You can convert decimals to fractions if and only if they are either terminating or recurring decimals.

In this tutorial, we'll learn how to convert decimals into fractions. But before we can dive into it, there’s an important question to be answered.

The short answer is, no.

Terminating decimals – A terminating decimal is one which can be expressed using a finite number of digits. In other words, if the decimal digits come to a definite stop, it’s a terminating decimal. For example — $\hspace{0.2em} 3.8 \hspace{0.2em}$, $\hspace{0.2em} 1.564 \hspace{0.2em}$ etc.

Recurring decimals – A recurring decimal is one in which a digit or a group of digits repeats continuously and indefinitely. For example — $\hspace{0.2em} 1.33... \hspace{0.2em}$, $\hspace{0.2em} 1.618... \hspace{0.2em}$ etc.

But what about the third kind of decimals — those that are neither terminating nor repeating?

You cannot convert a non-terminating non-repeating decimal into a fraction.

Let me explain the method using an example.

Example

Convert $\hspace{0.2em} 0.675 \hspace{0.2em}$ into a fraction.

Step 1. Write the number with a denominator as $\hspace{0.2em} 1 \hspace{0.2em}$.

$0.675 = \frac{0.675}{1}$

Step 2. Multiply the top and bottom by $\hspace{0.2em} 1000 \hspace{0.2em}$. (Actually, it's $\hspace{0.2em} 1 \hspace{0.2em}$ followed by as many $0$s as there are decimal digits in the given number. In this example, there are 3 decimal digits, so $\hspace{0.2em} 1000 \hspace{0.2em}$.)

$\begin{align*} \frac{0.675}{1} &= \frac{0.675 \times 1000}{1 \times 1000} \\[1.3em] &= \frac{675}{1000} \end{align*}$

After multiplication, you will have the given number without the decimal point on top.

Step 3. Simplify the fraction.

$\frac{675}{1000} = \frac{27}{40}$

If you don’t need to show all the work, you can just do this.

Put the number without the decimal at the top and $\hspace{0.2em} 1 \hspace{0.2em}$ followed by zeros (as many as there are decimal digits in the given number) at the bottom. Then simplify.

Example

Convert $\hspace{0.2em} 6.25 \hspace{0.2em}$ into a fraction.

Solution

Here, we have a non-zero number before the decimal. For such cases, I want to show you a smarter way. But first, let’s convert the decimal using what we learned above.

As we did for the previous example, we write the number without the decimal point. And under it, we write $\hspace{0.2em} 1 \hspace{0.2em}$ followed by two $0$s (since there are two decimal digits in the given number).

$\begin{align*} \frac{3.25}{1} &= \frac{3.25 \times 100}{1 \times 100} \\[1.3em] &= \frac{325}{100} \end{align*}$

Then, we simplify the fraction.

$\frac{325}{100} = \frac{13}{4}$

Finally, because we have an improper (or top-heavy) fraction, we convert it into a mixed number.

$\frac{13}{4} = 3\frac{1}{4}$

Now, there is nothing wrong with this approach but we can save some time and effort by taking a slightly different approach.

Step 1. Separate the whole number part and the decimal part.

$3.25 = 3 + 0.25$

Step 2. Ignore the whole number part for a moment and convert the decimal into a fraction using the same steps from earlier examples.

$\begin{align*} \frac{0.25}{1} &= \frac{0.25 \times 100}{1 \times 100} \\[1.3em] &= \frac{25}{100} \\[1.3em] &= \frac{1}{4} \end{align*}$

Step 3. Put the whole number and the fraction together to get the mixed number.

So,

$\begin{align*} 3 + {\color{Red} 0.25} &\, = \, 3 + {\color{Red} \frac{1}{4}} \\[1.3em] &\, = \,3 {\color{Red} \frac{1}{4}} \end{align*}$

Done.

In case you are wondering how this method is easier or faster, let me explain real quick. By ignoring the whole number part for step 2, we make both the top and bottom numbers smaller and hence easier to work with.

And then, in the last step, we don’t have to worry about converting the improper fraction into a mixed number.

Not convinced? Try converting $\hspace{0.2em} 379.5 \hspace{0.2em}$ using both methods, and you’ll see the difference.

Alright, now we know how to convert terminating decimals into fractions but what about repeating decimals?

Example

Convert $\hspace{0.2em} 0.3\overline{24} \hspace{0.2em}$ into a fraction.

Solution

Before we look at the actual step, let me rewrite the given number so you can visualize what’s happening as we move through the steps.

To convert a repeating decimal into a fraction, you follow these steps.

Step 1. Assign a variable, say $\hspace{0.2em} x \hspace{0.2em}$, to the given number.

$x = 0.3\overline{24}$

Step 2. Multiply the equation with a suitable power of $\hspace{0.2em} 10 \hspace{0.2em}$ so that the non-repeating part of the decimal digits comes to the left of the decimal point. Label this as equation $\hspace{0.2em} (1) \hspace{0.2em}$.

In this case, we have one non-repeating digit – $\hspace{0.2em} 3 \hspace{0.2em}$. So, we will multiply by $\hspace{0.2em} 10^1 \hspace{0.2em}$ or $\hspace{0.2em} 10 \hspace{0.2em}$.

$10x = 3.\overline{24} \hspace{0.25cm} \rule[0.1cm]{1cm}{0.1em} \hspace{0.15cm} (1)$

Step 3. Multiply the starting equation with a suitable power of $\hspace{0.2em} 10 \hspace{0.2em}$ so that one block of repeating digits also shifts to the left of the decimal. This is your equation $\hspace{0.2em} (2) \hspace{0.2em}$.

In the given number, there are two repeating digits – $\hspace{0.2em} 2 \hspace{0.2em}$ and $\hspace{0.2em} 4 \hspace{0.2em}$. We want these along with the non-repeating digit, $\hspace{0.2em} 3 \hspace{0.2em}$, to move ahead of the decimal. So altogether three digits, which means we multiply by $\hspace{0.2em} 10^3 \hspace{0.2em}$, or $\hspace{0.2em} 1000 \hspace{0.2em}$.

$1000x = 324.\overline{24} \hspace{0.25cm} \rule[0.1cm]{1cm}{0.1em} \hspace{0.15cm} (2)$

As one block of repeating digits moves ahead of the decimal, another block takes its place.

Step 4. Subtract equation $\hspace{0.2em} (2) \hspace{0.2em}$ from equation $\hspace{0.2em} (1) \hspace{0.2em}$ and solve for $\hspace{0.2em} x \hspace{0.2em}$.

$\hspace{0.2em} 1000x \hspace{0.2em}$ | $\hspace{0.2em} = \hspace{0.2em}$ | $324$ | $.\overline{24}$ | |

$-$ | $\hspace{0.2em} 10x \hspace{0.2em}$ | $\hspace{0.2em} = \hspace{0.2em}$ | $3$ | $.\overline{24}$ |

$\hspace{0.2em} 990x \hspace{0.2em}$ | $\hspace{0.2em} = \hspace{0.2em}$ | $321$ | $.00$ |

Dividing both sides by $\hspace{0.2em} 990 \hspace{0.2em}$, we get —

$\begin{align*} 990x &= 321 \\[1.3em] x &= \frac{321}{990} \end{align*}$

And finally, we simplify the fraction on the right to get our answer.

$x = \frac{107}{330}$

Example

Convert $\hspace{0.2em} 0.\overline{4} \hspace{0.2em}$ into a fraction.

Solution

Step 1. Again, we start by assigning $\hspace{0.2em} x \hspace{0.2em}$ to the given numbers.

$x = 0.\overline{4}$

Step 2. This time there is no non-repeating decimal digit, so we can skip this step. The equation above will be our equation $\hspace{0.2em} (1) \hspace{0.2em}$.

Step 3. There is one repeating digit, and we want it to go in front of the decimal. So we multiply the equation by $\hspace{0.2em} 10 \hspace{0.2em}$.

$10x = 4.\overline{4}$

Step 4. Now, we just need to subtract equation $\hspace{0.2em} (2) \hspace{0.2em}$ from $\hspace{0.2em} (1) \hspace{0.2em}$ and solve for x.

$\hspace{0.2em} 10x \hspace{0.2em}$ | $\hspace{0.2em} = \hspace{0.2em}$ | $4$ | $.\overline{4}$ | |

$-$ | $\hspace{0.2em} x \hspace{0.2em}$ | $\hspace{0.2em} = \hspace{0.2em}$ | $0$ | $.\overline{4}$ |

$\hspace{0.2em} 9x \hspace{0.2em}$ | $\hspace{0.2em} = \hspace{0.2em}$ | $4$ | $.0$ |

Dividing both sides by $\hspace{0.2em} 9 \hspace{0.2em}$, we get —

$\begin{align*} 9x &= 4 \\[1.3em] x &= \frac{4}{9} \end{align*}$

So $\hspace{0.2em} 0.\overline{4} \hspace{0.2em}$ expressed as a fraction is $\hspace{0.2em} \frac{4}{9} \hspace{0.2em}$.

To convert a negative decimal into a fraction, just put aside the negative sign for a moment, convert the decimal into a fraction, and put back the negative sign.

Here’s an example.

Convert $\hspace{0.2em} -0.25 \hspace{0.2em}$ into a fraction.

Solution

Ignoring the negative sign for a moment, we have

$\begin{align*} 0.25 &= \frac{25}{100} \\[1.3em] &= \frac{1}{4} \end{align*}$

And now, we replace the negative sign.

${\color{Red} -} 0.25 = {\color{Red} -} \frac{1}{4}$

And with that, we come to the end of this tutorial on how to convert decimals into fractions. Until next time.

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