Year 9 · Unit 1 Real numbers and indices · Lesson 2 of 7

Turning a repeating decimal into a fraction

A decimal that goes on forever can be an exact fraction. Line a repeating decimal up against a copy of itself, subtract, and the endless tail cancels.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 1) Does 3/40 end or repeat? Write it as a decimal if it ends.
    AnswerIt ends: 40 = 2 × 2 × 2 × 5 has only 2s and 5s. 3/40 = 0.075.
  2. (Year 7, Unit 5, lesson 1) Write 12/18 in its simplest form.
    Answer2/3. Divide the top and the bottom by 6.
  3. (Year 8, Unit 4, lesson 3) Solve 9x = 63.
    Answerx = 7, because 63 ÷ 9 = 7.
Words used in this lesson
repeating decimal
a decimal in which a block of digits repeats forever. For example: 0.4545….
block
the group of digits that repeats. For example: 45 in 0.4545….
tail
the digits that go on after the point. For example: the 4545… in 0.4545….
simplest form
a fraction where the top and the bottom share no factor except 1. For example: 5/11.
denominator
the bottom number of a fraction. For example: 11 in 5/11.
subtract
take one number away from another. For example: 100x − x.

Can something endless be exact?

Lesson 1 showed that 1/3 = 0.333… goes on forever and is still a fraction. A robot takes 0.4545… of an hour to charge. Is that an exact amount of time? Now go the other way. You are given the decimal and must find the fraction. Predict first: 0.7777… has the block 7. What fraction do you think it is? Then slide the block and the number of digits, and watch what the subtraction does.

2
45
100xmultiply x by
99x = 45after subtracting
5/11x in simplest form

x = 0.4545... and 100x = 45.4545.... The tails match, so 100x − x = 45, 99x = 45 and x = 45/99 = 5/11.

Show me the steps
  1. x = 0.333... (block 3): 10x − x = 3, so 9x = 3 and x = 3/9 = 1/3.
  2. x = 0.777... (block 7): 9x = 7, so x = 7/9, already in simplest form.
  3. x = 0.4545... (block 45): 100x − x = 45, so 99x = 45 and x = 45/99 = 5/11.
  4. x = 0.2727... (block 27): 99x = 27, so x = 27/99 = 3/11.
  5. x = 0.0505... (block 05): 99x = 5, so x = 5/99. The 0 belongs to the block.
  6. x = 0.999... (block 9): 9x = 9, so x = 9/9 = 1. 0.999... and 1 are the same number.

The idea is to make a second copy of the number with the tail shifted so that it is identical to the first. Subtracting removes the tail completely, and what is left is a plain whole number. For the prediction: with x = 0.7777…, 10x = 7.777…, so 10x − x = 7, 9x = 7 and x = 7/9. For the block 45: 100x has 45 before the point and the same 4545… after it, so 100x − x = 45 exactly.

To write a repeating decimal as a fraction: call it x, multiply by 10 for a one-digit block or 100 for a two-digit block, subtract x, and solve. When the block starts straight after the point, a block of one digit goes over 9 and a block of two digits goes over 99. Then simplify.

A consequence: 0.999… = 9/9 = 1. They are two names for one number.

A common mistake: writing 0.2777… as 27/99. That is 0.2727…, because 27/99 repeats both digits. In 0.2777… only the 7 repeats, so the 2 has to be moved before the point first. Multiply by 10 and by 100 and subtract, as in the second worked example.

Worked example: the whole decimal repeats

Write 0.636363… as a fraction in simplest form.

  1. Let x = 0.6363… The block 63 has two digits, so multiply by 100: 100x = 63.6363…
  2. Subtract: 100x − x = 63, so 99x = 63 and x = 63/99.
  3. Divide the top and the bottom by 9: 7/11. Check: 7 ÷ 11 = 0.6363…

Worked example: only part of it repeats

Write 0.8333… (only the 3 repeats) as a fraction in simplest form.

  1. Let x = 0.8333… Then 10x = 8.333… and 100x = 83.333… Now the tails after the point match.
  2. Subtract: 100x − 10x = 83 − 8 = 75, so 90x = 75 and x = 75/90.
  3. Divide the top and the bottom by 15: 5/6. Check: 5 ÷ 6 = 0.8333…

Your go. The first step is given. Write 0.7222… (only the 2 repeats) as a fraction in simplest form.

  1. Let x = 0.7222… Then 10x = 7.222… and 100x = 72.222…
  2. Subtract and simplify, then check.
Check the answer

100x − 10x = 72 − 7 = 65, so 90x = 65 and x = 65/90 = 13/18. Check: 13 ÷ 18 = 0.7222…

Check before you go on

1. Which fraction is equal to 0.181818… (the block 18 repeats)?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A runner covers 0.444… km in one lap and 0.222… km in the next. Write each as a fraction and add them. In simplest form, what is the denominator of the sum?

3. Write 0.121212… as a fraction in simplest form. What is its denominator?

Level 2 · Building

4. Write 0.4666… (only the 6 repeats) as a fraction in simplest form. What is its denominator?

5. Write 1.222… (the 2 repeats) as an improper fraction with denominator 9. What is the numerator?

6. To write 0.272727… as a fraction, Neil multiplies by 100. Why 100?

Level 3 · Stretch

7. Write 0.01111… (only the 1 repeats) as a fraction in simplest form. What is its denominator?

8. 8/33 is a repeating decimal with a block of two digits. What is the sum of the two digits in the block?

Something to think about

Write 0.3454545… (only the block 45 repeats, after the 3) as a fraction. Which two multipliers line the tails up?

One way to think about it

Multiply by 10 and by 1000: 10x = 3.454545… and 1000x = 345.454545… have the same tail, so 1000x − 10x = 345 − 3 = 342, 990x = 342 and x = 342/990 = 19/55.

Exit ticket

9. Which fraction equals 0.08888… (only the 8 repeats)?