Recurring Decimals as Exact Fractions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Endless decimals, exact fractions

Mathematics · Number Representation & Place Value · ages 14-15
Name ______________________   Date ____________

  1. Let x = 0.666... (6 repeating forever). Subtracting x from 10x gives kx = 6. What is the whole number k?

    Answer: ______________

  2. Let x = 0.555... (5 repeating forever). Why does 10x - x leave a whole number?

    • Multiplying by 10 stops the decimal.
    • The digits after the point become zeros.
    • The matching decimal tails cancel.
    • Subtraction rounds the decimal to 5.

  3. What is 0.777... (7 repeating forever) as a fraction in lowest terms?

    • 7/10
    • 7/9
    • 3/4

  4. Amara says 0.999... (9 repeating forever) is exactly equal to 1. Is Amara right?

    Circle one:   True   False

  5. What is 0.454545... (45 repeating forever) as a fraction in lowest terms?

    • 5/11
    • 4/11
    • 45/100

  6. In 0.2070707... (the block 07 repeats after the digit 2), the non-recurring digits after the point are 20.

    Circle one:   True   False

  7. Which statement correctly predicts the decimal produced by dividing 27 by 60?

    • It recurs because 60 has a factor of 3.
    • It terminates: 27/60 reduces to 9/20.
    • It terminates because 60 is even.
    • It recurs because 27 has a factor of 3.

  8. Which fraction gives a terminating decimal when you divide its numerator by its denominator?

    • 2/3
    • 5/11
    • 3/8

  9. What is 0.16333... (only the 3 repeating forever) as a fraction in lowest terms?

    • 163/999
    • 147/100
    • 49/300

  10. Tom sets x = 0.16333... (only 3 repeats). He multiplies by 100, then claims the decimal tails cancel in 100x - x and writes 99x = 16. Is his subtraction valid?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Endless decimals, exact fractions W1-mt_Mb-sXIZDWd-s1

  1. 9 · 10x - x = (10 - 1)x = 9x. The equation is 9x = 6, so k = 9.
  2. The matching decimal tails cancel. · 10x = 5.555... and x = 0.555... have equal decimal tails. Subtracting cancels those tails and leaves 5.
  3. 7/9 · Let x = 0.777... . Then 10x = 7.777... . Subtracting gives 9x = 7, so x = 7/9.
  4. True · Let x = 0.999... . Subtracting x from 10x gives 9x = 9. Dividing by 9 gives x = 1 exactly.
  5. 5/11 · The two-digit block needs a shift by 100. Subtraction gives 99x = 45, so x = 45/99 = 5/11.
  6. False · Only 2 comes before the stated repeating block. The next zero belongs to the block 07, so the non-recurring part is not 20.
  7. It terminates: 27/60 reduces to 9/20. · Reduce 27/60 by 3 to get 9/20. Since 20 = 2 x 2 x 5, the reduced denominator has only 2s and 5s, so the decimal terminates.
  8. 3/8 · All three fractions are in lowest terms. Only 3/8 has a denominator made entirely from 2s and 5s: 8 = 2 x 2 x 2.
  9. 49/300 · Use 100x = 16.333... and 1000x = 163.333... . Subtraction gives 900x = 147, so x = 147/900 = 49/300.
  10. False · The decimal parts 0.333... and 0.16333... differ. Their subtraction leaves 0.17, so 99x = 16.17. Use 1000x - 100x for matching tails.
Worksheet · LightMySky