Study notes

Fractions

Mixed numbers and improper fractions, equivalent fractions, comparing fractions, and adding/subtracting related fractions.

Learn it step by step

A mixed number has a whole part and a fraction part

$2 \frac{1}{3}$ means 2 whole ones plus $\frac{1}{3}$ more. An improper fraction like $\frac{7}{3}$ shows the same amount, but with the numerator bigger than the denominator.

Convert between mixed numbers and improper fractions

To change $2 \frac{1}{3}$ to an improper fraction: multiply the whole number by the denominator, add the numerator, keep the same denominator. $2 \times 3 + 1 = 7$ , so $2 \frac{1}{3} = \frac{7}{3}$ .

Equivalent fractions are the same size

Multiplying (or dividing) the numerator and denominator by the same number does not change the value: $\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$ .

\frac{1}{2}

is the same amount shaded as

\frac{2}{4}

\frac{3}{6}

Compare fractions using a common denominator

To compare $\frac{2}{3}$ and $\frac{3}{4}$ , rewrite both with denominator 12: $\frac{8}{12}$ and $\frac{9}{12}$ . Since $9 > 8$ , $\frac{3}{4}$ is bigger.

Add or subtract related fractions

Related fractions have one denominator that is a multiple of the other, like $\frac{1}{2}$ and $\frac{1}{4}$ . Rewrite the smaller-denominator fraction to match, then add or subtract the numerators: $\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$ .

1/2

1/4

\frac{1}{2}

is the same size as 2 out of 4 quarter-parts.

Find a fraction of a set or quantity

"Of" in a fraction problem means multiply, but for whole objects it is easier to divide first, then multiply. To find $\frac{3}{5}$ of 20 sweets: $20 \div 5 = 4$ per part, then $4 \times 3 = 12$ .

Worked examples

Change

3 \frac{2}{5}

to an improper fraction.

Multiply the whole number by the denominator: $3 \times 5 = 15$ .
Add the numerator: $15 + 2 = 17$ .
$3 \frac{2}{5} = \frac{17}{5}$ .

Change

\frac{11}{4}

to a mixed number.

Divide 11 by 4: $11 \div 4 = 2$ remainder $3$ .
The whole number is 2 and the remainder becomes the new numerator.
$\frac{11}{4} = 2 \frac{3}{4}$ .

Work out

\frac{2}{3} + \frac{1}{6}

6 is a multiple of 3, so rewrite $\frac{2}{3}$ as $\frac{4}{6}$ .
Add: $\frac{4}{6} + \frac{1}{6} = \frac{5}{6}$ .
The answer is $\frac{5}{6}$ .

There are 24 marbles.

\frac{3}{8}

of them are red. How many marbles are red?

Divide 24 into 8 equal parts: $24 \div 8 = 3$ .
Multiply by the number of parts wanted: $3 \times 3 = 9$ .
9 marbles are red.

Mind map

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