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.