Round Decimals in Word Problems and Fix Mistakes | Grade 5 Math | LevelKite

You already know how to round a decimal to any place. In this lesson, the problem will not always name the place. You will read the situation to find it.

Two terms will help. The exact value is the number before you round it. The rounding place is the place you round to, such as the ones place or the hundredths place.

Rounding swaps the exact value for a nearby number that is easier to use. How near it must be depends on the situation. So before you round, ask: What is the smallest unit this situation asks for? That unit tells you the rounding place.

Money is a good example. There are $100$ cents in a dollar, so $1$ cent is one hundredth of a dollar, or $0.01$ dollar. When a price is written in dollars, rounding to the nearest cent means rounding to the hundredths place.

In the table below, look at the first two rows. The same exact price gives two different rounded prices, because the two situations ask for different places.

How the words of a situation point to a rounding place

The situation says	Smallest unit	Rounding place	Example
to the nearest cent	$1$ cent, or $0.01$ dollar	hundredths	$5.128$ dollars rounds to $5.13$ dollars
to the nearest dollar	$1$ dollar	ones	$5.128$ dollars rounds to $5$ dollars
to the nearest whole kilogram	$1$ kilogram	ones	$18.64$ kilograms rounds to $19$ kilograms
to the nearest tenth of a second	$0.1$ second	tenths	$9.584$ seconds rounds to $9.6$ seconds

A rounded value is close to the exact value, but it is usually not equal to it. Some problems ask which way the number moved: up or down.

To see this, think about where the exact value sits. Take $5.128$ and the hundredths place. The number $5.128$ is more than $5.12$ and less than $5.13$ . These two numbers are its neighbors: the closest numbers below it and above it that have no digits after the rounding place.

5.12 < 5.128 < 5.13

Any exact value with a digit other than $0$ after the rounding place sits between two neighbors like this. Rounding moves it to one of them.

Round up and you land on the neighbor above. The rounded value is more than the exact value.
Round down and you land on the neighbor below. The rounded value is less than the exact value.

If every digit after the rounding place is $0$ , nothing moves. The rounded value equals the exact value, as when $4.30$ rounds to $4.3$ at the tenths place.

You can check by comparing the two numbers. Write $5.13$ as $5.130$ so that both numbers show thousandths. Both have $5$ ones, and $130$ thousandths is more than $128$ thousandths. So $5.13$ is more than $5.128$ .

Worked example

Round a price to the nearest cent

A shop works out the exact price of a piece of ribbon as $2.374$ dollars. The shop charges to the nearest cent. How much does the shop charge? Is that more or less than the exact price?

Find the rounding place from the situation. The price is in dollars, and 1 cent is one hundredth of a dollar. So the nearest cent means the hundredths place.
$1 \text{ cent} = 0.01 \text{ dollar}$
Find the hundredths digit in the exact price and the digit just to its right. The hundredths digit is 7. The digit to its right, in the thousandths place, is 4. Only this digit decides which way to round.
$2.3\underline{7}4$
The deciding digit 4 is less than 5. That means 2.374 is nearer to the neighbor below, 2.37, than to the neighbor above, 2.38. Round down: keep the 7 hundredths and drop the 4.
$2.374 \text{ rounds to } 2.37$
Compare the rounded price with the exact price. Write 2.37 as 2.370 so both numbers show thousandths. Both have 2 ones, and 370 thousandths is less than 374 thousandths.
$2.370 < 2.374$

Answer

The shop charges $2.37$ dollars, which is a little less than the exact price of $2.374$ dollars.

Worked example

Two rounded times from one exact time

A timer shows that a toy car takes $2.748$ seconds to roll down a ramp. Call this the exact time. The class data table lists times to the nearest hundredth of a second. The class poster shows times to the nearest tenth of a second. What time goes in the table, and what time goes on the poster? Is each one more or less than the exact time?

Start with the data table. It needs the hundredths place. In 2.748 the hundredths digit is 4 and the digit just to its right is 8. Since 8 is 5 or more, round up: 4 hundredths become 5 hundredths, and the 8 is dropped.
$2.7\underline{4}8 \text{ rounds to } 2.75$
Compare with the exact time. Write 2.75 as 2.750. Both numbers have 2 ones, and 750 thousandths is more than 748 thousandths. Rounding up gave a time that is more than the exact time.
$2.750 > 2.748$
Now the poster. It needs the tenths place. Go back to the exact time, 2.748, because the table time 2.75 has already been moved up. The tenths digit is 7 and the digit just to its right is 4. Since 4 is less than 5, round down: keep 7 tenths and drop the digits after it.
$2.\underline{7}48 \text{ rounds to } 2.7$
Compare with the exact time. Write 2.7 as 2.700. Both numbers have 2 ones, and 700 thousandths is less than 748 thousandths. Rounding down gave a time that is less than the exact time.
$2.700 < 2.748$

Answer

The data table lists $2.75$ seconds, which is more than the exact time. The poster shows $2.7$ seconds, which is less than the exact time.

Key concept

The situation picks the place. The exact value picks the direction.

To round in a word problem, follow three moves.

Find the place. Read the situation. The smallest unit it asks for tells you the rounding place: the nearest cent means hundredths of a dollar, and the nearest whole unit means ones.
Use the exact value once. Look only at the digit just right of the rounding place. If it is $5$ or more, round up. If it is less than $5$ , round down.
Compare. Rounding up gives a value that is more than the exact value. Rounding down gives a value that is less, unless every digit you drop is $0$ . Then the two values are equal.

Now you are the checker. Here is a test that works whenever the exact value sits between two neighbors. A correct answer is always one of those two neighbors. It is the nearer one, or the neighbor above when the exact value is exactly halfway between them.

When an answer fails the test, look for one of three common mistakes.

Wrong place. The work rounded to a different place than the task asked for.
Digits to the left changed. Rounding down keeps the digits to the left of the rounding place. Rounding up adds $1$ in the rounding place only. So digits to the left change in just one case: the digit in the rounding place is $9$ and it rounds up. Then $9 + 1$ makes $10$ : write $0$ and add $1$ to the next place on the left, as when $2.96$ rounds to $3.0$ at the tenths place.
Two steps. The work rounded a number that was already rounded, instead of using the exact value once.

Three rounding mistakes and how to fix them

Task	Mistaken answer	What went wrong	Correct answer
Round $6.283$ to the nearest tenth	$6.28$	Wrong place. The work rounded to hundredths. The neighbors at the tenths place are $6.2$ and $6.3$ .	$6.3$ , because the hundredths digit is $8$
Round $5.27$ to the nearest tenth	$6.3$	A digit to the left changed. Rounding up adds $1$ tenth only, so the $5$ ones stay.	$5.3$ , because the hundredths digit is $7$
Round $3.446$ to the nearest tenth	$3.5$	Two steps. The work went from $3.446$ to $3.45$ to $3.5$ instead of using the exact value once.	$3.4$ , because the hundredths digit is $4$

Worked example

Find and correct a rounding mistake

A dog has a mass of $23.482$ kilograms. A form asks for the mass to the nearest whole kilogram. Jada writes $23.5$ kilograms. Find Jada's mistake and correct it.

Name the place the situation needs. The form asks for whole kilograms, so the rounding place is the ones place. At the ones place, the neighbors of the exact mass are 23 below it and 24 above it.
$23 < 23.482 < 24$
Test Jada's answer. A correct answer must be one of the two neighbors, and 23.5 is neither 23 nor 24. Her answer still has a digit in the tenths place, so she rounded to the nearest tenth. That is the wrong place.
Correct the work from the exact mass. Do not round 23.5 again, because that would be rounding in two steps and would give 24. In 23.482 the ones digit is 3 and the digit just to its right is 4. Since 4 is less than 5, round down: keep the 3 ones and drop the digits after the decimal point.
$2\underline{3}.482 \text{ rounds to } 23$
Check the new answer. The halfway point between 23 and 24 is 23.5. The exact mass 23.482 is below 23.5, so 23 is the nearer neighbor. It is also less than the exact mass, which fits rounding down.
$23 < 23.482 < 23.5$

Answer

Jada rounded to the tenths place, but the form needs the ones place. The correct mass for the form is $23$ kilograms, which is less than the exact mass of $23.482$ kilograms.

Here is the big idea. In a word problem, the situation tells you the rounding place: the nearest cent means hundredths of a dollar, and the nearest whole unit means ones. The exact value tells you which way to round, and you use it only once. After you round, compare. Rounding up gives more than the exact value. Rounding down gives less, unless you dropped only zeros. When you check someone's work, find the two neighbors that the exact value sits between. A correct answer is always one of them.