114. Recurring Decimals and Rounding

Learning Intentions

• To know the meaning of the terms terminating decimal and recurring decimal
• To understand the different notations for recurring decimals (involving dots and dashes)
• convert a fraction to a recurring decimal using division
• round decimals to a given number of decimal places by first finding the critical digit

Pre-requisite Summary

• Know that decimals use place value columns such as tenths, hundredths and thousandths
• Be able to divide whole numbers and decimals using written division
• Understand that a fraction can be written as a division
• Know that some fractions convert to exact decimals
• Be able to identify the digit in a given decimal place
• Understand that rounding depends on the value of the digit to the right of the required place
• Know that place value determines the size of each digit in a decimal
• Be familiar with writing equivalent numerical forms of the same value

Worked Examples

Worked Example 1

State whether each decimal is terminating or recurring:
a) $0.45$
b) $0.333\dots$
c) $2.1875$

Worked Example 2

Write each recurring decimal using recurring decimal notation:
a) $0.666\dots$
b) $0.121212\dots$
c) $1.454545\dots$

Worked Example 3

Convert each fraction to a decimal using division:
a) ${\displaystyle \frac{1}{4}}$
b) ${\displaystyle \frac{1}{3}}$
c) ${\displaystyle \frac{2}{11}}$

Worked Example 4

Convert each fraction to a decimal using division:
a) ${\displaystyle \frac{5}{6}}$
b) ${\displaystyle \frac{7}{9}}$
c) ${\displaystyle \frac{4}{15}}$

Worked Example 5

Round each decimal to the stated number of decimal places by first identifying the critical digit:
a) $3.746$ to $2$ decimal places
b) $8.2519$ to $3$ decimal places
c) $0.684$ to $1$ decimal place

Worked Example 6

Round each decimal to the stated number of decimal places by first identifying the critical digit:
a) $5.999$ to $2$ decimal places
b) $12.3048$ to $3$ decimal places
c) $7.145$ to $2$ decimal places

Problems

Problem 1

State whether each decimal is terminating or recurring:
a) $0.8$
b) $0.272727\dots$
c) $4.0625$

Problem 2

Write each recurring decimal using recurring decimal notation:
a) $0.777\dots$
b) $0.343434\dots$
c) $2.181818\dots$

Problem 3

Convert each fraction to a decimal using division:
a) ${\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{2}{3}}$
c) ${\displaystyle \frac{5}{11}}$

Problem 4

Convert each fraction to a decimal using division:
a) ${\displaystyle \frac{1}{6}}$
b) ${\displaystyle \frac{8}{9}}$
c) ${\displaystyle \frac{7}{15}}$

Problem 5

Round each decimal to the stated number of decimal places by first identifying the critical digit:
a) $4.583$ to $2$ decimal places
b) $6.1287$ to $3$ decimal places
c) $9.46$ to $1$ decimal place

Problem 6

Round each decimal to the stated number of decimal places by first identifying the critical digit:
a) $2.995$ to $2$ decimal places
b) $15.6724$ to $3$ decimal places
c) $3.845$ to $2$ decimal places

Exercises

Understanding and Fluency

1. State whether each decimal is terminating or recurring:
   a) $0.5$
   b) $0.181818\dots$
   c) $3.125$
   d) $0.777\dots$

2. State whether each decimal is terminating or recurring:
   a) $1.04$
   b) $2.363636\dots$
   c) $0.090909\dots$
   d) $4.2$

3. Write each recurring decimal using appropriate recurring decimal notation:
   a) $0.444\dots$
   b) $0.575757\dots$
   c) $3.121212\dots$

4. Write each recurring decimal using appropriate recurring decimal notation:
   a) $0.999\dots$
   b) $1.666\dots$
   c) $2.454545\dots$

5. Convert each fraction to a decimal using division:
   a) ${\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{1}{3}}$
   c) ${\displaystyle \frac{2}{5}}$
   d) ${\displaystyle \frac{4}{9}}$

6. Convert each fraction to a decimal using division:
   a) ${\displaystyle \frac{5}{8}}$
   b) ${\displaystyle \frac{2}{7}}$
   c) ${\displaystyle \frac{7}{12}}$
   d) ${\displaystyle \frac{1}{11}}$

7. Round each decimal to the stated number of decimal places:
   a) $4.267$ to $2$ decimal places
   b) $8.5314$ to $3$ decimal places
   c) $0.96$ to $1$ decimal place
   d) $7.1452$ to $2$ decimal places

8. Round each decimal to the stated number of decimal places:
   a) $2.684$ to $2$ decimal places
   b) $5.9991$ to $3$ decimal places
   c) $13.406$ to $1$ decimal place
   d) $9.375$ to $2$ decimal places

Reasoning

9. Explain why $0.25$ is a terminating decimal but $0.333\dots$ is a recurring decimal.

10. A student says that $0.121212\dots$ is terminating because the digits $12$ repeat in a pattern. Explain the mistake.

11. Noah says that $0.4\dot{5}$ means both digits repeat. Is he correct? Explain.

12. Explain why the critical digit is important when rounding a decimal.

13. A student rounds $6.482$ to $2$ decimal places and writes $6.5$. Describe the error.

Problem-solving

14. A calculator display shows $0.1666\dots$ after dividing. State whether the decimal is terminating or recurring, and write it in recurring decimal notation.

15. A fraction is divided and gives $0.454545\dots$. Write this decimal using recurring notation and identify the repeating block.

16. A measurement is $7.2864$ m. Round this measurement to $2$ decimal places.

17. A price is $\mathrm{\$}3.995$. Round the price to $2$ decimal places.

18. A student converts ${\displaystyle \frac{2}{3}}$ to a decimal using division. What decimal should they obtain, and what type of decimal is it?

19. A decimal rounded to $1$ decimal place is $5.7$. Give two possible original decimals.

Potential Misunderstandings

• Students may think a recurring decimal stops because the repeating digits are predictable
• Students may confuse terminating decimals with decimals that only show a few digits on a calculator
• Students may think any decimal with repeated digits is terminating
• Students may misunderstand recurring notation and not know which digit or block is repeating
• Students may think dots or dashes mean the decimal is approximate rather than recurring
• Students may stop the division too early when converting a fraction to a decimal
• Students may not recognise that a repeated remainder in division leads to a recurring decimal
• Students may confuse the last kept digit with the critical digit when rounding
• Students may look at the wrong digit when rounding to a given number of decimal places
• Students may think rounding always makes a number larger
• Students may forget that digits after the rounded place become zero in value or are removed from the written decimal
• Students may make errors when the rounded digit becomes $10$ and needs regrouping, such as in $5.999$