Multiplying and dividing fractions

You meet mul­ti­ply­ing and divid­ing with frac­tions far more often than you might think. Half of three quar­ters of a litre of milk, two thirds of a class of 36 stu­dents, how many $\displaystyle \frac{3}{4}$-metre pieces can be cut from a rib­bon: each of these is a prod­uct or a quo­tient of frac­tions. The same rules work for ratio­nal num­bers below zero, such as a fall in tem­per­a­ture or an amount owed, so this les­son treats both together.

By the end you should be able to mul­ti­ply any two frac­tions, can­cel before you mul­ti­ply, divide by turn­ing the sec­ond frac­tion over, and explain why each rule works rather than just remem­ber­ing it.

Frac­tions can be mul­ti­plied and divided.

This part is eas­ier than adding.

First, here are the words this les­son uses.

The bot­tom num­ber says how many pieces the cake was cut into.

The top num­ber says how many of those pieces you have.

In $\displaystyle \frac{3}{4}$ the cake was cut into 4 pieces. You have 3 of them.

Adding two frac­tions needs the same bot­tom num­ber in both.

Mul­ti­ply­ing does not need that. Any two frac­tions will do.

Mul­ti­ply­ing two frac­tions

Pic­ture half a cake on a plate.

You want one third of that half.

Cut the half into three equal pieces. Take one piece.

Now think about the whole cake. It would give six pieces of that size.

So your piece is $\displaystyle \frac{1}{6}$ of the cake.

One third of $\displaystyle \frac{1}{2}$ is $\displaystyle \frac{1}{6}$.

Now look at the num­bers.

The two tops are 1 and 1. The two bot­toms are 3 and 2.

One third of a half

$$\displaystyle \begin{aligned}&\frac{1}{3} \times \frac{1}{2} \\ &= \frac{1 \times 1}{3 \times 2} \\ &= \frac{1}{6}\end{aligned}$$

The tops were mul­ti­plied. The bot­toms were mul­ti­plied. Noth­ing else hap­pened.

Remem­ber. To mul­ti­ply two frac­tions, mul­ti­ply the tops. Then mul­ti­ply the bot­toms. In sym­bols this is $\displaystyle \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$.

One third of a half means one third times a half.

The small word of is doing the mul­ti­ply­ing.

Frac­tions below zero

A frac­tion can sit below zero as well.

Draw a line with 0 in the mid­dle.

To the right of 0 is what you have.

To the left of 0 is what you owe.

$\displaystyle -\frac{5}{3}$ is five thirds on the left of 0.

Think of it as owing five thirds of a cake.

The minus sign only tells you which side of 0 you are on.

The dig­its work in the same way on both sides.

A worked exam­ple

Here is an exam­ple from the chap­ter.

Mul­ti­ply $\displaystyle -\frac{5}{3}$ by $\displaystyle \frac{7}{4}$

$$\displaystyle \begin{aligned}&-\frac{5}{3} \times \frac{7}{4} \\ &= -\frac{5 \times 7}{3 \times 4} \\ &= -\frac{35}{12}\end{aligned}$$

The tops give $5 \times 7 = 35$.

The bot­toms give $3 \times 4 = 12$.

There was one minus sign going in.

So there is one minus sign com­ing out.

Here is the rea­son. You started with some­thing you owe.

Tak­ing $\displaystyle \frac{7}{4}$ of a debt still leaves you owing.

So the answer stays on the left of 0.

Can­cel before you mul­ti­ply

Some­times a top and a bot­tom can both be cut up by the same num­ber.

You can cut both of them down before you mul­ti­ply.

That is called can­celling.

It keeps the num­bers small and the work­ing short.

Mul­ti­ply $\displaystyle -\frac{5}{3}$ by $\displaystyle \frac{9}{4}$

$$\displaystyle \begin{aligned}&-\frac{5}{3} \times \frac{9}{4} \\ &= -\frac{5 \times 9}{3 \times 4} \\ &= -\frac{5 \times 3}{1 \times 4} \\ &= -\frac{15}{4}\end{aligned}$$

The 3 below goes into the 9 above.

$9 \div 3 = 3$, so the 9 becomes 3. The 3 becomes 1.

You can also mul­ti­ply first and can­cel at the end.

That gives $\displaystyle -\frac{45}{12}$.

Both 45 and 12 can be cut into groups of 3.

3 is the biggest num­ber that goes into both.

Cut­ting both by 3 leaves $\displaystyle -\frac{15}{4}$.

Peo­ple write that biggest num­ber as H.C.F for short.

Both roads reach $\displaystyle -\frac{15}{4}$.

Three frac­tions at once

The chap­ter has one longer exam­ple.

It has three frac­tions and two minus signs.

Can­cel as much as you can before you mul­ti­ply.

Mul­ti­ply $\displaystyle -\frac{35}{111}$ by $\displaystyle \frac{37}{45}$ by $\displaystyle -\frac{99}{56}$

$$\displaystyle \begin{aligned}&-\frac{35}{111} \times \frac{37}{45} \times -\frac{99}{56} \\ &= \frac{35 \times 37 \times 99}{111 \times 45 \times 56} \\ &= \frac{5 \times 37 \times 99}{111 \times 45 \times 8} \\ &= \frac{5 \times 1 \times 99}{3 \times 45 \times 8} \\ &= \frac{5 \times 11}{3 \times 5 \times 8} \\ &= \frac{11}{24}\end{aligned}$$

Each line cuts one pair down.

35 and 56 both go into groups of 7. They become 5 and 8.

37 goes into 111 three times. They become 1 and 3.

99 and 45 both go into groups of 9. They become 11 and 5.

The last 5 above and the 5 below can­cel too.

That leaves 11 on top and $3 \times 8 = 24$ below.

There were two minus signs going in.

A minus sign flips a num­ber to the other side of 0.

Two flips take you back where you started.

So two minus signs give a plus.

A whole num­ber times a frac­tion

A whole num­ber is a frac­tion too.

Write 6 as $\displaystyle \frac{6}{1}$.

A bot­tom num­ber of 1 means the cakes were never cut.

Six whole cakes is six pieces of size one.

Then mul­ti­ply in the usual way.

Mul­ti­ply $-6$ by $\displaystyle \frac{24}{25}$

$$\displaystyle \begin{aligned}&-6 \times \frac{24}{25} \\ &= -\frac{6}{1} \times \frac{24}{25} \\ &= -\frac{6 \times 24}{1 \times 25} \\ &= -\frac{144}{25}\end{aligned}$$

$6 \times 24 = 144$ goes on top. The bot­tom stays 25.

One minus sign went in, so one comes out.

More worked exam­ples

Can­celling across: $\displaystyle \frac{3}{4} \times \frac{8}{9}$

The 4 below goes into the 8 above twice. The 3 above goes into the 9 below three times.

$$\displaystyle \begin{aligned}\frac{3}{4} \times \frac{8}{9} &= \frac{1 \times 2}{1 \times 3} \\ &= \frac{2}{3}\end{aligned}$$

Two neg­a­tives: $\displaystyle -\frac{2}{5} \times -\frac{15}{8}$

Two minus signs go in, so the answer is pos­i­tive. Then 5 goes into 15 three times, and 2 goes into 8 four times.

$$\displaystyle \begin{aligned}-\frac{2}{5} \times -\frac{15}{8} &= \frac{2 \times 15}{5 \times 8} \\ &= \frac{1 \times 3}{1 \times 4} \\ &= \frac{3}{4}\end{aligned}$$

A frac­tion of an amount

A class has 36 stu­dents and $\displaystyle \frac{2}{3}$ of them walk to school. How many walk?

$$\displaystyle \begin{aligned}\frac{2}{3} \times 36 &= \frac{2 \times 36}{3} \\ &= \frac{72}{3} \\ &= 24\end{aligned}$$

Twenty-four stu­dents walk. Tak­ing a third of 36 gives 12, and two thirds is twice that.

Zero

Zero times any frac­tion gives 0.

So $\displaystyle 0 \times \frac{1}{2} = 0$.

Take the half zero times. You end up with noth­ing.

Why the answer can get smaller

This part often feels wrong at first.

Mul­ti­ply­ing used to make num­bers grow.

$3 \times 4 = 12$ is big­ger than 3 and big­ger than 4.

But $\displaystyle \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$.

That is smaller than both of them.

Here is the rea­son.

$3 \times 4$ means three lots of 4.

You take the 4 three whole times. The pile grows.

$\displaystyle \frac{1}{3} \times \frac{1}{2}$ means one third of a half.

You do not take the half even once.

You take only a part of it.

A part of a cake is smaller than the cake.

Leave debts aside for the rest of this part.

Every­thing here is about amounts you have, above 0.

So watch the sec­ond num­ber.

$\displaystyle \frac{15}{2}$ is seven and a half. That is big­ger than 6.

$\displaystyle \frac{24}{25}$ is a lit­tle less than 1.

So $\displaystyle 6 \times \frac{24}{25}$ is a lit­tle less than 6.

Six lots of 25 make 150, and $\displaystyle \frac{144}{25}$ has only 144.

So $\displaystyle \frac{144}{25}$ is smaller than 6, just as we expected.

Remem­ber. Start with a num­ber above 0. Mul­ti­ply it by more than 1 and the answer grows. Mul­ti­ply by 1 and noth­ing changes. Mul­ti­ply by a num­ber between 0 and 1 and the answer shrinks.

Below 0 the words swap over, so take care there.

$\displaystyle -\frac{5}{3} \times \frac{9}{4}$ gives $\displaystyle -\frac{15}{4}$, which sits fur­ther left.

The rec­i­p­ro­cal

Turn a frac­tion upside down.

The new frac­tion is called its rec­i­p­ro­cal.

Remem­ber. The rec­i­p­ro­cal of $\displaystyle \frac{a}{b}$ is $\displaystyle \frac{b}{a}$. The top and the bot­tom swap places.

A whole num­ber has a rec­i­p­ro­cal too.

7 is the same as $\displaystyle \frac{7}{1}$.

Turn it over to get $\displaystyle \frac{1}{7}$.

A num­ber times its rec­i­p­ro­cal always gives 1.

Look at $\displaystyle \frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1$.

The top is $2 \times 3$. The bot­tom is $3 \times 2$.

What­ever frac­tion you pick, the same two num­bers get mul­ti­plied twice.

So the answer is always a num­ber over itself.

A num­ber over itself is 1.

Zero has no rec­i­p­ro­cal.

Turn­ing $\displaystyle \frac{0}{1}$ over would give $\displaystyle \frac{1}{0}$.

You can­not divide by 0.

Here is why. Ask how many pieces of noth­ing fit into one cake.

You could keep putting them in for ever.

The cake would never fill up. So there is no answer.

Divid­ing frac­tions

Divid­ing asks a ques­tion.

It asks how many of these fit into that.

Take $\displaystyle 1 \div \frac{1}{4}$.

How many quar­ters fit into one whole cake? Four of them fit.

So $\displaystyle 1 \div \frac{1}{4} = 4$.

Look at what just hap­pened.

You turned $\displaystyle \frac{1}{4}$ over and got 4.

Now try two whole cakes.

How many quar­ters fit into 2? Eight of them fit.

And $2 \times 4 = 8$.

So you turn the quar­ter over to get 4.

Then you mul­ti­ply by the 2 you started with.

The flip and the mul­ti­ply each do a job.

Small pieces fit in many times.

So a small num­ber to divide by gives a big answer.

Turn­ing a frac­tion over makes a small num­ber big.

Here is a sec­ond way to see it.

Divid­ing by 2 is the same as tak­ing half.

Tak­ing half means mul­ti­ply­ing by $\displaystyle \frac{1}{2}$.

And $\displaystyle \frac{1}{2}$ is the rec­i­p­ro­cal of 2.

The two jobs are really one job.

Remem­ber. To divide, turn the sec­ond frac­tion over. Then mul­ti­ply. In sym­bols, $\displaystyle \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$, as long as $c$ is not 0.

Two ways to write it

The chap­ter often writes divi­sion with­out a $\div$ sign.

It puts one frac­tion above the other instead.

The same job writ­ten two ways

$$\displaystyle \frac{a}{b} \div \frac{c}{d} = \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \times \frac{d}{c}$$

The long line in the mid­dle is the divide sign.

Both ways mean the same thing.

The steps

1. If a num­ber has a whole part and a frac­tion part, change it into a sin­gle frac­tion first.
2. Turn the sec­ond frac­tion over.
3. Change the $\div$ sign to a $\times$ sign.
4. Can­cel where a top and a bot­tom can both be cut by the same num­ber.
5. Mul­ti­ply the tops, then mul­ti­ply the bot­toms.

Divide $\displaystyle \frac{2}{3}$ by $\displaystyle \frac{5}{6}$

$$\displaystyle \begin{aligned}&\frac{2}{3} \div \frac{5}{6} \\ &= \frac{2}{3} \times \frac{6}{5} \\ &= \frac{2 \times 6}{3 \times 5} \\ &= \frac{12}{15} \\ &= \frac{4}{5}\end{aligned}$$

$\displaystyle \frac{12}{15}$ can still be made sim­pler.

Both 12 and 15 can be cut into groups of 3.

Cut the top and the bot­tom by 3. That leaves $\displaystyle \frac{4}{5}$.

Cut­ting both by the same num­ber does not change how much you have.

It only counts the same cake in big­ger pieces.

$\displaystyle \frac{12}{15}$ and $\displaystyle \frac{4}{5}$ are the same amount.

Now no num­ber goes into 4 and 5 any more.

A frac­tion like that is in its sim­plest form.

Divide $\displaystyle \frac{2}{3}$ by $2$

$$\displaystyle \begin{aligned}&\frac{2}{3} \div 2 \\ &= \frac{2}{3} \times \frac{1}{2} \\ &= \frac{2}{3 \times 2} \\ &= \frac{2}{6} \\ &= \frac{1}{3}\end{aligned}$$

Two thirds of a cake shared by two peo­ple.

Each one gets one third. The answer feels right.

Look at why only the bot­tom changed.

Divid­ing by 2 means mul­ti­ply­ing by $\displaystyle \frac{1}{2}$.

The top is mul­ti­plied by 1, so it stays as it was.

The bot­tom is mul­ti­plied by 2, so only the bot­tom grows.

Remem­ber. Divid­ing by a whole num­ber other than 0 changes only the bot­tom. In sym­bols, $\displaystyle \frac{a}{b} \div c = \frac{a}{bc}$, as long as $c$ is not 0.

Mixed num­bers

$\displaystyle 1\frac{2}{3}$ is a mixed num­ber.

It has a whole part and a frac­tion part.

You can­not turn it over as it stands.

Change it into a sin­gle frac­tion first.

One whole cake cut into thirds gives three thirds.

You already have two thirds more.

That makes five thirds in all.

So $\displaystyle 1\frac{2}{3}$ is $\displaystyle \frac{5}{3}$.

The work­ing below is the short way of count­ing those five thirds.

Mul­ti­ply the whole part by the bot­tom. Then add the top.

Change the mixed num­bers

$$\displaystyle \begin{aligned}1\frac{2}{3} &= \frac{1 \times 3 + 2}{3} = \frac{5}{3} \\ 7\frac{8}{9} &= \frac{7 \times 9 + 8}{9} = \frac{71}{9}\end{aligned}$$

Now divide in the usual way.

Divide $\displaystyle 1\frac{2}{3}$ by $\displaystyle 7\frac{8}{9}$

$$\displaystyle \begin{aligned}&1\frac{2}{3} \div 7\frac{8}{9} \\ &= \frac{5}{3} \div \frac{71}{9} \\ &= \frac{5}{3} \times \frac{9}{71} \\ &= \frac{5 \times 9}{3 \times 71} \\ &= \frac{45}{213} \\ &= \frac{15}{71}\end{aligned}$$

Both 45 and 213 can be cut into groups of 3.

$45 \div 3 = 15$.

$213 \div 3 = 71$.

You could can­cel ear­lier instead.

The 3 below goes into the 9 above, leav­ing 3.

Then the top is $5 \times 3 = 15$.

The bot­tom is $1 \times 71 = 71$.

That is $\displaystyle \frac{15}{71}$ again, reached with smaller num­bers.

More divi­sion exam­ples

A neg­a­tive divi­sor: $\displaystyle \frac{3}{4} \div -\frac{9}{10}$

Turn the sec­ond frac­tion over, keep­ing its minus sign, and mul­ti­ply. One minus sign goes in, so one comes out.

$$\displaystyle \begin{aligned}\frac{3}{4} \div -\frac{9}{10} &= \frac{3}{4} \times -\frac{10}{9} \\ &= -\frac{3 \times 10}{4 \times 9} \\ &= -\frac{1 \times 5}{2 \times 3} \\ &= -\frac{5}{6}\end{aligned}$$

Here 3 went into 9 three times, and 2 went into both 10 and 4.

A mixed num­ber divided by a whole num­ber: $\displaystyle 2\frac{1}{4} \div 3$

$$\displaystyle \begin{aligned}2\frac{1}{4} \div 3 &= \frac{9}{4} \div 3 \\ &= \frac{9}{4 \times 3} \\ &= \frac{9}{12} \\ &= \frac{3}{4}\end{aligned}$$

A word prob­lem

A rib­bon is $\displaystyle 3\frac{3}{4}$ metres long. How many pieces of $\displaystyle \frac{5}{8}$ metre can be cut from it?

$$\displaystyle \begin{aligned}3\frac{3}{4} \div \frac{5}{8} &= \frac{15}{4} \times \frac{8}{5} \\ &= \frac{3 \times 2}{1 \times 1} \\ &= 6\end{aligned}$$

Six pieces. To check, mul­ti­ply back: $\displaystyle 6 \times \frac{5}{8} = \frac{30}{8} = \frac{15}{4}$, which is the length of the rib­bon. Mul­ti­ply­ing the answer by the divi­sor should always give back the num­ber you started with.

The rules together

The first three rules are printed in the chap­ter.

The last one fol­lows from what a rec­i­p­ro­cal is.

Prac­tice

Work these out on paper.

Give each answer in its sim­plest form.

Answers

Three of them are worth a sec­ond look.

In ques­tion 2 the 7 above and the 7 below can­cel.

In ques­tion 5 one minus sign goes in, so one comes out.

In ques­tion 9 change $\displaystyle 2\frac{1}{2}$ into $\displaystyle \frac{5}{2}$ first.

Change $\displaystyle 1\frac{1}{4}$ into $\displaystyle \frac{5}{4}$ as well.

Two answers come out as whole num­bers, not frac­tions.

Remem­ber. Mul­ti­ply­ing needs no com­mon bot­tom num­ber. Divid­ing needs one flip, then a mul­ti­ply.

Com­mon mis­takes

• Look­ing for a com­mon bot­tom num­ber before mul­ti­ply­ing. That step belongs to adding. For mul­ti­ply­ing, just mul­ti­ply tops and bot­toms.
• Turn­ing the wrong frac­tion over. In $\displaystyle \frac{2}{3} \div \frac{5}{6}$ only the sec­ond frac­tion, $\displaystyle \frac{5}{6}$, is turned over. The first stays as it is.
• Turn­ing a mixed num­ber over as it stands. $\displaystyle 1\frac{2}{3}$ must first become $\displaystyle \frac{5}{3}$; its rec­i­p­ro­cal is $\displaystyle \frac{3}{5}$.
• Can­celling two tops together. You may can­cel a top against a bot­tom, never a top against another top.
• Los­ing a minus sign. Count the minus signs going in: one gives a neg­a­tive answer, two give a pos­i­tive answer.
• Writ­ing the whole num­ber on the bot­tom. In $\displaystyle 4 \times \frac{3}{8}$ the 4 mul­ti­plies the top, because $\displaystyle 4 = \frac{4}{1}$.

Key terms

Numer­a­tor

The top num­ber of a frac­tion: how many pieces you have.

Denom­i­na­tor

The bot­tom num­ber: how many equal pieces the whole was cut into. It is never 0.

Ratio­nal num­ber

A num­ber that can be writ­ten as $\displaystyle \frac{p}{q}$ with $p$ and $q$ inte­gers and $q \neq 0$.

Can­celling

Divid­ing a top and a bot­tom by the same num­ber to keep the work­ing small.

Sim­plest form

A frac­tion whose top and bot­tom share no fac­tor other than 1.

Rec­i­p­ro­cal

The frac­tion turned upside down. A num­ber times its rec­i­p­ro­cal is 1.

Mixed num­ber

A whole num­ber and a proper frac­tion writ­ten together, such as $\displaystyle 1\frac{2}{3}$.

H.C.F.

High­est com­mon fac­tor: the biggest num­ber that divides two num­bers exactly.

Answers

1. $\displaystyle \frac{1}{2} \times \frac{3}{5} = \frac{1 \times 3}{2 \times 5} = \frac{3}{10}$
2. $\displaystyle \frac{2}{7} \times \frac{7}{3} = \frac{2 \times 1}{1 \times 3} = \frac{2}{3}$
3. $\displaystyle 4 \times \frac{3}{8} = \frac{12}{8} = \frac{3}{2}$
4. $\displaystyle 0 \times \frac{9}{4} = 0$
5. $\displaystyle -\frac{5}{6} \times \frac{3}{10} = -\frac{1 \times 1}{2 \times 2} = -\frac{1}{4}$
6. $\displaystyle \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}$
7. $\displaystyle \frac{5}{6} \div 5 = \frac{5}{30} = \frac{1}{6}$
8. $\displaystyle \frac{2}{5} \div \frac{4}{5} = \frac{2}{5} \times \frac{5}{4} = \frac{10}{20} = \frac{1}{2}$
9. $\displaystyle 2\frac{1}{2} \div 1\frac{1}{4} = \frac{5}{2} \times \frac{4}{5} = \frac{20}{10} = 2$