# Converting Decimals To Fractions

So, how to convert decimals to fractions?

It appears, that we have enough knowledge to do that.

**Steps for converting decimals to fractions.**

- It is known, that we can represent any number as number divided by 1: $$$a=\frac{a}{1}$$$. Do this.
- Simultaneously multiply numerator and denominator by 10, until you get a whole number in the numerator.
- Reduce the fraction.

**Example 1**. Convert 0.8 into fraction.

First we write it as $$$\frac{0.8}{1}$$$.

Since numerator is not integer, we multiply both numerator and denominator by 10 (using equivalence of fractions): $$$\frac{{{0.8}\cdot{\color{red}{{{10}}}}}}{{{1}\cdot{\color{red}{{{10}}}}}}=\frac{{8}}{{10}}$$$.

Now, numerator is integer, so we proceed to the next step.

Reduce fraction: $$$\frac{{8}}{{10}}=\frac{{{4}\cdot{\color{blue}{{{2}}}}}}{{{5}\cdot{\color{blue}{{{2}}}}}}=\frac{{4}}{{5}}$$$.

**Answer**: $$${0.8}=\frac{{4}}{{5}}$$$.

Next example.

**Example 2**. Convert 0.25 into fraction.

First we write it as $$$\frac{{0.25}}{{1}}$$$.

Since numerator is not integer, we multiply both numerator and denominator by 10: $$$\frac{{{0.25}\cdot{\color{red}{{{10}}}}}}{{{1}\cdot{\color{red}{{{10}}}}}}=\frac{{2.5}}{{10}}$$$.

Numerator is still not integer, so we multiply it by 10 once more: $$$\frac{{2.5}}{{10}}=\frac{{{2.5}\cdot{\color{red}{{{10}}}}}}{{{10}\cdot{\color{red}{{{10}}}}}}=\frac{{25}}{{100}}$$$.

Now, numerator is integer, so we proceed to the next step.

Reduce fraction: $$$\frac{{25}}{{100}}=\frac{{{\color{blue}{{{25}}}}}}{{{4}\cdot{\color{blue}{{{25}}}}}}=\frac{{1}}{{4}}$$$.

**Answer**: $$${0.25}=\frac{{1}}{{25}}$$$.

You probably already noticed, that instead of multiplying by 10 each time, we could use powers of 10. Since in 0.25 there are 2 digits to the right of the dot, than, to obtain integer we could multiply by $$${{10}}^{{2}}={100}$$$.

**Example 3**. Convert 0.00345 into fraction.

First we write it as $$$\frac{{0.00345}}{{1}}$$$.

There are 5 digits after the dot, so we multiply numerator and denominator by $$${{10}}^{{5}}={100000}$$$: $$$\frac{{2.5}}{{10}}=\frac{{{0.00345}\cdot{\color{red}{{{100000}}}}}}{{{1}\cdot{\color{red}{{{100000}}}}}}=\frac{{345}}{{100000}}$$$.

Reduce fraction: $$$\frac{{345}}{{10000}}=\frac{{{69}\cdot{\color{blue}{{{5}}}}}}{{{20000}\cdot{\color{blue}{{{5}}}}}}=\frac{{69}}{{20000}}$$$.

**Answer**: $$${0.00345}=\frac{{69}}{{20000}}$$$.

If you have decimal, whose integer part is greater than 0, like 5.78, then separate it like $$${5}+{0.78}$$$ and convert 0.78 into fraction (it is $$$\frac{{39}}{{50}}$$$). Now, you can write it as mixed number: $$${5.78}={5}\frac{{39}}{{50}}$$$.

**Exercise 1.** Convert 0.32 to fraction.

**Answer**: $$$\frac{{8}}{{25}}$$$.

Next exercise.

**Exercise 2.** Convert 0.037 to fraction.

**Answer**: $$$\frac{{37}}{{1000}}$$$. Note, that fraction is irreducible.

Next exercise.

**Exercise 3.** Convert -0.85 to fraction.

**Answer**: $$$-\frac{{17}}{{20}}$$$.

Next exercise.

**Exercise 5.** Convert 7.2 to fraction.

**Answer**: $$${7}\frac{{1}}{{5}}$$$.

Last exercise.

**Exercise 5.** Convert -9.12 to fraction.

**Answer**: $$$-{9}\frac{{3}}{{25}}$$$. Hint: ignore minus sign, perform conversion, and then place minus in front of the result.