Study Guide: Percents
What Does Percent Mean?
Percent (per cent) is a Latin term, meaning ‘out of one hundred’. One hundred was chosen because it is so easy to calculate when using the decimal system (base-10). The symbol % is used to show that the number is a percent.
Percentages are used to describe parts of a whole. When using percentages, the whole is always made equal to one hundred. For example, 45% represents “45 out of 100”, or “45 percent of the total amount”.
We could say, “It snows 20 days out of every 100 days”, or we could say, “It snowed 20% of the time.”
How Percents are Used
Knowing how to calculate percents makes many situations easier to understand. Scientists use percents as a quick and accurate way to explain complex ideas. For example, “Following cleanup of the toxic spill, total populations of 20% of the species are increasing.”
Types of Percentage Problems
There are three main types of percentage problems:
Finding the Percent
Finding the Part
Finding the Whole
1. Finding the Percentage
Method 1: Start with 1%
Suppose you want to buy a several trees to help replant a forest that was clearcut and then abandoned by a large corporation. You checked local stores, and one offers a 20% discount off the list price of $500. How much will your trees cost at this price?
In this case, the whole is $500—the cost of the trees before the discount. The percentage to find is 20%. Once you find the discount, subtract it from the original price to get the discounted price.
\begin{align} \$500 \div 100 &= \text{$}5 &&\textit{1. Find value of 1%.} \\\ \$5 × 20 &= \$100 &&\textit{2. Multiply by percentage. Discount is \$100.}\\\ \$500 − 20 \text{%} &= &&\textit{3. Subtract discount from original price.} \\\ 500 − \$100 &= \text{$}400 &&\textit{Solution.} \end{align}
Method 2: Divide the Part by the Whole
If we want to calculate the percentage of days it rained in a month, we use the number of days in that month as the whole.
Let’s say we want to know the percentage of days it rained in April. April has 30 days. If it rained 15 days in April:
\begin{align} 15 \div 30 &= 0.5 &&\textit{1. Divide the part by the whole.} \\\ 0.5 \times 100 &= 50\% &&\textit{2. Multiply by 100, and add the percent symbol.} \\[3ex] \end{align}
Thus, in April it rained 50% of the time.
2. Finding the Part
It might help to remember that “times” means “of”. For example, “10 percent of 200 is 20.”
$$\text{PERCENT} \textit{ of } \text{WHOLE} = \text{PART} $$ $$\text{10%} \textit{ of } \text{200 is } \text{20} $$ $$\text{PERCENT} \times \text{WHOLE} = \text{PART} $$
Example: What is 50% of 25?
We have the percentage and the whole. We need to find the part.
\begin{align} 50 \text{%} \times 25 & = 0.5 \times 25 &&\textit{Multiply the percent by the whole.}\\\ & = 12.5 \end{align}
Thus, 12.5 is 50% of 25.
2. Finding the Percentage
$$\text{PERCENT} = \dfrac{\text{PART}}{\text{WHOLE}}$$
Example: What percent of 5 is 2?
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Divide the part by the whole. $$ 2 \div 5 = 0.4 $$
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Multiply the decimal by 100, and add the percentage symbol. $$ 0.4 \times 100 = 40 \text{%} $$
Thus, 2 is 40% of 5.
3. Finding the Whole
$$\text{WHOLE} = \dfrac{\text{PART}}{\text{PERCENT}}$$
Example: 45% of what is 2?
\begin{align} 45\text{%} &= 0.45 &&\textit{1. Convert the percent into a decimal.}\\\ 2 \div 0.45 &= 4.4 &&\textit{2. Part dIvided by the decimal.} \end{align}
Thus, 45% of 4.4 is 2.
How to calculate percentage change
A percentage change is the degree of change over time.
For a percentage increase:
- Find the absolute value of the New Value - Old Value.
- Subtract the result by the Old Value.
- Multiply by 100, and add the percent symbol.
$$ \dfrac{(\text{New Value} - \text{Old Value})}{\text{Old Value}} \times 100 = \text{ %}$$
For a percentage decrease:
- Find the absolute value of the Old Value - New Value.
- Subtract the result by the Old Value.
- Multiply by 100, and add the percent symbol.
$$ \dfrac{(\text{Old Value} - \text{New Value})}{\text{Old Value}} \times 100 = \text{ %}$$
Example: What is the price increase?
A gadget cost \$100 last year but now costs \$125. What is the percentage of the price increase?
\begin{align} 125 - 100 &= 25 && \textit{1. Subtract the old price from the new price.} \\\ 25 \div 100 &= 0.25 && \textit{2. Divide the result by the old price.} \\\ 0.25 \times 100 &= 25\text{%} && \textit{3. Multiply by 100, and add the percent symbol.} \end{align}
Thus, the cost of the gadget increased 25%.
A different gadget cost \$100 last year but now costs only \$75. To determine the price decrease:
\begin{align} 100 - 75 &= 25 && \textit{1. Subtract the new price from the old price.} \\\ 25 \div 100 &= 0.25 && \textit{2. Divide the result by the old price.} \\\ 0.25 \times 100 &= 25\text{%} && \textit{3. Multiply by 100, and add percent symbol.} \end{align}
Thus, this gadget costs 25% less than before.
Calculating percentage differences
We can use percentages to compare two different items that are related in some way. For example, we might want to determine how much a product cost last year versus how much a similar product costs this year. This calculation would give us the percent difference between the two product costs.
Here’s the formula for calculating a percentage difference. In this formula, Value1 is the cost of one product, and Value2 is the cost of the other product.
$$ \dfrac{|Value1 - Value2|}{(Value1 + Value2) \div 2} \times 100 = \text{ %}$$
Example: Comparing the cost of two products
A product cost \$25 last year and a similar product costs \$30 this year. To determine the percentage difference:
\begin{align} |30 - 25| &= 5 &&\textit{1. Find Absolute Value of Cost Difference.} \\[2ex] \dfrac{25 + 30}{2} &= 27.5 &&\textit{2. Find the Cost Average.} \\[2ex] 5 \div 27.5 &= 0.18 &&\textit{3. Divide Cost Difference by Cost Average.} \\[2ex] 0.18 \times 100 &= 18 \text{%} &&\textit{4. Multiply by 100, and add percent symbol.} \end{align}
Thus, the cost of the product from this year is 18% more than the cost of the product from last year.