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 we'll han­dle both together.

By the end you should be able to mul­ti­ply any two frac­tions, can­cel before you mul­ti­ply, and divide by turn­ing the sec­ond frac­tion over. Bet­ter still, you'll be able to explain why each rule works instead of just remem­ber­ing it.

Here's some good news to start with: mul­ti­ply­ing and divid­ing frac­tions is actu­ally eas­ier than adding them.

A quick reminder of the words first. The bot­tom num­ber says how many pieces the cake was cut into, and the top num­ber says how many of those pieces you have. In $\displaystyle \frac{3}{4}$ the cake was cut into 4 pieces and you have 3 of them.

When you add two frac­tions, you need the same bot­tom num­ber in both. Mul­ti­ply­ing does­n't care about that at all. Any two frac­tions will do.

Mul­ti­ply­ing two frac­tions#

Pic­ture half a cake sit­ting on a plate, and sup­pose you want one third of that half. You cut the half into three equal pieces and take one.

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. In other words, one third of $\displaystyle \frac{1}{2}$ is $\displaystyle \frac{1}{6}$.

Now look closely at the num­bers. The two tops are 1 and 1, and 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 and the bot­toms were mul­ti­plied. That's all; noth­ing else hap­pened.

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

Notice that one third of a half means one third times a half. That lit­tle word "of" is qui­etly 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, and to the left of 0 is what you owe.

So $\displaystyle -\frac{5}{3}$ 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 behave in exactly the same way on both sides.

A worked exam­ple#

Let's try one 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 $5 \times 7 = 35$ and the bot­toms give $3 \times 4 = 12$. One minus sign went in, so one minus sign comes out.

Why? You started with some­thing you owe, and 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 divided by the same num­ber. When that hap­pens, you can cut both of them down before you mul­ti­ply. This is called can­celling, and it keeps the num­bers small and the work­ing short.

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

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

Here the 3 below goes into the 9 above. Since $9 \div 3 = 3$, the 9 becomes 3 and the 3 becomes 1.

You could 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, and 3 is the biggest num­ber that goes into both (peo­ple call that biggest num­ber the H.C.F for short). Cut­ting both by 3 leaves $\displaystyle -\frac{15}{4}$. Either road, you arrive at $\displaystyle -\frac{15}{4}$.

Three frac­tions at once#

The chap­ter has one longer exam­ple, with three frac­tions and two minus signs. Don't panic at the big num­bers. Just 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, so they become 5 and 8. 37 goes into 111 three times, so they become 1 and 3. 99 and 45 both go into groups of 9, so they become 11 and 5. Finally the last 5 above and the 5 below can­cel too, leav­ing 11 on top and $3 \times 8 = 24$ below.

And the signs? Two minus signs went in. A minus sign flips a num­ber to the other side of 0, and 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, in dis­guise. Write 6 as $\displaystyle \frac{6}{1}$. A bot­tom num­ber of 1 means the cakes were never cut, so six whole cakes is sim­ply 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 and 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 $\displaystyle 0 \times \frac{1}{2} = 0$. If you take the half zero times, you end up with noth­ing on your plate.

Why the answer can get smaller#

Many stu­dents find this part feels wrong at first. Mul­ti­ply­ing always used to make num­bers grow: $3 \times 4 = 12$ is big­ger than 3 and big­ger than 4. But $\displaystyle \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$, which is smaller than both of them. What's going on?

$3 \times 4$ means three lots of 4. You take the 4 three whole times, so the pile grows. $\displaystyle \frac{1}{3} \times \frac{1}{2}$, on the other hand, means one third of a half. You don't take the half even once; you take only a part of it, and a part of a cake is always smaller than the cake.

Let's leave debts aside for the rest of this part and only talk about amounts you have, above 0. Keep your eye on the sec­ond num­ber.

$\displaystyle \frac{15}{2}$ is seven and a half, which is big­ger than 6. But $\displaystyle \frac{24}{25}$ is a lit­tle less than 1, so $\displaystyle 6 \times \frac{24}{25}$ should come out a lit­tle less than 6. Does it? Six lots of 25 make 150, and $\displaystyle \frac{144}{25}$ has only 144. So $\displaystyle \frac{144}{25}$ is indeed 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. For exam­ple, $\displaystyle -\frac{5}{3} \times \frac{9}{4}$ gives $\displaystyle -\frac{15}{4}$, which sits fur­ther left.

The rec­i­p­ro­cal#

Turn a frac­tion upside down, and the new frac­tion you get 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 $\displaystyle \frac{7}{1}$, so turn it over to get $\displaystyle \frac{1}{7}$.

Here's a lovely fact: 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 $2 \times 3$ and the bot­tom is $3 \times 2$. What­ever frac­tion you pick, the same two num­bers get mul­ti­plied on top and below, so the answer is always a num­ber over itself, and a num­ber over itself is 1.

Zero, though, has no rec­i­p­ro­cal. Turn­ing $\displaystyle \frac{0}{1}$ over would give $\displaystyle \frac{1}{0}$, and you can­not divide by 0. To see why, ask how many pieces of noth­ing fit into one cake. You could keep putting them in for ever and the cake would never fill up, so there is no answer.

Divid­ing frac­tions#

Divid­ing always asks a ques­tion: 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. 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, and $2 \times 4 = 8$. So you turn the quar­ter over to get 4, then mul­ti­ply by the 2 you started with. The flip and the mul­ti­ply each do a job.

It makes sense, too. Small pieces fit in many times, so divid­ing by a small num­ber gives a big answer, and turn­ing a frac­tion over is exactly what makes a small num­ber big.

Here is a sec­ond way to see it. Divid­ing by 2 is the same as tak­ing half, and 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, putting 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 exactly the same thing, so don't let the stacked ver­sion scare you.

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, so divide the top and the bot­tom by 3 and you are left with $\displaystyle \frac{4}{5}$.

Cut­ting both by the same num­ber does­n't change how much you have; it only counts the same cake in big­ger pieces. So $\displaystyle \frac{12}{15}$ and $\displaystyle \frac{4}{5}$ are the same amount. Now no num­ber goes into 4 and 5 any more, and a frac­tion like that is in its sim­plest form.

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

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

Think of two thirds of a cake shared between two friends. Each one gets one third, so the answer feels right.

Notice that 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, while 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 can't turn it over as it stands, so change it into a sin­gle frac­tion first.

One whole cake cut into thirds gives three thirds, and 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 just 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: $45 \div 3 = 15$ and $213 \div 3 = 71$.

A smarter move is to can­cel ear­lier. The 3 below goes into the 9 above, leav­ing 3. Then the top is $5 \times 3 = 15$ and the bot­tom is $1 \times 71 = 71$. That is $\displaystyle \frac{15}{71}$ again, reached with much friend­lier 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, and 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, and $\displaystyle 1\frac{1}{4}$ into $\displaystyle \frac{5}{4}$ as well.

Did you notice? 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#

Show 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. $\displaystyle 2\frac{1}{2} \div 1\frac{1}{4} = \frac{5}{2} \times \frac{4}{5} = \frac{20}{10} = 2$