Skip to content

Equal fractions, comparing fractions

Fractions, grade 6 Lesson 3 of 22Learning

Equal fractions, comparing fractions

The same number can be written as different fractions: 34\displaystyle \frac{3}{4} and 68\displaystyle \frac{6}{8} are the same number.

Equal fractions

Reducing
Six eighths of a cake and three quarters of a cake – exactly the same amount of cake.

Divide the numerator and the denominator by the same number, and the fraction does not change – the parts only get bigger, and there are fewer of them: 68=6÷28÷2=34\displaystyle {\frac{6}{8}}={\frac{6\div2}{8\div2}}={\frac{3}{4}}. This is called reducing. On the number line, 68\displaystyle \frac{6}{8} and 34\displaystyle \frac{3}{4} are at the same point:

Expanding

And the other way: multiply the numerator and the denominator by the same number – that is expanding: 34=3×24×2=68\displaystyle {\frac{3}{4}}={\frac{3\times2}{4\times2}}={\frac{6}{8}}. Here too the fraction does not change – the parts only get smaller, and there are more of them.

Reducing all the way

To reduce all the way, look for the biggest number that divides both the numerator and the denominator: 1218=12÷618÷6=23\displaystyle {\frac{12}{18}}={\frac{12\div6}{18\div6}}={\frac{2}{3}}. A fraction that cannot be reduced any more, like 23\displaystyle \frac{2}{3}, is in its simplest form.

Exercise 1

Reduce all the way.

a

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        Comparing fractions

        Same denominator

        When the denominators are the same, the parts are the same size – and the fraction with more parts is bigger: 57>37\displaystyle \frac{5}{7}>\frac{3}{7}.

        Same numerator
        A third of a cake is more than a fifth of a cake.

        When the numerators are the same, there are as many parts – what decides is their size. The bigger the denominator, the smaller the parts: 23>25\displaystyle \frac{2}{3}>\frac{2}{5}.

        Different numerators and denominators

        Expand both fractions to a common denominator – a number both denominators go into – and compare the numerators: 23=812\displaystyle {\frac{2}{3}}={\frac{8}{12}} and 34=912\displaystyle {\frac{3}{4}}={\frac{9}{12}}, so 34>23\displaystyle \frac{3}{4}>\frac{2}{3}.

        Sometimes it is easier to compare with 1: 65=115\displaystyle {\frac{6}{5}}={1\frac{1}{5}} and 76=116\displaystyle {\frac{7}{6}}={1\frac{1}{6}}, and a fifth is more than a sixth – so 65>76\displaystyle \frac{6}{5}>\frac{7}{6}.

        Exercise 2

        Fill in >\displaystyle >, <\displaystyle < or =\displaystyle {}={}, and explain how you decided:

        49 □ 7935 □ 3856 □ 3446 □ 6998 □ 109\displaystyle \frac{4}{9}\ \square\ \frac{7}{9} \qquad \frac{3}{5}\ \square\ \frac{3}{8} \qquad \frac{5}{6}\ \square\ \frac{3}{4} \qquad \frac{4}{6}\ \square\ \frac{6}{9} \qquad \frac{9}{8}\ \square\ \frac{10}{9}