Multiplying and dividing decimals

Dec­i­mals turn up when­ever we mea­sure or pay. Petrol costs so many rupees for each litre, cloth is sold by the metre, and a recipe may need $0.25$ of a kilo­gram of sugar. To find the cost of $2.5$ metres of cloth, or how many $0.6$ m pieces a rib­bon makes, you must mul­ti­ply or divide dec­i­mals. The good news is that you already know how: you mul­ti­ply and divide as with whole num­bers, and one short rule tells you where the point goes.

A dec­i­mal is a num­ber with a point in it. The Rs $2.50$ on a price label is a dec­i­mal. The part after the point is less than one whole.

You have already added and sub­tracted dec­i­mals. Now you will mul­ti­ply and divide them. A jug, a ruler and some money will show you why the rules work.

Mul­ti­ply­ing dec­i­mals

A jug holds $8$ glasses of water. Fill it half full. Half is $0.5$. You now have $4$ glasses. Half a jug is less than a full jug. So $0.5 \times 8 = 4$, and $4$ is smaller than $8$.

A dec­i­mal below $1$ is only part of one whole. So mul­ti­ply­ing by it gives you a smaller answer. A dec­i­mal above $1$ works the other way. $1.2 \times 0.4 = 0.48$, and $0.48$ is big­ger than $0.4$.

See it on a ruler first

Take a strip $1$ metre long. Take $0.3$ of the strip. That piece is $30$ cm. Now take $0.2$ of that piece. You get $6$ cm.

Six cen­time­tres out of a whole metre is $0.06$ of the metre. So $0.2 \times 0.3 = 0.06$. The rule below is the short way to that same answer.

A hun­dred square shows the same thing. Let the whole square stand for $1$. Shade $0.3$ of it one way and $0.2$ of it the other way. The part shaded both ways is $0.2$ of $0.3$.

The method

The method has two steps. First mul­ti­ply the num­bers as if the points were not there. Then count the dig­its after the point in both num­bers.

A digit is one sin­gle num­ber sign, like the $2$ or the $3$. In $0.2$ there is one digit after the point. Your answer needs as many dig­its after the point as the two num­bers have between them.

The chap­ter's exam­ple

Work out $0.2 \times 0.3$. Cover the points and mul­ti­ply $2 \times 3$. You get $6$. Now count. $0.2$ has $1$ digit after the point. $0.3$ has $1$ digit after the point. So you count $1 + 1 = 2$ dig­its.

The answer needs two dig­its after the point. Start at the right of $6$ and count two places. You need a zero to fill the gap. The answer is $0.06$.

Mul­ti­ply­ing $0.2 \times 0.3$

$$\begin{aligned}2 \times 3 &= 6 \\ 1 + 1 &= 2 \text{ digits after the point} \\ 0.2 \times 0.3 &= 0.06\end{aligned}$$

Why the rule works

Cut one rupee into $10$ equal parts. Each part is one tenth. Two of them is $0.2$ of the rupee.

Now cut each tenth into $10$ again. You get $100$ tiny parts of a rupee. Each one is a hun­dredth. Hun­dredths sit in the sec­ond place after the point.

A frac­tion is a way of writ­ing part of some­thing. $\displaystyle \frac{2}{10}$ means two parts out of ten. So $0.2$ is $\displaystyle \frac{2}{10}$, and $0.3$ is $\displaystyle \frac{3}{10}$.

Mul­ti­ply the top num­bers: $2 \times 3 = 6$. Mul­ti­ply the bot­tom num­bers: $10 \times 10 = 100$. So the answer is $\displaystyle \frac{6}{100}$.

$\displaystyle \frac{6}{100}$ is six hun­dredths. Hun­dredths take two places after the point. That gives $0.06$. The count­ing rule is this frac­tion work made short.

Why $0.2 \times 0.3$ gives $0.06$

$$\displaystyle \begin{aligned}0.2 \times 0.3 &= \frac{2}{10} \times \frac{3}{10} \\ &= \frac{6}{100} \\ &= 0.06\end{aligned}$$

Remem­ber. Mul­ti­ply as if there were no points. Then count the dig­its after the point in both num­bers. Your answer gets that many dig­its after the point.

One more of the same shape

The chap­ter gives one exam­ple only. The sum below is ours. It is built to the same shape as the chap­ter's.

Work out $0.5 \times 0.3$. First mul­ti­ply $5 \times 3$. You get $15$. Each num­ber has one digit after the point. So the answer needs $1 + 1 = 2$ dig­its after the point. The answer is $0.15$.

Mul­ti­ply­ing $0.5 \times 0.3$

$$\begin{aligned}5 \times 3 &= 15 \\ 1 + 1 &= 2 \text{ digits after the point} \\ 0.5 \times 0.3 &= 0.15\end{aligned}$$

You can check that in your head. $0.5$ is a half. Half of $0.3$ is $0.15$. So the answer is right.

Harder sums with the same rule

The rule works for any dec­i­mals, not only tenths. Here are three more, each a lit­tle harder.

Mul­ti­ply­ing $0.4 \times 3$

$$\begin{aligned}4 \times 3 &= 12 \\ 1 + 0 &= 1 \text{ digit after the point} \\ 0.4 \times 3 &= 1.2\end{aligned}$$

A whole num­ber has no dig­its after the point, so it adds noth­ing to the count.

Mul­ti­ply­ing $2.5 \times 1.3$

$$\begin{aligned}25 \times 13 &= 325 \\ 1 + 1 &= 2 \text{ digits after the point} \\ 2.5 \times 1.3 &= 3.25\end{aligned}$$

Check by esti­mat­ing: $2.5$ is between $2$ and $3$, and $1.3$ is a lit­tle more than $1$. So the answer should be a lit­tle more than $2.5$. $3.25$ fits.

Mul­ti­ply­ing $0.25 \times 0.4$

$$\begin{aligned}25 \times 4 &= 100 \\ 2 + 1 &= 3 \text{ digits after the point} \\ 0.25 \times 0.4 &= 0.100 = 0.1\end{aligned}$$

Count the three places first, and only then drop the zeros at the right-hand end. If you drop them first, you will put the point in the wrong place.

Divid­ing a dec­i­mal by a whole num­ber

A whole num­ber is a num­ber with no point in it, like $4$ or $6$.

The worked exam­ple and the rule in this sec­tion are our own, set out in the chap­ter's style.

Four chil­dren share Rs $9.36$ equally. How much does each child get? Here you divide a dec­i­mal by a whole num­ber.

Rs $9.36$ is $9$ rupees and $36$ paise. Share the $9$ rupees first. Each child gets $2$ rupees, and $1$ rupee is left over.

Change that rupee into $100$ paise. Now you have $136$ paise to share. Each child gets $34$ paise. So each child gets Rs $2.34$.

Shar­ing Rs $9.36$ between $4$ chil­dren (ours, not the chap­ter's)

$$\begin{aligned}9 \div 4 &= 2 \text{ rupees each, } 1 \text{ rupee left} \\ 100 + 36 &= 136 \text{ paise left to share} \\ 136 \div 4 &= 34 \text{ paise each} \\ 9.36 \div 4 &= 2.34\end{aligned}$$

You can check by mul­ti­ply­ing back. $2.34 \times 4 = 9.36$. That is the money you started with.

Divid­ing $7.5$ by $3$

$$\begin{aligned}7.5 \div 3 &= 2.5\end{aligned}$$

Three goes into $7$ two times, with $1$ left over. Carry that $1$ across the point, so the $5$ tenths become $15$ tenths. Three goes into $15$ five times. Write the point in the answer straight above the point in $7.5$, and you have $2.5$. Check: $2.5 \times 3 = 7.5$.

Why the point does not move

The rupees stayed rupees, and the paise stayed paise. So each place in the answer means the same as the place above it.

That is why the point in the answer sits straight under the point in the num­ber. You divide just as you do with whole num­bers.

Remem­ber. When you divide by a whole num­ber, the point does not move. Keep the point in the answer straight under the point in the num­ber.

Divid­ing by another dec­i­mal

The worked exam­ple and the rule in this sec­tion are our own as well.

A rib­bon is $4.8$ m long. You cut it into pieces $0.6$ m long. How many pieces do you get? This time you divide by a dec­i­mal.

It is hard to share some­thing into pieces of $0.6$. Shar­ing into pieces of $6$ is much eas­ier. So change the $0.6$ into a whole num­ber first.

Move the point in $0.6$ one place to the right. It becomes $6$. Move the point in $4.8$ one place as well. It becomes $48$.

Move the point the same num­ber of places in both num­bers. Then divide in the usual way.

Divid­ing $4.8$ by $0.6$ (ours, not the chap­ter's)

$$\begin{aligned}&4.8 \div 0.6 \\ &= 48 \div 6 \\ &= 8\end{aligned}$$

You get $8$ pieces. Check it on a ruler. Eight pieces of $0.6$ m come to $8 \times 0.6 = 4.8$ m.

Mov­ing the point two places

Some­times the num­ber you divide by has two dig­its after the point. Then move both points two places.

Divid­ing $1.35$ by $0.15$

$$\begin{aligned}&1.35 \div 0.15 \\ &= 135 \div 15 \\ &= 9\end{aligned}$$

Check: $9 \times 0.15 = 1.35$. Mov­ing both points two places mul­ti­plies both num­bers by $100$, so the answer stays the same.

Why you may move both points

Rs $0.6$ is $60$ paise. Move the point one place right and you get Rs $6$. That is ten times as much.

Mov­ing the point one place right always makes a num­ber ten times big­ger. Every digit slides up one place.

A divi­sion is also a frac­tion. $4.8 \div 0.6$ means $\displaystyle \frac{4.8}{0.6}$. You mul­ti­plied the top by ten. You mul­ti­plied the bot­tom by ten as well.

Mul­ti­ply the top and the bot­tom by the same num­ber, and the frac­tion is still the same amount. $\displaystyle \frac{1}{2}$ and $\displaystyle \frac{10}{20}$ are the same amount. So the answer does not change.

Why $4.8 \div 0.6$ becomes $48 \div 6$

$$\displaystyle \begin{aligned}\frac{4.8}{0.6} &= \frac{4.8 \times 10}{0.6 \times 10} \\ &= \frac{48}{6} \\ &= 8\end{aligned}$$

Remem­ber. Move the point in both num­bers by the same num­ber of places. Move it until the num­ber you divide by is whole. Then divide.

Prac­tice

1. Share Rs $6.25$ between $5$ chil­dren. How much does each child get?
2. A rib­bon is $3.6$ m long. You cut it into pieces $0.4$ m long. How many pieces do you get?
3. A strip is $1$ m long. Take $0.9$ of the strip, then take $0.3$ of that piece. How much of the metre is that?

Com­mon mis­takes

• Lin­ing up the points when mul­ti­ply­ing, as in addi­tion. For mul­ti­ply­ing you count dig­its instead.
• Count­ing dig­its in only one num­ber. Count the dig­its after the point in both num­bers and add the counts.
• For­get­ting the fill­ing zero. $0.2 \times 0.3$ is $0.06$, not $0.6$.
• Mov­ing the point in only one num­ber when divid­ing by a dec­i­mal. Move it the same num­ber of places in both.
• Expect­ing mul­ti­ply­ing to always make a num­ber big­ger. Mul­ti­ply­ing by a dec­i­mal below $1$ makes it smaller.

Key terms

Dec­i­mal

A num­ber with a point in it, such as $2.34$.

Digit after the point

A digit to the right of the dec­i­mal point; each one is a place of tenths, hun­dredths and so on.

Tenth

One of ten equal parts of a whole, writ­ten $0.1$.

Hun­dredth

One of a hun­dred equal parts of a whole, writ­ten $0.01$.

Whole num­ber

A num­ber with no point in it, such as $4$.

Esti­mate

A rough answer worked out with easy num­bers, used to check where the point goes.

Answers

Word prob­lems

1. $6.25 \div 5 = 1.25$, so each child gets Rs $1.25$.
2. $3.6 \div 0.4 = 36 \div 4 = 9$ pieces.
3. $0.9 \times 0.3 = 0.27$ of the metre, which is $27$ cm.

Sums

1. $0.6 \times 0.7 = 0.42$
2. $1.2 \times 0.4 = 0.48$
3. $8.4 \div 7 = 1.2$
4. $7.35 \div 5 = 1.47$
5. $0.96 \div 0.3 = 9.6 \div 3 = 3.2$
6. $12.5 \div 0.5 = 125 \div 5 = 25$