KS3 · Maths

Converting between fractions and decimals (terminating and recurring)

1/3 + 1/3 + 1/3 = 1. But 0.3 + 0.3 + 0.3 = 0.9. Where did the missing 0.1 go? Can you tell if a decimal stops before dividing?

Sort it before you divide

Will the decimal stop, or go on for ever?

A terminating decimal stops: 7/100 = 0.07. A recurring decimal repeats for ever: 7/30 = 0.2333… Can you tell which you'll get without dividing? Sort with the rule most people try first, then switch to Rule 2 and watch which fractions jump sides.

Rule 1 says: if the denominator is a multiple of 10, the decimal stops. Sort each fraction the way Rule 1 says. Don't worry yet about whether it's right.

Still to sort

Rule 1 says: it stops (0)

The denominator is a multiple of 10.

Rule 1 says: it recurs (0)

The denominator is not a multiple of 10.

8 of 8 still to sort.

Rule 2 works on any fraction: simplify, factorise the denominator, look for primes other than 2 and 5.

Watch out: "It ends in a 0" is not the test. A multiple of 10 (like 30 or 60) is not the same as a power of 10 (10, 100, 1000…).

Writing a decimal that never ends

You can't write infinitely many digits, so we use dots instead. A dot goes over the first and the last digit of the repeating block, and every digit between them repeats too. So 0.3̇ = 0.333… and 0.1̇7̇ = 0.171717… Now try a trickier one.

0.16̇5̇ has dots over the 6 and the 5 only, not the 1. Which of these is it, written out?

Where recurring decimals come from

Problem

Write 1/8 and 1/11 as decimals by short division. Watch the remainders.

The shortcut: aim for 10, 100 or 1000

3/10 = 0.3 231/1000 = 0.231

When the denominator is a power of 10, there's nothing to work out. Read it off place value: 3 tenths is 0.3, and 231 thousandths is 0.231.

1 / 5

Now go backwards

From decimal to simplified fraction

Write 0.824 and 5.68 as fractions in their simplest form.

0.824 has three decimal places, so its last digit, 4, is in the thousandths column.Every terminating decimal can be written over a power of 10. The number of decimal places tells you which one.
missing step

Which line is step 2?

0.824 = 824/10000.824 = 824/1000.824 = 824/10 000

The calculator trap

What is the calculator really telling you?

Type 1 ÷ 6 into a scientific calculator. The screen shows 0.1666666667.

Which of these is closest to what you think?

It stops after the 7, so 1/6 is a terminating decimal.The 7 at the end is a real digit of 1/6. Somewhere the 6s turn into a 7.The 6s go on for ever. The calculator ran out of space and rounded the last one.0.1666666667 is so close to 1/6 that I can use either one in my working.

How sure are you?

I'm sureI think soI'm guessing

It stops after the 7, so 1/6 is a terminating decimal.

This is a common misconception

The calculator stopped, not the decimal. A calculator can only show a limited number of digits. 1/6 = 0.1666… with the 6s going on for ever (0.16̇), so it's recurring. Rule 2 agrees: 6 = 2 × 3, and that 3 means it recurs. Trust the denominator, not the screen.

The 7 at the end is a real digit of 1/6. Somewhere the 6s turn into a 7.

This is a common misconception

The 6s never turn into anything. The calculator ran out of room and rounded its last digit, so the 6 became a 7. That 7 isn't part of 1/6 at all.

The 6s go on for ever. The calculator ran out of space and rounded the last one.

This matches the evidence

Exactly. The display is a rounded snapshot of a recurring decimal. So to decide whether a fraction terminates, simplify it and check the denominator. Don't trust where the screen happens to stop.

0.1666666667 is so close to 1/6 that I can use either one in my working.

This is a common misconception

Close isn't the same as exact, and small gaps add up. 1/3 + 1/3 + 1/3 = 1. But round each third to 0.3 and you get 0.3 + 0.3 + 0.3 = 0.9. That's where the missing 0.1 went! A fraction keeps the exact value. A rounded recurring decimal doesn't.

Calculators round. Fractions are exact. Use the denominator to decide whether a fraction terminates, and keep the fraction when you need an accurate answer.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Same number, two ways of writing it. And a trick that tells you whether a decimal will stop, before you even start dividing.

What you need to know

A fraction and a decimal can be the same number written two ways: 7/2 = 3.5 and 3/8 = 0.375.
The line in a fraction means divide: 2/5 means 2 ÷ 5, which is 0.4.
A terminating decimal has a finite number of digits after the point (92.2 has one, 193.3894 has four). A recurring decimal has infinitely many, in a repeating pattern.
To predict which: simplify the fraction, then write the denominator as a product of primes. Only 2s and/or 5s means it terminates. Any other prime means it recurs.
Every terminating decimal can be written as a fraction over 10, 100, 1000…, and then simplified.

The big picture

A fraction and a decimal can be the same number: 3/8 = 0.375. To turn a fraction into a decimal, divide the numerator by the denominator, or rewrite it over 10, 100 or 1000. Some decimals terminate and some recur for ever, and you can tell which before dividing. Simplify the fraction, then factorise the denominator: only 2s and 5s means it terminates. To go back, write the decimal over a power of 10 and simplify.

Key points

1Short division with trailing zeros turns a fraction into a decimal. If the remainder reaches 0, the decimal terminates. If a remainder repeats, it recurs.

2Powers of 10 are the target: 3/8 = 375/1000 = 0.375. You get there by multiplying by a fraction equal to 1, like 125/125.

3A multiple of 10 is not a power of 10. 7/100 = 0.07 terminates, but 7/30 = 0.2333… recurs because 30 = 2 × 3 × 5.

4Dots go over the first and last digits that repeat: 0.1̇7̇ = 0.171717… and 0.16̇5̇ = 0.1656565…

5Decimal to fraction: the number of decimal places chooses the power of 10, then simplify using the HCF. 0.824 = 824/1000 = 103/125.

6For a decimal bigger than 1, split off the whole number: 5.68 = 5 + 0.68 = 5 17/25 = 142/25.

7Calculators round the last digit they show. A rounded recurring decimal isn't exact, but a fraction is.

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Worked example

Problem

Does 3/12 give a terminating or a recurring decimal? If it terminates, write it as a decimal.

⚠ Watch out

Thinking a denominator that's a multiple of 10 always gives a terminating decimal. 30 is a multiple of 10, but 30 = 2 × 3 × 5, and that 3 makes 7/30 = 0.2333… recur. Check the prime factors of the simplified denominator, not whether it ends in 0.

🧠

Memory hook

10 = 2 × 5. Only 2s and 5s can build 10, 100 or 1000. So once the fraction is simplified, only 2s and 5s in the denominator make a decimal stop.

✓

Check yourself

Without dividing: does 3/6 terminate? Then write 0.65 as a simplified fraction. (3/6 = 1/2, and 2 is a prime factor of 10, so yes. 0.65 = 65/100 = 13/20.)

Flashcards

(14)

What does the line in a fraction mean?

Divide. 2/5 means 2 ÷ 5, which is 0.4.

What is a terminating decimal?

One with a finite number of digits after the decimal point. 92.2 has one and 193.3894 has four.

Name two numbers that are NOT terminating decimals.

1.9̇ (1.999… for ever) and π. Both have an infinite number of decimal places.

What does 0.4̇73̇ mean?

0.473473473… The dots on the 4 and the 3 mark the start and end of the repeating block, so the 7 between them repeats too.

During short division, how can you tell a decimal will recur?

A remainder comes back. For 1/3 the remainder is 1 at every step, so 1/3 = 0.333… = 0.3̇.

Write 1/12 as a decimal.

0.08333…, written 0.083̇. The remainder 4 keeps coming back, so the 3 repeats.

How do you turn 13/5 into a decimal without dividing?

Multiply by 2/2 to get a power of 10: 13/5 = 26/10 = 2.6.

Why are 2 and 5 the only primes that let a decimal terminate?

They're the prime factors of 10 (10 = 2 × 5). Every power of 10 is built only from 2s and 5s, so only those denominators can be turned into 10, 100, 1000…

Why must you simplify before checking the denominator?

A factor can cancel. 3/15 looks like it has a 3 in the denominator, but 3/15 = 1/5, which terminates.

Does 3/(2 × 5²) terminate? What about 1/(7 × 11)?

3/(2 × 5²) terminates, because its denominator is only 2s and 5s. 1/(7 × 11) recurs, because 7 and 11 aren't prime factors of 10.

Write 0.4891 as a fraction.

4891/10 000. Four decimal places means ten-thousandths, 10⁴.

Are 0.56 and 0.560 different numbers?

No. Trailing zeros after the last digit don't change the value. Both equal 14/25.

Write 12.35 as an improper fraction.

12.35 = 12 7/20. Since 12 = 240/20, that's 247/20.

Which fractions are 0.4̇ and 0.7̇?

0.4̇ = 4/9 and 0.7̇ = 7/9.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Will the decimal stop, or go on for ever?

From decimal to simplified fraction

What is the calculator really telling you?

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