Repeating decimals and fractions

Overview

A decimal may end, repeat forever, or continue without a repeating pattern. Terminating and repeating decimals are rational numbers because they can be written as fractions. This topic teaches learners how to move between decimal form and fraction form, especially when a digit or block of digits repeats.

This skill is important for comparing quantities, simplifying answers, changing between percentages and fractions, and recognizing rational numbers in different forms.

The key difference is between "keeps going" and "keeps going in a repeated pattern". A recurring decimal never ends, but it is still organized. That repeated organization is what allows it to be changed into a fraction exactly.

For example:

• $0.25$ ends, so it is terminating.
• $0.252525\ldots$ repeats the block $25$, so it is recurring.
• $0.25025002500025\ldots$ may continue, but no fixed repeated block is shown, so it should not be treated as recurring unless more information is given.

Prerequisites

• Place value in decimals.
• Equivalent fractions and simplifying fractions.
• Multiplying and dividing by powers of $10$.
• Basic algebraic subtraction of equal quantities.
• Understanding rational numbers as numbers of the form $\frac{a}{b}$, where $b \ne 0$.

Learning Scope

This page covers terminating decimals, recurring decimals, recurring notation, and conversion from recurring decimals to fractions.

It does not fully cover percentages, advanced surds, or calculator approximation methods. Decimal-place rounding belongs mainly to Approximations, rounding, significant figures, and decimal places.

Subtopics

Terminating Decimals

A terminating decimal ends after a finite number of decimal places. It can be written as a fraction with denominator $10$, $100$, $1000$, and so on, then simplified.

For example:

$$ 0.45=\frac{45}{100}=\frac{9}{20} $$

Key insight: every terminating decimal is rational.

Place value tells the first fraction form:

• One decimal place means tenths.
• Two decimal places mean hundredths.
• Three decimal places mean thousandths.

Examples:

$$ 0.7=\frac{7}{10} $$

$$ 0.08=\frac{8}{100}=\frac{2}{25} $$

$$ 1.35=1+\frac{35}{100}=1+\frac{7}{20}=\frac{27}{20} $$

Misconception note: The number of decimal places decides the denominator before simplifying. For $0.08$, the first denominator is $100$, not $10$, because the $8$ is in the hundredths place.

Repeating Decimals

A repeating decimal, also called a recurring decimal, has a digit or block of digits that repeats forever.

Examples include:

• $0.\overline{3}=0.333\ldots$
• $0.\overline{27}=0.272727\ldots$
• $1.2\overline{5}=1.25555\ldots$

The bar shows the repeating part. If a block repeats, the bar should cover the whole block.

Key insight: every repeating decimal is rational, even though it never ends.

The dots are important. The notation $0.\overline{3}$ means:

$$ 0.333333\ldots $$

It does not mean $0.3$. Similarly:

$$ 0.1\overline{23}=0.123232323\ldots $$

Here the digit $1$ does not repeat; only the block $23$ repeats.

Changing a Simple Recurring Decimal to a Fraction

For a decimal like $0.\overline{6}$, let the decimal be $x$, multiply by $10$ because one digit repeats, then subtract.

$$ \begin{aligned} x &= 0.\overline{6} \\ 10x &= 6.\overline{6} \\ 10x-x &= 6.\overline{6}-0.\overline{6} \\ 9x &= 6 \\ x &= \frac{6}{9}=\frac{2}{3} \end{aligned} $$

So $0.\overline{6}=\frac{2}{3}$.

Why the subtraction works: $10x$ and $x$ have the same recurring tail, so subtracting removes the infinite repeated part.

$$ 6.6666\ldots - 0.6666\ldots = 6 $$

This avoids trying to write infinitely many digits.

Changing a Repeating Block to a Fraction

If two digits repeat, multiply by $100$. If three digits repeat, multiply by $1000$.

For $0.\overline{18}$:

$$ \begin{aligned} x &= 0.\overline{18} \\ 100x &= 18.\overline{18} \\ 100x-x &= 18 \\ 99x &= 18 \\ x &= \frac{18}{99}=\frac{2}{11} \end{aligned} $$

Key insight: choose the power of $10$ so that the repeating parts line up before subtracting.

The number of repeating digits controls the multiplier:

• One repeating digit: multiply by $10$.
• Two repeating digits: multiply by $100$.
• Three repeating digits: multiply by $1000$.

For $0.\overline{047}$, three digits repeat, so multiply by $1000$:

$$ \begin{aligned} x &= 0.\overline{047} \\ 1000x &= 47.\overline{047} \\ 1000x-x &= 47 \\ 999x &= 47 \\ x &= \frac{47}{999} \end{aligned} $$

The zero in the recurring block matters. The block is $047$, not just $47$.

Decimals With a Non-Repeating Start

Some recurring decimals have non-repeating digits before the repeated part, such as $0.1\overline{6}=0.1666\ldots$.

Use two multiplications: one to move past the non-repeating part and one to align the repeating part.

For $0.1\overline{6}$:

$$ \begin{aligned} x &= 0.1666\ldots \\ 10x &= 1.666\ldots \\ 100x &= 16.666\ldots \\ 100x-10x &= 16.666\ldots-1.666\ldots \\ 90x &= 15 \\ x &= \frac{15}{90}=\frac{1}{6} \end{aligned} $$

The two multiplications have different purposes:

• $10x$ moves the decimal point past the non-repeating digit $1$.
• $100x$ moves one more place so the repeating $6$ tails line up.

For a decimal such as $0.23\overline{7}$, there are two non-repeating digits and one repeating digit. You would use $100x$ and $1000x$.

Checking a Converted Fraction

After converting a recurring decimal to a fraction, divide the numerator by the denominator or reason with known fractions to check the result.

For example:

$$ \frac{1}{6}=0.1666\ldots=0.1\overline{6} $$

The check confirms that the conversion above is sensible.

Key Terms

• Decimal: a number written using a decimal point.
• Terminating decimal: a decimal that ends.
• Repeating decimal: a decimal with a digit or block of digits that repeats forever.
• Recurring digit: a digit that repeats in a decimal.
• Recurring block: a group of digits that repeats in a decimal.
• Fraction: a number written as $\frac{a}{b}$, where $b \ne 0$.
• Rational number: a number that can be written as a fraction of two integers.
• Simplest form: a fraction form where numerator and denominator have no common factor greater than $1$.

Worked Examples

Example 1: Convert a terminating decimal

Write $0.375$ as a fraction in simplest form.

$$ \begin{aligned} 0.375 &= \frac{375}{1000} \\ &= \frac{375 \div 125}{1000 \div 125} \\ &= \frac{3}{8} \end{aligned} $$

Therefore, $0.375=\frac{3}{8}$.

Example 2: Convert a one-digit recurring decimal

Write $0.\overline{4}$ as a fraction.

$$ \begin{aligned} x &= 0.\overline{4} \\ 10x &= 4.\overline{4} \\ 10x-x &= 4 \\ 9x &= 4 \\ x &= \frac{4}{9} \end{aligned} $$

Therefore, $0.\overline{4}=\frac{4}{9}$.

Example 3: Convert a two-digit recurring decimal

Write $0.\overline{72}$ as a fraction in simplest form.

$$ \begin{aligned} x &= 0.\overline{72} \\ 100x &= 72.\overline{72} \\ 100x-x &= 72 \\ 99x &= 72 \\ x &= \frac{72}{99}=\frac{8}{11} \end{aligned} $$

Therefore, $0.\overline{72}=\frac{8}{11}$.

Example 4: Convert a mixed recurring decimal

Write $2.\overline{3}$ as a fraction.

$$ \begin{aligned} x &= 2.\overline{3} \\ 10x &= 23.\overline{3} \\ 10x-x &= 23.\overline{3}-2.\overline{3} \\ 9x &= 21 \\ x &= \frac{21}{9}=\frac{7}{3} \end{aligned} $$

Therefore, $2.\overline{3}=\frac{7}{3}$.

Example 5: Convert a recurring decimal with a non-repeating start

Write $0.4\overline{7}$ as a fraction.

Let:

$$ x=0.4777\ldots $$

There is one non-repeating digit, $4$, and one repeating digit, $7$. Multiply by $10$ to move past the non-repeating digit:

$$ 10x=4.777\ldots $$

Then multiply by $100$ to align the repeating tails:

$$ 100x=47.777\ldots $$

Subtract:

$$ \begin{aligned} 100x-10x &= 47.777\ldots-4.777\ldots \\ 90x &= 43 \\ x &= \frac{43}{90} \end{aligned} $$

Therefore, $0.4\overline{7}=\frac{43}{90}$.

Example 6: Convert a recurring decimal greater than 1

Write $3.1\overline{2}$ as a fraction.

Let:

$$ x=3.1222\ldots $$

There is one non-repeating digit after the decimal point and one repeating digit, so use $10x$ and $100x$:

$$ \begin{aligned} 10x &= 31.222\ldots \\ 100x &= 312.222\ldots \end{aligned} $$

Subtract:

$$ \begin{aligned} 100x-10x &= 312.222\ldots-31.222\ldots \\ 90x &= 281 \\ x &= \frac{281}{90} \end{aligned} $$

Therefore, $3.1\overline{2}=\frac{281}{90}$.

Example 7: Correct an error with bar notation

A learner writes:

$$ 0.\overline{36}=0.36666\ldots $$

This is incorrect. The bar over $36$ means the whole block $36$ repeats:

$$ 0.\overline{36}=0.363636\ldots $$

The decimal $0.36666\ldots$ should be written as:

$$ 0.3\overline{6} $$

Common Mistakes

• Treating a recurring decimal as terminating. Correction: $0.\overline{3}$ means $0.333\ldots$, not $0.3$.
• Placing the bar over the wrong digits. Correction: $0.\overline{27}$ means $27$ repeats, not only $7$.
• Multiplying by the wrong power of $10$. Correction: multiply by $10$ for one repeating digit, $100$ for two, and $1000$ for three.
• Subtracting before the recurring parts line up. Correction: align identical recurring tails first.
• Forgetting to simplify the final fraction.
• Ignoring non-repeating digits before the recurring part. Correction: use two powers of $10$ when the repeat does not start immediately after the decimal point.
• Dropping zeros inside a recurring block. Correction: $0.\overline{09}$ repeats $09$, so it is $0.090909\ldots$, not $0.999\ldots$.
• Thinking "never ends" means "irrational". Correction: recurring decimals never end but are rational because they can be written as fractions.
• Writing a rounded calculator display as an exact recurring decimal. Correction: a calculator result such as $0.3333333333$ is a finite display unless recurrence is stated.

Practice Tasks

1. Write $0.6$ as a fraction in simplest form.
2. Write $0.125$ as a fraction in simplest form.
3. Write $0.08$ as a fraction in simplest form.
4. Expand the notation $0.\overline{8}$ using dots.
5. Expand the notation $0.\overline{35}$ using dots.
6. Expand the notation $0.4\overline{12}$ using dots.
7. Convert $0.\overline{5}$ to a fraction.
8. Convert $0.\overline{12}$ to a fraction.
9. Convert $0.\overline{09}$ to a fraction.
10. Convert $1.\overline{6}$ to a fraction.
11. Convert $0.2\overline{7}$ to a fraction.
12. Convert $0.13\overline{5}$ to a fraction.
13. Convert $2.0\overline{4}$ to a fraction.
14. Explain why every recurring decimal is rational.
15. A learner says $0.\overline{9}=\frac{9}{10}$. Correct the learner's method.
16. A learner says $0.\overline{24}=0.24444\ldots$. Correct the notation.
17. Which multiplier should be used first to convert $0.\overline{426}$? Explain.
18. Which two multipliers should be used to convert $0.12\overline{8}$? Explain.
19. Check whether $\frac{43}{90}$ is equal to $0.4\overline{7}$.
20. Write one terminating decimal and one recurring decimal that are both rational numbers.

Generated Question Layer

• Direct conversion: terminating decimals to simplified fractions.
• Recurring notation: identify and expand recurring digits or blocks.
• Algebraic conversion: convert one-digit, two-digit, and mixed recurring decimals to fractions.
• Comparison questions: decide which of two decimal or fraction forms is larger.
• Error-analysis questions: diagnose wrong recurring bar placement or wrong multiplier.
• Application questions: convert measurement or percentage decimals into fractions for exact calculation.
• Multiplier-selection questions: choose $10$, $100$, $1000$, or two powers of $10$ before doing the conversion.
• Notation-repair questions: correct wrongly placed recurring bars and expanded decimal forms.
• Verification questions: check a converted fraction by division or by reversing the algebra.

Learner Aid Opportunities

• diagram: place-value model showing tenths, hundredths, and thousandths as fractions.
• chart: recurring decimal patterns such as $0.\overline{3}$, $0.\overline{6}$, $0.\overline{9}$, and their fractions.
• animation: recurring tails lining up before subtraction.
• interactive: learners choose the correct multiplier for a recurring block and complete missing algebra steps.
• LLM tutor: prompts learners to explain why the subtraction removes the recurring part.
• step builder: guided boxes for $x$, first multiplication, second multiplication, subtraction, simplification, and check.
• misconception drill: distinguish terminating, recurring, rounded, and non-recurring decimals.
• notation highlighter: learners mark exactly which digit or block repeats before converting.
• progressive practice set: one-digit repeats, two-digit repeats, repeats with leading zero, repeats after a non-repeating start, then mixed numbers.

Exam-Derived Signals

• topic_frequency_2021_2025.json reports $2$ primary mapped records for this topic across 2021-2025, appearing in 2024 and 2025.
• Primary records include csee_041_2024_p1_q01_b_ii on expressing a class count in decimal form and csee_041_2025_p1_q01_a on expressing $0.1136$ as a simple fraction. These mappings are unreviewed; the 2025 record is flagged for missing marks.
• Several secondary records connect this topic to percentages, approximation, geometry, and statistics contexts. These are needs-manual-review signals and should not be treated as reviewed past-question links.
• exam_format_topic_crosswalk_2022.jsonl maps this topic to the official exam-format group "Numbers/Fractions, Decimals and percentages/Approximations" with $1$ item and $7.14\%$ weight.

Source And Review Notes

• Official syllabus alignment comes from data/curriculum_map.json and the official syllabus reference path raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Exam-derived signals are unreviewed extraction records from data/question_map_2021_2025.jsonl and data/topic_frequency_2021_2025.json.
• The worked examples are generated for learning support and are not copied from past papers.
• A reviewer should check notation choices for recurring bars against the renderer used by the wiki.