Decimals and Percentages
Learning objectives
- Convert between fractions, decimals and percents.
- Find a percent of a quantity and what percent one quantity is of another.
- Find the whole when a percent of it is known.
- Compute percent change, discounts and successive changes.
Fractions, decimals and percents
A percent is a fraction with denominator 100: \(p\%=\frac{p}{100}\). The word means "per hundred".
Move the slider to shade \(p\) of the 100 squares. The page shows the same number as a percent, a fraction and a decimal, and finds that percent of a whole.
\(\frac34=0.75=75\%\) and \(\frac18=0.125=12.5\%\).
To turn a decimal into a percent, multiply by 100: \(0.07=7\%\).
Percent of a quantity
Find \(35\%\) of 80.
\[35\%\cdot 80=0.35\cdot 80=28\]
What percent is 18 of 24?
\[\frac{18}{24}=\frac34=75\%\]
15 is \(30\%\) of a number. Find the number.
We have \(0.30\cdot x=15\), so \(x=\frac{15}{0.30}=50\).
Percent change
If a quantity changes from \(a\) to \(b\), the percent change is
\[\frac{b-a}{a}\cdot 100\%\]
A positive result is an increase and a negative result is a decrease.
A price goes from 40 to 50, and later back from 50 to 40. Find both percent changes.
\[\frac{50-40}{40}\cdot 100\%=25\%,\qquad \frac{40-50}{50}\cdot 100\%=-20\%\]
The two changes are not opposites, because each percent is taken of a different starting value.
A jacket costs 60. It is discounted by \(20\%\), then \(10\%\) tax is added. What is the final price?
\[60\cdot 0.80=48,\qquad 48\cdot 1.10=52.8\]
Percent changes do not simply add up. An increase of \(20\%\) followed by a decrease of \(20\%\) gives \(1.2\cdot 0.8=0.96\), which is a \(4\%\) decrease overall.
Exercises
Write \(\frac58\) as a decimal and as a percent.
Show solution
\(\frac58=0.625=62.5\%\)
Find \(12\%\) of 250.
Show solution
\(0.12\cdot 250=30\)
What percent is 45 of 60?
Show solution
\(\frac{45}{60}=\frac34=75\%\)
After a \(25\%\) discount a jacket costs 45. What was the original price?
Show solution
The price is \(75\%\) of the original: \(0.75x=45\), so \(x=60\).
A population grows from 8000 to 9200. Find the percent increase.
Show solution
\(\frac{9200-8000}{8000}\cdot 100\%=\frac{1200}{8000}\cdot 100\%=15\%\)
A price rises by \(10\%\) and then falls by \(10\%\). Is the final price the same as the start?
Show solution
No: \(1.10\cdot 0.90=0.99\), so the final price is \(1\%\) lower than the start.
Summary
- A percent is a fraction out of 100; fractions, decimals and percents are three ways to write one number.
- To find a percent of a quantity, multiply by the decimal form.
- Percent change is the change divided by the old value, times 100%.
- Successive percent changes multiply; they do not add.