Calculating with percents

As we discussed in pre-algebra, percent is a ratio that compares a number to 100. Percent means per hundred. Percent is usually expressed with the percent symbol %.

Percent problems are usually solved by using proportions.


Example

In a classroom 14 of the 21 students are female. How many percent does that correspond to?

We know that the ratio of girls to all students is

$$\frac{14}{21}$$

And we know that this ratio is a proportion to a ratio with the denominator 100.

$$\frac{14}{21}=\frac{x}{100}$$

As we saw in the last section from here we can calculate x

$$x=100\cdot \frac{14}{21}$$

$$x=\frac{1400}{21}$$

$$x\approx 67$$

i.e. 67% of the students in the class are female.

One of the ratios in these proportions is always a comparison of two numbers (above 14/21). This numbers are called the percentage (14) and the base (21). The other ratio is called the rate and always has the denominator 100.

$$\frac{percentage}{base}=rate$$

Another way of saying this is that

$$percent=\frac{part}{whole}$$

Percent of change, or p%, indicates how much a quantity has increased or decreased in comparison with the original amount. It's calculated as:

$$percent\: of\: change=\frac{amount\: of\: increase\: or\: decrease}{old\: amount}$$


Example

Johnny is at the store where there is a big sign telling him that there is a $4.99 discount on a shirt that originally costs $39.99. But how big is the discount in percent?

$$\frac{\$ 4.99}{\$ 39.99}\approx 0.12$$

$$0.12=12\%$$

The prize of the shirt has decreased by 12%.


Video lesson

A prize increases from $500 to $585. How big is the increase in percent?