Dividing fractions

Fractions, grade 6 Lesson 12 of 22Learning

Dividing fractions

Ten divided by a half

Ten divided by a half – two stories

\displaystyle 10\div\frac{1}{2}

– how much is it? This calculation also has two stories.

The first story: how many halves are there in 10? Cut each cake into two halves.

And count the halves: two in each cake, and in ten cakes – twenty.

There are 20 halves in 10 cakes:

\displaystyle {10\div\frac{1}{2}}={20}

The second story: half a tray holds 10 cakes. How many cakes does a whole tray hold?

A whole tray has two halves, and each half holds 10 cakes – together 20 cakes.

Why is it the same? In both stories every whole has two halves: two half-cakes in a cake, two half-trays in a tray. So in both we multiply by 2 – dividing by a half is multiplying by 2.

Step 1 of 7

Dividing by a unit fraction

Division as “how many times does it fit?”

$\displaystyle 6\div\frac{1}{5}$ asks: how many fifths are there in 6? In each whole there are 5 fifths, so in 6 wholes there are $\displaystyle {6\times5}={30}$ : $\displaystyle {6\div\frac{1}{5}}={30}$ . Dividing by a unit fraction (a fraction with 1 on top) is multiplying by its denominator.

Dividing by any fraction

$\displaystyle 6\div\frac{2}{5}$ : how many times does $\displaystyle \frac{2}{5}$ fit into 6? $\displaystyle \frac{2}{5}$ is twice $\displaystyle \frac{1}{5}$ , so it fits half as many times: $\displaystyle {30\div2}={15}$ . As a calculation: $\displaystyle {6\div\frac{2}{5}}={6\times5\div2}={6\times\frac{5}{2}}={15}$ .

Multiplying by the reciprocal

\displaystyle \frac{5}{2}

is the reciprocal of

\displaystyle \frac{2}{5}

: the fraction upside down.

Dividing by a fraction is multiplying by its reciprocal: $\displaystyle {\frac{3}{4}\div\frac{5}{6}}={\frac{3}{4}\times\frac{6}{5}}$ .

Multiplying both numbers of a division

Multiply both numbers of a division by the same number, and the result stays the same: $\displaystyle {6\div2}={60\div20}={3}$ . Multiply both by 6, and the number you divide by becomes whole: $\displaystyle {\frac{3}{4}\div\frac{5}{6}}={\left(\frac{3}{4}\times6\right)\div\left(\frac{5}{6}\times6\right)}={\left(\frac{3}{4}\times6\right)\div5}={\frac{3}{4}\times6\times\frac{1}{5}}={\frac{3}{4}\times\frac{6}{5}}$

Dividing mixed numbers

Write them as fractions first, then multiply by the reciprocal: $\displaystyle {2\frac{1}{4}\div1\frac{1}{2}}={\frac{9}{4}\div\frac{3}{2}}={\frac{9}{4}\times\frac{2}{3}}={\frac{18}{12}}={1\frac{1}{2}}$ .

Exercise 1

Divide.
Write each result in its lowest terms, or as a mixed number.

How many thirds in 4?

/ – 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

Exercise 2

An adult’s step is $\displaystyle \frac{3}{4}$ m long.
How many steps does it take to cross a yard 60 m long?

/ – 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