Percent means "per hundred" The word percent comes from "per centum", per hundred. So \(45\%\) means \(45\) out of every \(100\) :
\[45\% = \frac{45}{100} = 0.45\]
RuleConverting percents
Percent to decimal : divide by \(100\) ,\(8\% = 0.08\) and \(150\% = 1.5\) .Decimal to percent : multiply by \(100\) ,\(0.375 = 37.5\%\) .Fraction to percent : convert to a decimal first. \(\frac{3}{8} = 0.375 = 37.5\%\) .Percents can be more than \(100\%\) :\(150\%\) of its height, it's one and a half times as tall as before.
Finding a percent of a number Some percents are easy to do in your head. \(10\%\) is one tenth, so \(10\%\) of \(80\) is \(8\) .\(5\%\) is half of that, \(4\) ,\(20\%\) is double, \(16\) .\(10\%\) and \(5\%\) is a great way to check your work.
The percent equation Every basic percent problem fits one equation:
\[\text{part} = \text{percent} \times \text{whole}\]
You can solve for whichever piece is missing.
ExampleFinding the percent
You got \(42\) questions right out of \(48\) .
\[42 = p \times 48 \qquad p = \frac{42}{48} = 0.875 = 87.5\%\]
ExampleFinding the whole
\(18\) students walk to school. That's \(15\%\) of all the students in the grade. How many students are in the grade?
\[18 = 0.15 \times w \qquad w = \frac{18}{0.15} = 120\]
Discounts, tax and tips ExampleA discount
A jacket priced at \(64\) dollars is \(25\%\) off. What's the sale price?
The discount is \(0.25 \cdot 64 = 16\) dollars, so the sale price is \(64 - 16 = 48\) dollars.
Shortcut: after \(25\%\) off you pay \(75\%\) of the price, and \(0.75 \cdot 64 = 48\) .
ExampleSales tax
A video game costs \(40\) dollars and the sales tax is \(7\%\) .
Tax: \(0.07 \cdot 40 = 2.80\) dollars. Total: \(40 + 2.80 = 42.80\) dollars. Or in one step: \(1.07 \cdot 40 = 42.80\) .
A tip works just like tax: a \(20\%\) tip on a \(35\) -dollar meal is \(0.2 \cdot 35 = 7\) dollars.
Percent change RulePercent increase or decrease
\[\text{percent change} = \frac{\text{amount of change}}{\text{original amount}}\]
If the amount went up, it's a percent increase ; if it went down, a percent decrease .
ExamplePercent increase
A town's population grew from \(2{,}500\) to \(2{,}800\) .
The change is \(300\) ,\(2{,}500\) :
\[\frac{300}{2{,}500} = 0.12 = 12\%\]
Watch outAlways divide by the original
A price goes from \(50\) to \(40\) dollars: a \(10\) -dollar drop, which is \(\frac{10}{50} = 20\%\) of the original. Going back from \(40\) to \(50\) is the same \(10\) dollars, but it's \(\frac{10}{40} = 25\%\) of the new starting point.
That's why a \(20\%\) increase followed by a \(20\%\) decrease doesn't get you back where you started: \(100 \to 120 \to 96\) .
Practice Practice
Write \(0.6\) ,\(\frac{7}{20}\) and \(2.5\) as percents.
Show Hide the solution\(0.6 = 60\%\) ,\(\frac{7}{20} = 0.35 = 35\%\) and \(2.5 = 250\%\) .
Practice
A bike costs \(250\) dollars. It's on sale for \(30\%\) off, and then \(8\%\) sales tax is added to the sale price. What's the final cost?
Show Hide the solutionSale price: \(0.7 \cdot 250 = 175\) dollars.
With tax: \(1.08 \cdot 175 = 189\) dollars.
Quick check
What is \(5\%\) as a decimal?
A \(0.5\) B \(0.05\) C \(5.0\) D \(0.005\)
Divide by \(100\) :\(5 \div 100 = 0.05\) .
Quick check
A shirt's price rises from \(20\) to \(25\) dollars. What's the percent increase?
A \(5\%\) B \(20\%\) C \(25\%\) D \(125\%\)
The change is \(5\) dollars, and \(\frac{5}{20} = 0.25 = 25\%\) .\(20\%\) ,