GCSE · Maths · AQA · Spec 8300 · Foundation

Exact calculation with fractions

Three thirds make exactly one whole. So if one third is 0.33, do three lots of 0.33 make 1? Work it out, then see where it really lands.

Maths · Number

Do three thirds make one?

Each tag is one third, or one third rounded to a decimal. Multiply each one by 3, then drag its tag to your answer on the line.

Maths · Number

Keep it in fractions

Work out 5/12 + 1/4. Give your answer as a fraction in its simplest form.

  1. 5/12 + 1/4
  2. missing step
Which line is step 2?

Maths · Number

Where is the exact answer lost?

Work out 5/6 × 2/5 + 1/9. Give your answer as a fraction in its simplest form.

A student's working — which line goes wrong?

Maths · Number

Your turn: one exact calculation

Work out 3/4 − 2/3 × 3/8. Give your answer as a fraction in its simplest form, and show every line of your working. [3 marks]

0 words · your answer stays on this page and is not sent anywhere.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A fraction is an exact number. Keep it a fraction all the way through, and your answer stays exact.

What you need to know

  • Calculating exactly means your answer is the precise number itself — not a decimal that is only close to it. Sometimes that number is whole: 3 × 1/3 is exactly 1. When a question also says "give your answer as a fraction in its simplest form", that is an extra instruction about how to write it, so follow it.
  • A fraction such as 1/3 is exact. Its decimal carries on for ever (0.333…), so any decimal you write down for it, such as 0.33, is a different, rounded number.
  • Keep every line of the working in fractions: make the denominators the same to add or subtract, multiply the numerators and the denominators to multiply, and never swap in a rounded decimal part-way through.
  • Finish by simplifying: divide the numerator and the denominator by their highest common factor, so the only factor they share is 1.
  • Rounding part-way through changes the numbers you are working with, so the answer is a different number — and writing that answer as a fraction at the end (0.99 = 99/100) does not bring the exact value back.

The big picture

Calculating exactly means the answer is the precise number itself, not a rounded decimal that is only close to it. This lesson shows on a number line why a rounded decimal of one third is a different number, then practises keeping every line of a calculation in fractions, simplifying at the end, and spotting the one line where a worked answer stops being exact.

Key points

1Exact answer = the precise number itself, not a rounded one (3 × 1/3 = 1 exactly). Simplest form only when the question asks for it.
20.33 is not 1/3: 3 × 0.33 = 0.99, but 3 × 1/3 = 1.
3Stay in fractions on every line; use the same denominator to add or subtract.
4Simplify at the end by the highest common factor.
5One rounded line makes every line after it approximate.

Worked example

Problem

Work out 2/3 × 3/5 − 1/6. Give your answer as a fraction in its simplest form.

⚠ Watch out

Changing a fraction to a rounded decimal part-way through — for example writing 2/3 as 0.67. Every later line then works with a different number, so the final answer is close to the exact fraction but not equal to it.

🧠

Memory hook

Fractions in, fractions all the way, fraction out — then simplify.

✓

Check yourself

Without looking back: work out 1/2 + 1/3 × 3/4, giving your answer as a fraction in its simplest form. Then explain why writing 1/3 as 0.33 would not give the exact answer.

Flashcards

(6)
What does 'calculate exactly' mean when the answer is a fraction?
Give the precise value itself — keep it as a fraction rather than a rounded decimal that is only close to it. If the question asks for simplest form, simplify it too.
Is 0.33 equal to 1/3?
No. 0.33 is 1/300 less than 1/3, so it is a different number: 3 × 0.33 = 0.99, but 3 × 1/3 = 1.
How do you keep a calculation with fractions exact?
Keep every line in fractions — use the same denominator to add or subtract — and never swap in a rounded decimal part-way through.
How do you write a fraction in its simplest form?
Divide the numerator and the denominator by their highest common factor, so the only factor they share is 1.
Why does rounding part-way through change the final answer?
The rounded value is a different number, so every later line works with it and the difference is carried to the end.
Is 9/12 in its simplest form?
No — 9 and 12 share a factor of 3, so 9/12 = 3/4.

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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