Fractions · Grade 5

What is an improper fraction and how do you convert it to a mixed number?

An improper fraction has a numerator larger than its denominator, so it holds more than one whole. The same amount can be written as a mixed number: a whole with a fraction beside it. You convert by dividing — how many times the denominator fits into the numerator — and back again by multiplying and adding.

Learning objectives

When the numerator is larger

If we took five quarters, we took more than four quarters — that is, more than one whole. A fraction like that, with a numerator larger than its denominator, is called improper.

There is nothing wrong with it. It simply writes an amount greater than one using only parts, and it is in fact the convenient form when you are calculating.

You can tell immediately which side of one a fraction falls on. Numerator smaller than denominator means less than a whole, equal means exactly one, larger means more than one — with no calculation at all.

Mixed numbers

The same amount can be written as a whole with a fraction beside it. Five quarters is one and a quarter, which is far easier to picture in real life.

To convert from improper to mixed, ask how many whole times the denominator fits into the numerator. That count is the whole part, and the remainder stays over the same denominator.

The other direction

To convert from mixed to improper, multiply the whole by the denominator and add the numerator. The result is the new numerator, and the denominator does not change.

The reason is simple: every whole is worth denominator-many parts. Two wholes in thirds are six thirds, and the third already there makes seven thirds.

Worked examples

  1. Convert 7 over 3 to a mixed number

    1. Ask how many times 3 fits into 7: twice
    2. Twice 3 is 6, leaving a remainder of 1
    3. The remainder stays over the denominator 3

    Answer: Two and a third

  2. Convert the mixed number 3 and a quarter to an improper fraction

    1. Multiply the whole by the denominator: 3 times 4 is 12
    2. Add the numerator: 12 plus 1 is 13
    3. The denominator stays 4

    Answer: 13 over 4

  3. Without calculating: is 9 over 11 more or less than one?

    1. Compare numerator with denominator: 9 is less than 11
    2. A numerator smaller than the denominator means we have not reached a whole

    Answer: Less than one

Common mistakes

Adding the whole and the numerator in a mixed number
Two and a third is not three thirds. Multiplying by the denominator is what translates wholes into parts, and only then can you add.
Changing the denominator during conversion
The denominator stays the same in both directions. If it changed, a conversion has been confused with expanding or reducing.
Treating an improper fraction as a mistake
It is a perfectly valid form, and the most convenient one for multiplying and dividing. Convert to a mixed number when the question asks, not by default.

What to remember

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