Fractions, decimals, & percentages FAQ

A terminating decimal is one that ends. For example, we can write ${\displaystyle \frac{33}{100}}$‍ as the decimal $0.33$‍, which has just two decimal places. A repeating decimal, on the other hand, goes on forever. For example, we can write ${\displaystyle \frac{1}{3}}$‍ as the decimal $0.3333\dots$‍, where the $3$‍ keeps repeating. Sometimes we use an overline to show which digits are repeating. So we could write ${\displaystyle \frac{1}{3}}$‍ as $0.\bar{3}$‍.

Every simplified rational fraction where the denominator has factors other than $2$‍ and $5$‍ will be a repeating decimal. Sometimes, the repeating part is longer than one digit. For example, we can write ${\displaystyle \frac{8}{11}}$‍ as the decimal $0.727272\dots$‍ or $0.\overline{72}$‍.

We can compare terminating and repeating decimals in a similar way as we compare two terminating decimals. We start from the largest place value, then compare each place value from left to right until we find one where the numbers differ.

For example, let's compare $0.67$‍ and $0.\bar{6}$‍. The two decimals both have $0$‍ ones and $6$‍ tenths. However, $0.67$‍ has $7$‍ hundredths, and $0.\bar{6}$‍ has $6$‍ hundredths. So $0.67>0.\bar{6}$‍.

Try it yourself with our Converting fractions to decimals exercise.

To find the percent increase or decrease, we need two numbers: the original number and the new number. We divide the difference between the two by the original number. We'll get our value in decimal or fraction form, and we can rewrite it as a percent from there.

For example, if we start with $20$‍ and increase to $30$‍, we'd find:

That was a positive change, so we had a $50\mathrm{\%}$‍ increase.

On the other hand, if we start with $20$‍ and decrease to $15$‍, we'd find:

That was a negative change, so we had a $25\mathrm{\%}$‍ decrease.

Try it yourself with our Percent problems exercise.

Writing equivalent forms of percent expressions can let us choose the form that makes the context clearest or that is easiest for us to calculate.

Suppose we wanted to find the price of a sewing machine after an $8\mathrm{\%}$‍ discount. If the sewing machine originally cost $m$‍ dollars, we could represent the price after the discount like this:

Writing it that way makes it clear that we're taking away a percentage. If we wanted to make it faster to calculate, we might write the same amount like this:

Then we only have one operation to calculate, but the subtraction is less obvious.

Other times, we use a different form to help us use mental math. For example, suppose that there were $60\text{cm}$‍ of rain last year, but this year, it rained $120\mathrm{\%}$‍ as much. We could write that as $60\cdot 1.20\text{cm}$‍, but some people find it easier to calculate $60\cdot {\displaystyle \frac{6}{5}}\text{cm}$‍. They both mean the same thing, so use the one that works best for you!

Try it yourself with our Equivalent expressions with percent problems exercise.