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Grade 7 · lesson 2 of 4

Turn a Repeating Digit into a Fraction

A repeating decimal can be converted using two equations whose repeating tails match.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Convert 0.111… (the digit 1 repeats forever) to a fraction. Candidate answers: (A) 1/10 (B) 1/9 (C) 1/99 (D) 2/9.
  • Independent check inputs: Convert 0.222… (the digit 2 repeats forever) to a fraction. Candidate answers: (A) 1/3 (B) 1/5 (C) 2/9 (D) 2/99.

Learn the idea

Rational numbers can be represented as ratios of integers with nonzero denominator. Their decimal forms terminate or repeat, and negative quantities can be compared by their positions to the left of zero. A repeating decimal can be converted using two equations whose repeating tails match. Multiplying by ten shifts a one-digit repeating block so subtraction cancels the infinite tail.

Here is the method on the supplied teaching case. Let x=0.111…. Then 10x=1.111…. Subtract x to get 9x=1, so x=1/9. This is an exact repeating decimal.

The dots mean the specified pattern repeats forever. A finite string such as 0.333 is different from 0.333…; do not replace an exact repeating fraction with an unexplained rounded value.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Convert 0.111… (the digit 1 repeats forever) to a fraction. Candidate answers: (A) 1/10 (B) 1/9 (C) 1/99 (D) 2/9. Solution: Let x=0.111…. Then 10x=1.111…. Subtract x to get 9x=1, so x=1/9. This is an exact repeating decimal. The final answer is 1/9. The dots mean the specified pattern repeats forever. A finite string such as 0.333 is different from 0.333…; do not replace an exact repeating fraction with an unexplained rounded value.

Your turn

Check your reasoning

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Helpful tips
  • A repeating decimal can be converted using two equations whose repeating tails match.
  • The dots mean the specified pattern repeats forever.
Convert 0.222… (the digit 2 repeats forever) to a fraction.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

2/9

Let x=0.222…. Then 10x=2.222…. Subtract x to get 9x=2, so x=2/9. This is an exact repeating decimal.

Take it further

Changed-input task: Convert 0.888… (the digit 8 repeats forever) to a fraction. Candidate answers: (A) 1 (B) 4/5 (C) 8/9 (D) 8/99. Show the mathematical reason, not only the selected answer. Expected result: 8/9. Let x=0.888…. Then 10x=8.888…. Subtract x to get 9x=8, so x=8/9. This is an exact repeating decimal.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Convert 0.333… (the digit 3 repeats forever) to a fraction.
Need a hint?
  • A repeating decimal can be converted using two equations whose repeating tails match.
  • The dots mean the specified pattern repeats forever.
2. Convert 0.444… (the digit 4 repeats forever) to a fraction.
Need a hint?
  • A repeating decimal can be converted using two equations whose repeating tails match.
  • The dots mean the specified pattern repeats forever.
3. Convert 0.555… (the digit 5 repeats forever) to a fraction.
Need a hint?
  • A repeating decimal can be converted using two equations whose repeating tails match.
  • The dots mean the specified pattern repeats forever.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

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These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.