Decimals that stop and decimals that repeat

A rupee is worth 100 paise. So 50 paise is half a rupee. You can write that half as $\displaystyle \frac{50}{100}$. You can also write it as Rs $0.50$. Both mean the same money.

A dec­i­mal is a num­ber with a point in it. $0.5$ and $1.25$ are dec­i­mals. The point splits the num­ber in two. Whole things sit on the left. Parts of a thing sit on the right.

Ten paise is $\displaystyle \frac{10}{100}$ of a rupee. That is the same as $\displaystyle \frac{1}{10}$ of a rupee. You write it as Rs $0.10$.

So a frac­tion and a dec­i­mal can hold the same value. $\displaystyle \frac{5}{10}$ is a frac­tion. Its dec­i­mal is $0.5$. They are two ways of writ­ing one amount.

Some frac­tions give a dec­i­mal that stops. Some give a dec­i­mal that never stops. This les­son shows you how to tell which is which.

You meet dec­i­mals on price tags, weigh­ing scales, ther­mome­ters and cricket run rates. Know­ing how they con­nect to frac­tions lets you move between the two with­out a cal­cu­la­tor. It also pre­pares you for the next big idea in this course: num­bers whose dec­i­mals never stop and never repeat, called irra­tional num­bers.

Tenths, hun­dredths and thou­sandths

Dec­i­mal places are built on ten. The first place after the point is tenths. The next place is hun­dredths. The next is thou­sandths. Each place is ten times smaller than the one before.

So a dec­i­mal is really a frac­tion over ten, or a hun­dred, or a thou­sand. $0.1$ is $\displaystyle \frac{1}{10}$. $0.01$ is $\displaystyle \frac{1}{100}$. $0.001$ is $\displaystyle \frac{1}{1000}$.

Money works the same way. One paise is $\displaystyle \frac{1}{100}$ of a rupee. So one paise is Rs $0.01$.

Say each num­ber slowly. $0.1$ is read as zero point one. $1.2$ is read as one point two. $0.01$ is read as zero point zero one. $0.001$ is read as zero point zero zero one.

Remem­ber. Read the dig­its after the point one at a time. $12.35$ is read as twelve point three five. It is not read as twelve point thirty five.

The same rule holds for longer num­bers. $1.235$ is read as one point two three five.

Noughts at the very front do no work. Noughts at the very end after the point do no work either. So $0001035.5300120000$ can be writ­ten as $1035.530012$.

Expanded form

Expanded form splits a num­ber into the value of each digit.

Expand $457.368$

$$\displaystyle \begin{aligned}457.368 &= 400 + 50 + 7 + \frac{3}{10} + \frac{6}{100} + \frac{8}{1000} \\ &= 400 + 50 + 7 + 0.3 + 0.06 + 0.008\end{aligned}$$

You read it as four hun­dred fifty seven and three tenths six hun­dredths and eight thou­sandths.

Place value

Place value tells you what a digit is worth where it sits. In $457.368$ the place value of 4 is $400$. The place value of 3 is $\displaystyle \frac{3}{10}$. The place value of 8 is $\displaystyle \frac{8}{1000}$.

Grams and rupees

Weight uses this idea every day. $1000$ grams make 1 kilo­gram. So 1 gram is $\displaystyle \frac{1}{1000}$ of a kilo­gram. That is $0.001$ kg.

Money works the same way. $100$ paise make 1 rupee. So 1 paise is $\displaystyle \frac{1}{100}$ of a rupee.

Write 2 kg 348 g in kilo­grams

$$\begin{aligned}2 \text{ kg } 348 \text{ g} &= 2 \text{ kg} + 348 \text{ g} \\ &= 2 \text{ kg} + 0.348 \text{ kg} \\ &= 2.348 \text{ kg}\end{aligned}$$

Write Rs 24 and 50 paise in rupees

$$\displaystyle \begin{aligned}24 \text{ rupees } 50 \text{ paise} &= 24 + \frac{50}{100} \\ &= 24 + 0.5 \\ &= \text{Rs } 24.5\end{aligned}$$

Why ten mat­ters

Ten, a hun­dred, a thou­sand. Each one is the one before with another nought on the end. These are called the pow­ers of ten.

Look at what they are made of. Ten is $2 \times 5$. A hun­dred is $2 \times 2 \times 5 \times 5$. A thou­sand is $2 \times 2 \times 2 \times 5 \times 5 \times 5$.

Only 2s and 5s, every time, how­ever far you go. That one fact dri­ves the whole rule below.

Sim­plest form comes first

The bot­tom of a frac­tion is the num­ber under the line. In $\displaystyle \frac{3}{8}$ the bot­tom is 8. It tells you how many equal pieces one whole was cut into.

A frac­tion is in its sim­plest form when no num­ber divides into both the top and the bot­tom.

$\displaystyle \frac{6}{15}$ is not in sim­plest form yet. 3 goes into 6, and 3 goes into 15. Divide both by 3 and you get $\displaystyle \frac{2}{5}$.

That is the same amount, writ­ten as small as it will go. Mak­ing a frac­tion smaller like this is called can­celling.

The test

Remem­ber. Put the frac­tion in its sim­plest form first. Then look at the bot­tom. If the bot­tom is made of 2s and 5s only, the dec­i­mal stops. If any other num­ber is in there, the dec­i­mal repeats for ever.

Here is the rea­son, in plain words. You want the bot­tom to become ten, or a hun­dred, or a thou­sand. You can only build those out of 2s and 5s.

A bot­tom of 8 is $2 \times 2 \times 2$. Feed it three 5s and it becomes $1000$. A bot­tom of 3 can never become a power of ten. No amount of mul­ti­ply­ing will clear that 3 away.

So test the tidy frac­tion, never the untidy one. $\displaystyle \frac{6}{15}$ looks like it fails, because its bot­tom holds a 3. But in sim­plest form the bot­tom is just 5.

Test $\displaystyle \frac{6}{15}$

$$\displaystyle \begin{aligned}\frac{6}{15} &= \frac{2}{5} \\ &= \frac{2 \times 2}{5 \times 2} \\ &= \frac{4}{10} \\ &= 0.4\end{aligned}$$

A frac­tion into a dec­i­mal, with no divid­ing

You can turn the bot­tom into a power of ten. Then you read the dec­i­mal straight off.

The two frac­tions below are our own exam­ples. Your book sets its own num­bers here. The method is the same one.

Take $\displaystyle \frac{3}{8}$. The bot­tom is $8 = 2 \times 2 \times 2$. It has three 2s and no 5s. Give it three 5s and it reaches $1000$. Three 5s mul­ti­ply to $125$.

Write $\displaystyle \frac{3}{8}$ as a dec­i­mal

$$\displaystyle \begin{aligned}\frac{3}{8} &= \frac{3 \times 125}{8 \times 125} \\ &= \frac{375}{1000} \\ &= 0.375\end{aligned}$$

What­ever you do to the bot­tom, you do to the top. Mul­ti­ply­ing both by 125 is really mul­ti­ply­ing by $\displaystyle \frac{125}{125}$. And $\displaystyle \frac{125}{125}$ is just 1.

Mul­ti­ply­ing by 1 never changes an amount. So $\displaystyle \frac{3}{8}$ and $\displaystyle \frac{375}{1000}$ are the same size. Only the writ­ing is dif­fer­ent.

Now pic­ture a mea­sur­ing jug with a hun­dred small marks up the side. Full is one whole. Cut the jug into 20 equal parts. Each part is 5 of those marks.

So $\displaystyle \frac{7}{20}$ of the jug is seven of those parts. That is $7 \times 5 = 35$ marks. The work­ing below just writes down what you counted.

The bot­tom is $20 = 2 \times 2 \times 5$. It has two 2s and one 5. One more 5 lifts it to $100$.

Write $\displaystyle \frac{7}{20}$ as a dec­i­mal

$$\displaystyle \begin{aligned}\frac{7}{20} &= \frac{7 \times 5}{20 \times 5} \\ &= \frac{35}{100} \\ &= 0.35\end{aligned}$$

35 marks out of a hun­dred is $0.35$.

Here is a harder one to try the method on. Take $\displaystyle \frac{5}{16}$. The bot­tom is $16 = 2 \times 2 \times 2 \times 2$, four 2s and no 5s. It needs four 5s, and four 5s mul­ti­ply to $625$.

Write $\displaystyle \frac{5}{16}$ as a dec­i­mal

$$\displaystyle \begin{aligned}\frac{5}{16} &= \frac{5 \times 625}{16 \times 625} \\ &= \frac{3125}{10000} \\ &= 0.3125\end{aligned}$$

Four 2s matched with four 5s make $10000$, so the dec­i­mal has four places.

When the dec­i­mal repeats

$\displaystyle \frac{1}{3}$ fails the test. Three is not a 2 and not a 5. So its dec­i­mal can­not stop. Here is what goes wrong.

Share one rupee between three chil­dren. Each child gets 33 paise. One paise is left over. Break that paise into ten smaller bits.

Share those bits out. Each child gets 3 more, and one bit is left over again. The left­over never goes away. It only gets smaller.

The bit left over is called the remain­der. It is what is still there after shar­ing. Now do the same thing as a divi­sion.

Divide 1 by 3

$$\begin{aligned}1 \div 3 &= 0 \text{ , remainder } 1 \\ 10 \div 3 &= 3 \text{ , remainder } 1 \\ 10 \div 3 &= 3 \text{ , remainder } 1 \\ 10 \div 3 &= 3 \text{ , remainder } 1 \\ &\text{and so on for ever}\end{aligned}$$

The remain­der is 1 every time. So the same step comes round again for ever. You get $0.333\ldots$ and the 3s carry on.

Write a bar over the dig­its that repeat. So $\displaystyle \frac{1}{3} = 0.\overline{3}$. The bar saves you writ­ing 3s all day.

$\displaystyle \frac{1}{6}$ fails too. Six is $2 \times 3$. That 3 spoils it, even though the 2 is fine. So $\displaystyle \frac{1}{6} = 0.1\overline{6}$.

Remem­ber. Your book uses longer words for these. A dec­i­mal that stops is called a ter­mi­nat­ing dec­i­mal. A dec­i­mal that repeats for ever is called a recur­ring dec­i­mal. The bot­tom of a frac­tion is called the denom­i­na­tor. Same things, longer names.

Going back: a dec­i­mal into a frac­tion

$1.2$ has one digit after the point. So it is counted in tenths. Write it over ten, then can­cel.

Write $1.2$ as a ratio­nal num­ber

$$\displaystyle \begin{aligned}1.2 &= \frac{12}{10} \\ &= \frac{6}{5}\end{aligned}$$

$1.25$ has two dig­its after the point. So it is counted in hun­dredths. Write it over a hun­dred.

Write $1.25$ as a ratio­nal num­ber

$$\displaystyle \begin{aligned}1.25 &= \frac{125}{100} \\ &= \frac{5}{4}\end{aligned}$$

So count the dig­its after the point. One digit means tenths. Two dig­its mean hun­dredths. Three mean thou­sandths.

That is the same lad­der you saw at the start. Each place is ten times smaller than the one before. The last digit tells you what to write the num­ber over.

A repeat­ing dec­i­mal into a frac­tion

A repeat­ing dec­i­mal has no last digit. So you can­not sim­ply write it over a hun­dred. There is a trick for these.

The num­ber below is our own exam­ple. Your book works this method on its own num­ber.

Give the num­ber a name. Call it $x$. Then mul­ti­ply by ten. That shifts every digit one place to the left.

One digit repeats here, just the 3. Ten moves the num­ber along by exactly one digit. So the two end­less tails line up under each other.

From the point onwards, $10x$ and $x$ then have the same end­less run of dig­its. Tak­ing one line from the other wipes that run out. A block of two repeat­ing dig­its would need a hun­dred instead.

Write $0.\overline{3}$ as a ratio­nal num­ber

$$\displaystyle \begin{aligned}\text{Let } x &= 0.\overline{3} \\ 10x &= 3.\overline{3} \\ 10x - x &= 3.\overline{3} - 0.\overline{3} \\ 9x &= 3 \\ x &= \frac{3}{9} = \frac{1}{3}\end{aligned}$$

Now a block of two repeat­ing dig­its. Here you mul­ti­ply by a hun­dred, because a hun­dred moves the num­ber along by two dig­its.

Write $0.\overline{27}$ as a ratio­nal num­ber

$$\displaystyle \begin{aligned}\text{Let } x &= 0.\overline{27} \\ 100x &= 27.\overline{27} \\ 100x - x &= 27 \\ 99x &= 27 \\ x &= \frac{27}{99} = \frac{3}{11}\end{aligned}$$

Check it with the test: the bot­tom is $11$, which is not made of 2s and 5s. So the dec­i­mal of $\displaystyle \frac{3}{11}$ must repeat, and it does.

So $0.\overline{3}$ is $\displaystyle \frac{1}{3}$. That matches the divi­sion you did ear­lier. The two ends of the les­son meet.

Ratio­nal num­bers

A ratio­nal num­ber is any num­ber you can write as one whole num­ber over another. $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{6}{5}$ are both ratio­nal.

You have just seen both kinds of dec­i­mal turn back into frac­tions. $1.2$ became $\displaystyle \frac{6}{5}$. And $0.\overline{3}$ became $\displaystyle \frac{1}{3}$.

Remem­ber. A dec­i­mal that stops is a ratio­nal num­ber. A dec­i­mal that repeats for ever is a ratio­nal num­ber too. Both can be writ­ten as one whole num­ber over another.

Prac­tice

These six ques­tions are ours, not your book's.

1. Write $\displaystyle \frac{1}{4}$ as a dec­i­mal.
2. Write $\displaystyle \frac{9}{20}$ as a dec­i­mal.
3. Write $\displaystyle \frac{1}{8}$ as a dec­i­mal.
4. Does $\displaystyle \frac{5}{6}$ stop or repeat?
5. Write $1.5$ as a ratio­nal num­ber.
6. Write $0.75$ as a ratio­nal num­ber.

Com­mon mis­takes

• Read­ing $12.35$ as "twelve point thirty five". Read the dig­its after the point one at a time.
• Test­ing a frac­tion before can­celling it. $\displaystyle \frac{6}{15}$ looks as if it repeats, but it is $\displaystyle \frac{2}{5} = 0.4$.
• Mul­ti­ply­ing only the bot­tom of a frac­tion. What­ever you do to the bot­tom, do to the top.
• Writ­ing 85 g as $0.85$ kg. A gram is a thou­sandth of a kilo­gram, so 85 g is $0.085$ kg.
• Putting the bar over the wrong dig­its. In $\displaystyle \frac{1}{6} = 0.1\overline{6}$ only the 6 repeats, so the bar sits over the 6 alone.
• Mul­ti­ply­ing by ten for a two-digit repeat­ing block. Use a hun­dred, so the end­less tails line up.

Key terms

Dec­i­mal

A num­ber writ­ten with a point, whole parts to the left and parts of a whole to the right.

Place value

What a digit is worth because of the place where it sits.

Denom­i­na­tor

The bot­tom of a frac­tion: how many equal pieces one whole is cut into.

Sim­plest form

A frac­tion whose top and bot­tom share no fac­tor except 1.

Ter­mi­nat­ing dec­i­mal

A dec­i­mal that stops.

Recur­ring dec­i­mal

A dec­i­mal in which a digit or block of dig­its repeats for ever, shown with a bar.

Ratio­nal num­ber

Any num­ber that can be writ­ten as one whole num­ber over another (the bot­tom not zero).

Answers

1. $\displaystyle \frac{1}{4} = \frac{1 \times 25}{4 \times 25} = \frac{25}{100} = 0.25$
2. $\displaystyle \frac{9}{20} = \frac{9 \times 5}{20 \times 5} = \frac{45}{100} = 0.45$
3. $\displaystyle \frac{1}{8} = \frac{1 \times 125}{8 \times 125} = \frac{125}{1000} = 0.125$
4. It repeats. $\displaystyle \frac{5}{6}$ is already in sim­plest form and $6 = 2 \times 3$ has a 3, so $\displaystyle \frac{5}{6} = 0.8\overline{3}$.
5. $\displaystyle 1.5 = \frac{15}{10} = \frac{3}{2}$
6. $\displaystyle 0.75 = \frac{75}{100} = \frac{3}{4}$