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B1.5 · Round terminating and repeating decimals

Learn to round terminating and repeating decimals through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Number

Use place value and the digit just to the right

A decimal can show a value exactly, or it can show a value with many digits. Rounding gives a nearby value with fewer decimal places or fewer digits. In this lesson, you will use place value to round both terminating decimals and repeating decimals. A terminating decimal ends. A repeating decimal continues with a digit or group of digits that repeats. The key idea is to look at the digit immediately to the right of the place you are rounding to.

What you will learn

  • Recognize terminating and repeating decimals.
  • Name the place value to which a decimal is being rounded.
  • Round terminating and repeating decimals using the same place-value rule.
  • Check whether a rounded value is reasonable.

1. Review decimal place value

A decimal point separates the whole-number part from the part less than one. Each place to the right of the decimal point is worth one tenth as much as the place before it. The first place is tenths, the second is hundredths, and the third is thousandths.
For example, in 5.3725.372, the 33 is in the tenths place, the 77 is in the hundredths place, and the 22 is in the thousandths place. The digits have different values because they are in different places.
A terminating decimal has a final digit. For example, 4.6254.625 terminates. A repeating decimal has a digit or group of digits that repeats forever. We can show that with dots, as in 2.1666…2.1666\ldots, where the 66 keeps repeating. The dots mean that the decimal continues in the same pattern.
  • Tenths are the first place after the decimal point.
  • Hundredths are the second place after the decimal point.
  • Thousandths are the third place after the decimal point.
  • Terminating means the decimal ends; repeating means a pattern continues.

2. The rounding rule

First decide what place the answer should end at. That place is the rounding place. Find its digit. Then look at the digit immediately to its right. This is the next digit.
If the next digit is 00, 11, 22, 33, or 44, keep the rounding digit the same. If the next digit is 55, 66, 77, 88, or 99, increase the rounding digit by one. After that, remove the digits to the right when rounding to a decimal place. The answer then has the requested number of decimal places.
For rounding to the nearest whole number, the rounding place is the ones place. The digit to its right is the tenths digit. For rounding to the nearest tenth, check the hundredths digit. For rounding to the nearest hundredth, check the thousandths digit.
A number line can help explain the rule. The number being rounded lies between two values at the requested place. It is closer to one of them. If it is exactly halfway, the next digit is 55, so the rule chooses the greater value.
0–4→keep;5–9→increase by 10\text{–}4\rightarrow\text{keep};\quad 5\text{–}9\rightarrow\text{increase by }1
  • Name the rounding place before looking at digits.
  • Check only the digit immediately to the right of that place.
  • Digits 00 through 44: keep the rounding digit.
  • Digits 55 through 99: increase the rounding digit by one.

3. Use the same rule for repeating decimals

A repeating decimal may not have a final digit, but it still has a digit in every place. You can identify the rounding place and the next digit just as you do with a terminating decimal.
For example, in 3.2888…3.2888\ldots, the hundredths digit is 88, and the thousandths digit is also 88. The repeating pattern supplies the next digit, even though the decimal continues.
Do not stop reading the decimal before the digit needed for rounding. If the decimal is 1.24999…1.24999\ldots and you are rounding to the nearest hundredth, the hundredths digit is 44 and the next digit is 99. The rounding digit increases by one. The continuing 99s do not change which digit you check first.
Sometimes increasing a digit changes the preceding place, as when rounding 6.986.98 to the nearest tenth. The tenths digit increases from 99 to 1010 tenths, so the result is 7.07.0. The zero shows that the answer is given to the nearest tenth.
  • Repeating decimals have digits in all decimal places.
  • Use the first digit after the rounding place, even if a pattern repeats.
  • Keep a trailing zero when it shows the requested place value, as in 7.07.0 rounded to the nearest tenth.

4. Check the result

After rounding, check two things. First, does the answer end at the requested place? Second, is the answer close to the original number?
For instance, rounding to the nearest tenth should give one digit after the decimal point. If the original number is between 4.24.2 and 4.34.3, its rounded answer should be one of those nearby tenths, not a number far away.
When you round a repeating decimal, do not try to write every repeating digit. Only the rounding digit and the next digit are needed. The same place-value rule works whether the decimal ends or continues.
  • Check the number of decimal places in your answer.
  • Check that the rounded value is near the original.
  • Use the digit right after the rounding place.

Worked example

Round a terminating decimal to the nearest hundredth

Round 8.7468.746 to the nearest hundredth.
Nearest hundredth
Nearest hundredthThe value $8.746$ is between $8.74$ and $8.75$ and is closer to $8.75$.+0.009999999999999787

The value $8.746$ is between $8.74$ and $8.75$ and is closer to $8.75$.

  1. Find the rounding place
    The hundredths place is the second place after the decimal point. In 8.7468.746, its digit is 44.
    8.74‾68.7\underline{4}6
  2. Check the next digit
    The digit immediately to the right is 66. Since 66 is at least 55, increase the hundredths digit by one.
    4+1=54+1=5
  3. Write the rounded number
    Keep two digits after the decimal point. The rounded value is 8.758.75.
    8.746≈8.758.746\approx 8.75
Answer: 8.758.75
Check: The answer has two decimal places. It is just above 8.748.74, which fits the fact that the thousandths digit was 66.

Worked example

Round a repeating decimal to the nearest tenth

Round 3.2888…3.2888\ldots to the nearest tenth.
Nearest tenth
Nearest tenthThe repeating decimal is closer to $3.3$ than to $3.2$.+0.09999999999999964

The repeating decimal is closer to $3.3$ than to $3.2$.

  1. Find the rounding place
    The tenths place is the first place after the decimal point. Its digit is 22.
    3.2‾888…3.\underline{2}888\ldots
  2. Check the next digit
    The hundredths digit is 88. It is at least 55, so increase the tenths digit from 22 to 33.
    2+1=32+1=3
  3. Write the rounded number
    The result has one digit after the decimal point. The repeating digits are not included in the rounded answer.
    3.2888…≈3.33.2888\ldots\approx 3.3
Answer: 3.33.3
Check: The number is between 3.23.2 and 3.33.3. Its hundredths digit is 88, so it rounds up to 3.33.3.

Worked example

Round a repeating decimal to the nearest whole number

Round 6.4999…6.4999\ldots to the nearest whole number.
Nearest whole number
Nearest whole number$6.4999\ldots$ is just below $6.5$, the halfway point between $6$ and $7$.67+1

$6.4999\ldots$ is just below $6.5$, the halfway point between $6$ and $7$.

  1. Find the rounding place
    To round to the nearest whole number, use the ones place. The ones digit is 66.
    6‾.4999…\underline{6}.4999\ldots
  2. Check the next digit
    The digit immediately to the right of the ones place is the tenths digit, 44. Since 44 is less than 55, keep the ones digit the same.
    4<54<5
  3. Write the rounded number
    Keep the ones digit and remove the decimal part. The rounded whole number is 66.
    6.4999…≈66.4999\ldots\approx 6
Answer: 66
Check: The decimal part begins with 44 tenths, so the value is below the halfway point of 6.56.5. The nearest whole number is 66.

Common mistakes and how to avoid them

Looking at the rounding digit instead of the digit to its right.
Correction: Keep the rounding place in view, then check the next digit.
Rounding every digit one at a time.
Correction: Use only the digit immediately after the rounding place to decide whether to keep or increase.
Thinking a repeating decimal cannot be rounded because it does not end.
Correction: Read the digit needed in the next place. The repeating pattern continues to provide that digit.
Removing a zero that shows the requested place.
Correction: Keep the zero when it shows the answer is rounded to a particular decimal place, such as 7.07.0 to the nearest tenth.

Lesson summary

  • Terminating decimals end; repeating decimals continue with a pattern.
  • Choose the rounding place and look at the digit immediately to its right.
  • If that digit is 00 to 44, keep the rounding digit. If it is 55 to 99, increase it by one.
  • Check that your answer has the requested place value and is close to the original number.

Check your understanding

Question 1

What is 5.2635.263 rounded to the nearest hundredth?
  1. 5.265.26
  2. 5.275.27
  3. 5.35.3
  4. 5.2635.263
Show answer and explanation
5.265.26
The hundredths digit is 66. The next digit is 33, so keep the 66. The answer is 5.265.26.

Question 2

What is 4.7777…4.7777\ldots rounded to the nearest tenth?
  1. 4.74.7
  2. 4.84.8
  3. 55
  4. 4.774.77
Show answer and explanation
4.84.8
The tenths digit is 77, and the hundredths digit is also 77. Increase the tenths digit by one to get 4.84.8.

Question 3

What is 9.519.51 rounded to the nearest whole number?
  1. 99
  2. 9.59.5
  3. 1010
  4. 1111
Show answer and explanation
1010
The ones digit is 99. The tenths digit is 55, so increase the ones digit by one. The result is 1010.

Key terms

Decimal point
The mark that separates the whole-number part from the part less than one.
Terminating decimal
A decimal that ends after a final digit.
Repeating decimal
A decimal in which a digit or group of digits continues in the same pattern.
Rounding place
The place value that the rounded answer is meant to show.

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