Turning decimals back into fractions

In the last les­son you turned frac­tions into dec­i­mals. Now you will go the other way and turn a dec­i­mal back into a frac­tion $\displaystyle \frac{p}{q}$. For a dec­i­mal that stops, this takes one line. For a recur­ring dec­i­mal there is a neat trick: give the num­ber a name, mul­ti­ply it by $10$ or $100$, and take one line away from the other so the end­less tails dis­ap­pear.

Ter­mi­nat­ing dec­i­mals#

Count the dig­its after the point. One digit means tenths, two mean hun­dredths, three mean thou­sandths. Write the num­ber with­out its point over $10$, $100$ or $1000$, then sim­plify.

Exam­ple 1#

Change $1.2$ into a ratio­nal num­ber.

$$\displaystyle 1.2 = \frac{12}{10} = \frac{6}{5}$$

Exam­ple 2#

Change $1.25$ into a ratio­nal num­ber.

$$\displaystyle 1.25 = \frac{125}{100} = \frac{5}{4}$$

Here $125 = 25 \times 5$ and $100 = 25 \times 4$, so we can­cel $25$.

Recur­ring dec­i­mals: the sub­trac­tion trick#

A recur­ring dec­i­mal has no last digit, so you can­not just write it over $100$. Here is the trick.

Exam­ple 3#

Change $0.\overline{3}$ into a ratio­nal num­ber.

Let $x = 0.33333\ldots$

One digit repeats, so mul­ti­ply by $10$. Every digit moves one place to the left.

$$10x = 3.33333\ldots$$

Now write $x$ under $10x$ and take it away. After the point, both lines have the same end­less 3s, so they can­cel out com­pletely.

$$\begin{array}{rcl} 10x &=& 3.33333\ldots \\ (-)\quad x &=& 0.33333\ldots \\ \hline 9x &=& 3.00000\ldots \end{array}$$

So $9x = 3$, which gives $\displaystyle x = \frac{3}{9} = \frac{1}{3}$.

$$\displaystyle 0.\overline{3} = \frac{1}{3}$$

This matches the divi­sion $1 \div 3 = 0.333\ldots$ from the last les­son.

Exam­ple 4#

Change $1.\overline{27}$ into a ratio­nal num­ber.

Let $x = 1.272727\ldots$

Two dig­its repeat, so mul­ti­ply by $100$. That moves every digit two places, and the tails line up again.

$$\begin{array}{rcl} 100x &=& 127.272727\ldots \\ (-)\quad x &=& 1.272727\ldots \\ \hline 99x &=& 126 \end{array}$$

So $\displaystyle x = \frac{126}{99}$. Both $126$ and $99$ can be divided by $9$: $126 = 9 \times 14$ and $99 = 9 \times 11$.

$$\displaystyle 1.\overline{27} = \frac{126}{99} = \frac{14}{11}$$

Remem­ber. One repeat­ing digit: mul­ti­ply by $10$. Two repeat­ing dig­its: mul­ti­ply by $100$. Three: mul­ti­ply by $1000$. The aim is to make the end­less tails line up so they can­cel.

When some dig­its do not repeat#

Some­times a few dig­its come before the repeat­ing block. In $0.2\overline{35}$ the $2$ does not repeat, but $35$ does: $0.2353535\ldots$

The trick still works. We make two num­bers that both have the repeat­ing part straight after the point, and sub­tract.

Exam­ple 5#

Change $0.2\overline{35}$ into a ratio­nal num­ber.

Let $x = 0.2353535\ldots$

Mul­ti­ply by $10$ to move the $2$ past the point. Mul­ti­ply by $1000$ to move the $2$ and one block of $35$ past the point.

$$\begin{array}{rcl} 1000x &=& 235.353535\ldots \\ (-)\quad 10x &=& 2.353535\ldots \\ \hline 990x &=& 233 \end{array}$$

So $\displaystyle x = \frac{233}{990}$. This can­not be sim­pli­fied fur­ther.

Exam­ple 6#

Change $1.5\overline{3}$ into a ratio­nal num­ber.

Let $x = 1.53333\ldots$

$$\begin{array}{rcl} 100x &=& 153.3333\ldots \\ (-)\quad 10x &=& 15.3333\ldots \\ \hline 90x &=& 138 \end{array}$$

So $\displaystyle x = \frac{138}{90}$. Divide the top and the bot­tom by $6$: $138 = 6 \times 23$ and $90 = 6 \times 15$.

$$\displaystyle 1.5\overline{3} = \frac{23}{15}$$

A short cut#

Look at the answers again. There is a pat­tern that saves time.

• Top: write all the dig­its up to the end of the first repeat­ing block, with­out the point. Take away the dig­its that come before the bar.
• Bot­tom: write one $9$ for each repeat­ing digit, then one $0$ for each digit after the point that does not repeat.

Check it on $1.5\overline{3}$. Top: $153 - 15 = 138$. Bot­tom: one repeat­ing digit gives one $9$, and one non-repeat­ing digit after the point gives one $0$. So the bot­tom is $90$, and we get $\displaystyle \frac{138}{90}$, the same as before.

Exam­ple 7#

Change $12.\overline{05}$ into a ratio­nal num­ber.

Short cut. Top: $1205 - 12 = 1193$. Bot­tom: two repeat­ing dig­its, no oth­ers after the point, so $99$.

$$\displaystyle 12.\overline{05} = \frac{1205 - 12}{99} = \frac{1193}{99}$$

Check by sub­trac­tion. Let $x = 12.050505\ldots$

$$\begin{array}{rcl} 100x &=& 1205.050505\ldots \\ (-)\quad x &=& 12.050505\ldots \\ \hline 99x &=& 1193 \end{array}$$

So $\displaystyle x = \frac{1193}{99}$. Both ways agree.

Exam­ple 8#

Change $0.1\overline{3}$ into a ratio­nal num­ber.

Here $0.1\overline{3} = 0.13333\ldots$. You may also see it writ­ten as $0.13\overline{33}$, with the bar over two 3s. It is the same num­ber, but the shorter bar is neater.

Short cut. Top: $13 - 1 = 12$. Bot­tom: one $9$ and one $0$, so $90$.

$$\displaystyle 0.1\overline{3} = \frac{12}{90} = \frac{2}{15}$$

If you used the longer bar, you would get $\displaystyle \frac{1333 - 13}{9900} = \frac{1320}{9900}$. That also sim­pli­fies to $\displaystyle \frac{2}{15}$. The same num­ber always ends at the same sim­plest frac­tion.

Exam­ple 9#

Change $0.02\overline{7}$ into a ratio­nal num­ber.

$0.02\overline{7} = 0.027777\ldots$. Two dig­its after the point do not repeat, and one digit repeats.

$$\displaystyle 0.02\overline{7} = \frac{27 - 2}{900} = \frac{25}{900} = \frac{1}{36}$$

Here $25$ goes into $900$ exactly $36$ times, so the frac­tion becomes $\displaystyle \frac{1}{36}$.

Remem­ber. Every ter­mi­nat­ing dec­i­mal and every recur­ring dec­i­mal can be writ­ten as $\displaystyle \frac{p}{q}$. So they are all ratio­nal num­bers.

Com­mon mis­takes#

• Mul­ti­ply­ing by $10$ when two dig­its repeat. Use $100$, so the tails line up.
• For­get­ting the non-repeat­ing dig­its. In $0.2\overline{35}$ you must first move the $2$ past the point.
• Leav­ing the answer unsim­pli­fied. $\displaystyle \frac{126}{99}$ should become $\displaystyle \frac{14}{11}$.
• Count­ing the whole-num­ber dig­its when mak­ing the bot­tom. In $12.\overline{05}$ only the dig­its after the point decide the 9s and 0s.

Prac­tice#

1. Change $1.4$ into a ratio­nal num­ber.
2. Change $12.02$ into a ratio­nal num­ber.
3. Change $25.12$ into a ratio­nal num­ber.
4. Change $0.00102$ into a ratio­nal num­ber.
5. Change $0.\overline{6}$ into a ratio­nal num­ber.
6. Change $2.\overline{3}$ into a ratio­nal num­ber.
7. Change $0.\overline{45}$ into a ratio­nal num­ber.
8. Change $0.1\overline{6}$ into a ratio­nal num­ber.
9. Change $0.3\overline{18}$ into a ratio­nal num­ber.
10. Change $2.1\overline{6}$ into a ratio­nal num­ber.

Answers#