046. Decimals, Fractions and Recurring Decimals

Learning Intentions

• convert decimals to fractions
• convert fractions to decimals
• To understand the symbols used to indicate recurring decimals

Pre-requisite Summary

• Understand place value in decimals, including tenths, hundredths and thousandths
• Know that a decimal can be written as a fraction with denominator $10$, $100$, $1000$, \dots
• Be able to simplify fractions using common factors
• Understand that division can be written as a fraction
• Know that some fractions give terminating decimals and some give repeating decimal patterns
• Be able to compare equivalent forms of the same number

Worked Examples

Worked Example 1

Convert each decimal to a fraction in simplest form:
a) $0.4$
b) $0.35$
c) $1.2$

Worked Example 2

Convert each decimal to a fraction in simplest form:
a) $0.08$
b) $2.75$
c) $0.125$

Worked Example 3

Convert each fraction to a decimal:
a) ${\displaystyle \frac{3}{10}}$
b) ${\displaystyle \frac{7}{100}}$
c) ${\displaystyle \frac{9}{4}}$

Worked Example 4

Convert each fraction to a decimal:
a) ${\displaystyle \frac{3}{5}}$
b) ${\displaystyle \frac{7}{8}}$
c) ${\displaystyle \frac{11}{20}}$

Worked Example 5

Write each recurring decimal using the correct notation and describe the repeating part:
a) $0.333\dots$
b) $0.121212\dots$
c) $2.454545\dots$

Worked Example 6

For each number, state whether the decimal is terminating or recurring:
a) ${\displaystyle \frac{1}{4}}$
b) ${\displaystyle \frac{1}{3}}$
c) ${\displaystyle \frac{5}{6}}$

Problems

Problem 1

Convert each decimal to a fraction in simplest form:
a) $0.6$
b) $0.45$
c) $1.5$

Problem 2

Convert each decimal to a fraction in simplest form:
a) $0.09$
b) $3.25$
c) $0.375$

Problem 3

Convert each fraction to a decimal:
a) ${\displaystyle \frac{4}{10}}$
b) ${\displaystyle \frac{9}{100}}$
c) ${\displaystyle \frac{7}{2}}$

Problem 4

Convert each fraction to a decimal:
a) ${\displaystyle \frac{4}{5}}$
b) ${\displaystyle \frac{3}{8}}$
c) ${\displaystyle \frac{13}{20}}$

Problem 5

Write each recurring decimal using the correct notation and describe the repeating part:
a) $0.666\dots$
b) $0.272727\dots$
c) $1.818181\dots$

Problem 6

For each number, state whether the decimal is terminating or recurring:
a) ${\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{2}{3}}$
c) ${\displaystyle \frac{7}{9}}$

Exercises

Understanding and Fluency

1. Convert each decimal to a fraction in simplest form:
   a) $0.2$
   b) $0.7$
   c) $0.9$

2. Convert each decimal to a fraction in simplest form:
   a) $0.25$
   b) $0.6$
   c) $0.75$

3. Convert each decimal to a fraction in simplest form:
   a) $1.4$
   b) $2.3$
   c) $3.05$

4. Convert each fraction to a decimal:
   a) ${\displaystyle \frac{1}{10}}$
   b) ${\displaystyle \frac{3}{10}}$
   c) ${\displaystyle \frac{7}{10}}$

5. Convert each fraction to a decimal:
   a) ${\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{3}{4}}$
   c) ${\displaystyle \frac{7}{5}}$

6. Convert each fraction to a decimal:
   a) ${\displaystyle \frac{9}{20}}$
   b) ${\displaystyle \frac{11}{25}}$
   c) ${\displaystyle \frac{7}{8}}$

7. Write each recurring decimal using the correct notation:
   a) $0.777\dots$
   b) $0.454545\dots$
   c) $3.121212\dots$

8. State whether each decimal is terminating or recurring:
   a) $0.5$
   b) $0.\dot{3}$
   c) $1.2\dot{7}$

Reasoning

9. Explain why $0.35={\displaystyle \frac{35}{100}}$ before simplification.

10. A student says $0.4={\displaystyle \frac{4}{100}}$. Explain the mistake.

11. Explain why ${\displaystyle \frac{3}{4}}$ gives a terminating decimal but ${\displaystyle \frac{1}{3}}$ gives a recurring decimal.

12. A student writes $0.272727\dots$ as $0.\dot{2}\dot{7}$. Explain whether this notation is correct and what it means.

Problem-solving

13. A measuring jug contains $0.75$ L of water. Write this amount as a fraction in simplest form.

14. A ribbon is ${\displaystyle \frac{7}{8}}$ m long. Write this length as a decimal.

15. A battery charge is shown as $0.6$ of full capacity. Write this as a fraction in simplest form.

16. A runner completes ${\displaystyle \frac{11}{20}}$ of a lap. Write this as a decimal.

17. A calculator displays $0.333\dots$. Write this using recurring decimal notation and state the repeating digit.

18. A number is ${\displaystyle \frac{5}{3}}$. Write it as a decimal and identify whether it is terminating or recurring.

Potential Misunderstandings

• Students may confuse the place value of the last decimal digit when converting a decimal to a fraction
• Students may forget to simplify the fraction after writing the decimal over a power of ten
• Students may think every fraction converts to a terminating decimal
• Students may stop a decimal expansion too early and assume it terminates
• Students may not recognise that a mixed or improper fraction can still be converted to a decimal
• Students may misunderstand recurring decimal notation and think the dots apply to only one digit when a block of digits repeats
• Students may write the denominator incorrectly when converting decimals with two or three decimal places
• Students may confuse $0.3$ with $0.\dot{3}$ even though one terminates and the other recurs

Next: 047. Percentages and Decimals