Rewrite an improper fraction as a whole number, and a whole number as a fraction

Learn this first

You should already be able to

· Read a fraction: a whole cut into equal parts, and how many of them you have.
· Know that a fraction whose numerator equals its denominator is one whole.
· Multiply two small whole numbers.

A whole number does not look like it has a denominator, and that is the whole difficulty. It does. Cut one whole into 4 equal parts and you have 44 — the same amount, written to show the pieces.

So 3 wholes are 3 lots of 4 quarters — and the three wholes have to be the same size, or their quarters are not the same quarter and cannot be counted together. Count the parts: 3 × 4 = 12 of them, each still a quarter. That is 124, and it is the same amount as 3 — not a bigger number, a different way of writing it.

The denominator says how the whole was cut, so it does not change when you count more wholes. Only the numerator does. That is why the question has to tell you which denominator to use: 3 is also 62 and 93, and all of them are right.

Worked examples

The picture above, done as a calculation.

Write 3 as a fraction with denominator 4.

1. One whole cut into 4 equal parts is 4/4.
2. There are 3 wholes, so there are 3 lots of 4 parts: 3 × 4 = 12.
3. Every part is still one 4th, so the denominator stays 4: 3 = 12/4.

Answer: 124

A different denominator, and a numerator that is not the obvious multiple — halves, not quarters.

Write 5 as a fraction with denominator 2.

1. One whole cut into 2 equal parts is 2/2.
2. There are 5 wholes, so there are 5 lots of 2 parts: 5 × 2 = 10.
3. Every part is still one 2th, so the denominator stays 2: 5 = 10/2.

Answer: 102

Try it, with help

Write 4 as a fraction with denominator 3.

Hint: One whole is 3/3. How many thirds are in 4 wholes?

Show the answer

123 — 4 wholes, each worth 33: 4 × 3 = 12 parts, each still one 3th.

Write 2 as a fraction with denominator 7.

Hint: The denominator is larger than the whole number this time. It still does not change.

Show the answer

147 — 2 wholes, each worth 77: 2 × 7 = 14 parts, each still one 7th.

If it went wrong

You added the denominator to the whole number instead of multiplying by it.

Count the parts in the picture rather than the wholes. 3 wholes of 4 quarters is not 3 + 4 pieces — it is 4 pieces, then 4 more, then 4 more. Adding gives you 7 quarters, which is not even two whole bars, so it cannot be 3.

Now try this: Write 2 as a fraction with denominator 5.

Show the answer

105 — 2 × 5 = 10, not 2 + 5. Two wholes of five fifths is ten fifths.

You multiplied the denominator by itself and left the whole number out.

Ask what each number is doing. The denominator says how big one piece is; the whole number says how many wholes you are counting. Both have to appear, so the multiplication is whole number × denominator — never the denominator twice.

Now try this: Write 4 as a fraction with denominator 3.

Show the answer

123 — 4 × 3 = 12. Not 3 × 3 — the 4 is how many wholes there are and it cannot be left out.

The numerator and the denominator are the wrong way round.

Check the size before checking the writing. Your answer should be the same amount as the whole number you started with, so it must be worth at least 1 — the numerator has to be the bigger one. If the top is smaller than the bottom, the fraction is less than one whole and cannot equal 3.

Now try this: Is 312 the same amount as 4? If not, what is?

Show the answer

No — 12/3. — 312 is less than one whole, so it cannot be 4. Turned round, 123 is 4 wholes of 3/3.

One whole is d/d. To write a whole number with denominator d, multiply the whole number by d for the numerator and keep d underneath — the amount does not change, only the way it is written.

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Topic quiz

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