 \div 0.6 = 72 \div 6$
Divide. 6 times 12 is 72, so 72 divided by 6 is 12. The division ends with nothing left over, so you do not need to round.
$72 \div 6 = 12$

Answer

The baker can make $12$ batches of bread.

How can you check a quotient? Every division asks a multiplication question. $7.2 \div 0.6$ asks: what number times $0.6$ makes $7.2$ ? So you can check a quotient by multiplying it by the divisor. The answer to a multiplication is the product. Compare the product with the dividend.

What you should see depends on the kind of quotient:

An exact quotient is the full answer. The division ended with nothing left over, and you kept every digit. When you multiply it by the divisor, the product equals the dividend.
For a rounded quotient, you rounded some digits off. So it is a little more or a little less than the exact quotient. When you multiply it by the divisor, the product does not equal the dividend. It should be close to the dividend.

If the product is far from the dividend, go back and look for a mistake.

Checking the two quotients you have found

Division	Quotient	Exact or rounded?	Quotient times divisor	Compare with the dividend
$7.2 \div 0.6$	$12$	Exact	$12 \times 0.6 = 7.2$	The product equals $7.2$ .
$7.4 \div 3$	$2.5$	Rounded	$2.5 \times 3 = 7.5$	The product is close to $7.4$ , not equal to it.

Key concept

Divide one place further, round, then check

To round a quotient to a place, divide one place further than that place. Use the extra digit to round, then drop it.

To check, multiply your quotient by the divisor and compare the product with the dividend. For an exact quotient, the product equals the dividend. For a rounded quotient, the product should be close to the dividend, not equal to it.

Worked example

Share, round, and check

Four friends share $9.3$ kilograms of apples equally. How many kilograms of apples does each friend get? Round your answer to the nearest tenth of a kilogram.

Decide what the problem asks for. You know the total: 9.3 kilograms of apples. You know the number of shares: 4, one for each friend. You need the size of one share, so divide the total by the number of shares.
$9.3 \div 4$
You need tenths, so divide as far as the hundredths place. 9 ones divided by 4 is 2 ones, with 1 one left over. Now you have 13 tenths. 13 tenths divided by 4 is 3 tenths, with 1 tenth left over. Now you have 10 hundredths. 10 hundredths divided by 4 is 2 hundredths, with 2 hundredths left over. Stop here.
$9.3 \div 4 = 2.32\ldots$
Round to the nearest tenth. The extra digit is 2, which is 4 or less. So the quotient is below 2.35, the halfway point between 2.3 and 2.4. Round down to 2.3 and drop the extra digit.
$2.32\ldots \text{ rounds to } 2.3$
Check by multiplying the quotient by the divisor. 2.3 is a rounded quotient, so the product should be close to the dividend, 9.3, not equal to it. 4 shares of 2.3 kilograms make 9.2 kilograms. That is only 0.1 away from 9.3, so the answer makes sense.
$2.3 \times 4 = 9.2$

Answer

Each friend gets about $2.3$ kilograms of apples.

Here is what to remember. To round a quotient, divide one place further than the place you need, and let the extra digit tell you which way to round. In a word problem, first decide whether you need the size of a share or the number of shares. Then check by multiplying your quotient by the divisor. An exact quotient gives a product equal to the dividend. A rounded quotient gives a product that is close to the dividend, not equal to it.