Decimals and Fractions

Links and useful resources

Concept summary and connections

Converting decimal to fraction
Converting fraction to decimal
Powers of 10 and decimals

Worked examples

Converting decimals to fractions:

First, do you recognize it? if so, you're done!

But, if you don't recognize it:

Write the decimal number as a whole number fraction with a power of ten in the denominator
Simplify the fraction

Example: Write 0.25 as a fraction

\begin{aligned} 0.25 & = 2 \cdot 10^{- 1} + 5 \cdot 10^{- 2} \\ = \frac{2}{10} + \frac{5}{100} \\ = \frac{20}{100} + \frac{5}{100} \\ = \frac{25}{100} & definition of decimal \\ = \frac{5}{20} & common factor 4 \\ = \frac{1}{4} & common factor 5 \end{aligned}

Converting fractions to decimals

First, do you already know it? If so, you're done! It's worth memorizing a bunch of common ones!

But, if you don't recognize it, the answer is simply to use long division and stop when you have enough digits to round to the correct place value.
Example: Write $\frac{3}{7}$ as a decimal to the hundredths place:

Use long division to compute 3 divided by 7 out to the thousandths place (one place value more than what you're looking for! It's important for rounding!)

Result: $0.428$
Now round to the hundredths place: $0.43$

Shortcut trick for making decimals from fractions (sometimes)

Sometimes, you can see a way to make the denominator into a power of ten easily. If you can do that, then you can just change the fraction's denominator to a power of ten and read off the decimal answer!

Example: Convert $1 \frac{3}{5}$ to a decimal:

The whole-number part doesn't change, so we can just ignore it for now.
Notice that $5 \cdot 20 = 100$ , so $\frac{3}{5} \frac{20}{20} = \frac{60}{100} = \frac{6}{10}$
The decimal part is $\frac{6}{10} = 0.6$ , so the answer is 1.6!