Year 7 · Unit 5 Fractions · Lesson 7 of 7

Why some decimals never stop

Every fraction is a division. Watch long division reach a remainder of 0, or meet a remainder it has seen before, and see which bottom numbers make a decimal stop.

Number and AlgebraScientific calculator

Warm-up

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

  1. (Unit 2, lesson 2) Write 3/20 as a percentage.
    Answer15%. 3/20 = 15/100, because 20 × 5 = 100.
  2. (Lesson 1 of this unit) Write 12/30 in simplest form.
    Answer2/5. Divide the top and the bottom by 6, the greatest common factor.
  3. (Unit 1, lesson 4) Write 40 as a product of prime factors.
    Answer2 × 2 × 2 × 5.
Words used in this lesson
terminating decimal
a decimal with a last digit: it stops. For example: 0.125 is terminating.
recurring decimal
a decimal with a block of digits that repeats for ever. For example: 0.2727... is recurring.
recurring dot
a dot over the first and last digit of the repeating block. For example: 0.2̇7̇ means 0.2727....
repeating block
the group of digits that repeats. For example: in 0.142857142857... the block is 142857.
remainder
what is left over in a division. For example: 10 ÷ 4 has a remainder of 2.
prime factor
a prime number that divides exactly. For example: the prime factors of 12 are 2 and 3.

Which decimals stop?

On a calculator, 1 ÷ 8 gives 0.125 and stops. 1 ÷ 3 gives 0.3333333333, and it looks as if it would never stop. Predict first: of 1/4, 1/5, 1/6 and 1/8, which ones stop, and which go on for ever? Then test them, and some more, in the figure, which does the long division one digit at a time.

7
3
0.428571428571...decimal
repeats: 428571stops or repeats
7prime factors of the bottom, simplest form

3/7: long division gives 0.428571428571... The remainder 3 comes back, so the block 428571 repeats for ever. The bottom number of 3/7 is 7: it has a prime factor other than 2 and 5.

Show me the steps
  1. 1/8 = 0.125: the remainders go 1, 2, 4 and then 0, so it stops.
  2. 1/6 = 0.1666...: after the first digit the remainder 4 comes back, so 6 repeats.
  3. 3/11 = 0.2727...: remainders 3, 8, then 3 again, so the block 27 repeats.
  4. 7/12 = 0.58333...: the bottom 12 = 2 × 2 × 3 has a 3, so it repeats.
  5. 6/15 = 0.4: in simplest form it is 2/5, and its bottom 5 is a single 5, so it stops.
  6. 1/16 = 0.0625: 16 = 2 × 2 × 2 × 2, only 2s, so it stops after 4 digits.

Every fraction is a division, and its decimal either stops or repeats. The numerator (top) is divided by the denominator (bottom). In long division the remainder is always less than the denominator, so there are only so many remainders. Either one is 0 and the decimal terminates, or a remainder comes back and the digits repeat in a block for ever: a recurring decimal.

A fraction in simplest form has a terminating decimal exactly when its bottom number has no prime factors other than 2 and 5, because then the bottom divides a power of 10: 7/40 = 175/1000.

We write a recurring decimal with a dot over the first and last digit of the block: 1/3 = 0.3̇, 3/11 = 0.2̇7̇, 1/7 = 0.1̇42857̇.

A recurring decimal is an exact number: 0.333... is 1/3, not a rounded copy of it.

Back to the calculator: 1/4 = 0.25, 1/5 = 0.2 and 1/8 = 0.125 stop, because 4, 5 and 8 are made only of 2s and 5s. Only 1/6 goes on for ever, because 6 = 2 × 3 has a 3.

A common mistake: “A decimal that never ends is only an approximation.” The calculator's display is the rounded copy. The fraction, and the decimal with its repeating block, are exact: 0.333... = 1/3.

Going further: back to a fraction

This part goes beyond the core of the lesson, which is fraction to decimal and the rule for which decimals stop. To turn a repeating decimal into a fraction, line it up with a shifted copy of itself so the tails match, and subtract them. Try one digit and two digits.

27
3/11as a fraction
99bottom: a 9 for each digit
0.2727...the decimal

x = 0.2727... Then 100x = 27.2727..., the tails match, and 100x − x = 27, so 99x = 27 and x = 27/99 = 3/11.

Show me the steps
  1. 0.7777...: one digit repeats, 10x − x = 7, so 9x = 7 and x = 7/9.
  2. 0.2727...: two digits, 100x − x = 27, so 99x = 27 and x = 27/99 = 3/11.
  3. 0.4545...: 99x = 45, so x = 45/99 = 5/11.
  4. 0.1212...: 99x = 12, so x = 12/99 = 4/33.
  5. 0.0505...: the block is 05, so 99x = 5 and x = 5/99.

Worked examples

Does 7/40 terminate? If so, write it as a decimal.

  1. 7/40 is in simplest form. 40 = 2 × 2 × 2 × 5: only 2s and 5s.
  2. So it terminates. 7 ÷ 40 = 0.175.
  3. Check: 0.175 × 40 = 7. 0.175.

Going further: write 0.4333... as a fraction.

  1. Let x = 0.4333... The 4 does not repeat, so first move it in front of the point: 10x = 4.333... Then move one block further: 100x = 43.333...
  2. The tails match, so 100x − 10x = 43 − 4 = 39, which is 90x = 39.
  3. x = 39/90 = 13/30.

Your go. The first step is given. Write 0.7272... as a fraction.

  1. Two digits repeat, so multiply by 100 and subtract x: 100x − x = 72, which is 99x = 72.
  2. Now write x as a fraction and simplify, then check.
Check the answer

x = 72/99. 72 and 99 both divide by 9, so x = 8/11. Check: 8 ÷ 11 = 0.7272...

With a calculator

A scientific calculator shows the quotient to its display width. 1 ÷ 7 gives 0.1428571429 on a ten-digit display. The last digit is a 9 only because the display rounds: the true digits are 0.142857142857..., and the tenth digit is 8, followed by 5, which rounds it up to 9. The calculator cannot show “for ever”, so look for the block yourself: 142857 repeats.

Estimate first. 1/7 is a little less than 1/6, which is about 0.17, and a little more than 1/8, which is 0.125, so an answer near 0.14 makes sense. The calculator's decimal is a rounded copy of the fraction: the fraction 1/7 is the exact number.

Check before you go on

1. Which of these fractions has a decimal that stops?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Write 5/16 as a decimal. Give every digit.

3. Which decimal is exactly 1/3?

Level 2 · Building

4. A calculator shows 1 ÷ 3 = 0.3333333333. A student writes 1/3 of an hour as 0.33 hours and says that is 33 minutes. How many minutes is exactly 1/3 of an hour?

5. Write 0.151515... as a fraction in simplest form. What is the bottom number?

6. 3/13 = 0.230769230769... The block 230769 repeats. What is the 20th digit after the decimal point?

Level 3 · Stretch

7. For how many of the fractions 1/12, 2/12, 3/12, ..., 11/12 does the decimal stop?

8. 0.8333... = ☐/6. What number goes in the box?

Something to think about

Before you try 9 in the figure, use the trick from the second figure on x = 0.999... by hand. What do you get? Is 0.999... a bit less than 1, or exactly 1?

One way to think about it

10x = 9.999..., and 10x − x = 9, so 9x = 9 and x = 1. Another way: 1/3 = 0.333..., and three thirds make 1, so 3 × 0.333... = 0.999... = 1. There is no number between 0.999... and 1, so they are the same number written two ways.

Exit ticket

9. 1/3 = 0.333... Which statement is true?