Table of Contents
Percent calculations connect percentages to actual quantities. In this chapter we focus on how to use given percentages in different kinds of calculations, and how to set up these calculations correctly.
Reading and Using Percentage Information
A percentage like “$15\%$ of 80” always means:
$$
15\% \times 80 = 0.15 \times 80
$$
The basic pattern is:
- To find $p\%$ of a number $N$:
$$
p\% \text{ of } N = \frac{p}{100} \times N
$$
Here:
- $p$ is the percent.
- $N$ is the “whole” or original amount.
You should already know how to convert between fractions, decimals, and percentages; here we just use those conversions inside problems.
Finding a Percentage of a Quantity
This is the most direct type of percent calculation: you know the percent and the whole, and you want the part.
Example steps:
- Convert the percent to a decimal (or fraction).
- Multiply by the whole.
Example:
- $25\%$ of 60:
$$
25\% \times 60 = 0.25 \times 60 = 15
$$
You can also use fractions:
$$
25\% = \frac{25}{100} = \frac{1}{4}, \quad \frac{1}{4} \times 60 = 15
$$
Finding What Percent One Number Is of Another
Here you know the part and the whole and you want the percent.
General pattern:
$$
\text{Percent} = \frac{\text{part}}{\text{whole}} \times 100\%
$$
Example:
If 12 out of 30 students like chess, what percent is that?
$$
\frac{12}{30} \times 100\% = 0.4 \times 100\% = 40\%
$$
Always make sure:
- “Part” is the smaller portion you are comparing.
- “Whole” is the total amount you compare against.
If needed, decide clearly what you are treating as the “whole.” For example, “What percent of the class are boys?” versus “What percent of the boys play soccer?” use different wholes.
Finding the Whole from a Percentage and a Part
Here you know the percent and the part, and you want the whole. This is a common step in word problems.
If $p\%$ of a number is $P$, then the whole $N$ satisfies:
$$
\frac{p}{100} \times N = P
$$
Solve for $N$:
$$
N = \frac{P}{p/100} = P \times \frac{100}{p}
$$
Example:
$30\%$ of some number is 18. What is the number?
We have:
$$
0.30 \times N = 18
$$
So:
$$
N = \frac{18}{0.30} = 60
$$
Think of this as “divide the part by the decimal form of the percent.”
Percent Increase and Percent Decrease
Percent change compares a change to the original amount.
Percent Change Formula
Suppose a quantity changes from an original value $O$ to a new value $N$.
- Find the change:
$$
\text{change} = N - O
$$ - Divide by the original:
$$
\text{relative change} = \frac{N - O}{O}
$$ - Convert to a percent:
$$
\text{percent change} = \frac{N - O}{O} \times 100\%
$$
- If $N > O$, the percent change is a percent increase.
- If $N < O$, the percent change is a percent decrease (often written as a positive number with the word “decrease”).
Percent Increase
From $O$ to $N$, with $N > O$:
$$
\text{percent increase} = \frac{N - O}{O} \times 100\%
$$
Example:
A price goes from \$50 to \$65.
Change:
$$
65 - 50 = 15
$$
Percent increase:
$$
\frac{15}{50} \times 100\% = 0.3 \times 100\% = 30\%
$$
Percent Decrease
From $O$ to $N$, with $N < O$:
$$
\text{percent decrease} = \frac{O - N}{O} \times 100\%
$$
Example:
A number drops from 80 to 60.
Change:
$$
80 - 60 = 20
$$
Percent decrease:
$$
\frac{20}{80} \times 100\% = 0.25 \times 100\% = 25\%
$$
Sometimes you may see the formula written with a sign:
$$
\text{percent change} = \frac{N - O}{O} \times 100\%
$$
A positive answer means increase; a negative answer means decrease.
Using Percentages to Find New Values
Often a problem gives an original value and a percent increase or decrease, and asks for the new value.
One-Step Method Using Multipliers
Turn the percentage change into a multiplier on the original.
For an increase of $p\%$:
- New value $= O \times (1 + p/100)$
For a decrease of $p\%$:
- New value $= O \times (1 - p/100)$
Here $O$ is the original value.
Example (increase):
A \$200 item increases by $8\%$.
$$
\text{new price} = 200 \times (1 + 0.08) = 200 \times 1.08 = 216
$$
Example (decrease):
A \$150 bill is reduced by $20\%$.
$$
\text{new bill} = 150 \times (1 - 0.20) = 150 \times 0.8 = 120
$$
The factors $1.08$ and $0.8$ are sometimes called “growth factor” and “decay factor.”
Successive Percentage Changes
When several percentage changes happen one after another, use multiplication step by step. Do not add the percentages unless the context justifies it; in general, successive percent changes combine multiplicatively.
Example:
A price first increases by $10\%$, then decreases by $10\%$. Start with \$100.
- After $10\%$ increase:
$$
100 \times 1.10 = 110
$$ - After $10\%$ decrease:
$$
110 \times 0.90 = 99
$$
The final price \$99 is not the original \$100. A $10\%$ increase and a $10\%$ decrease do not cancel each other exactly.
In general, for successive changes of $p\%$ and $q\%$:
- Overall factor $= (1 + p/100)(1 + q/100)$ (with $q$ negative for a decrease).
Percentage Points vs Percent
In some situations (often in statistics or finance), two different types of “percent change” language appear:
- Percent: relative change, using the percent change formula.
- Percentage points: simple difference between two percentage values.
Example:
A tax rate rises from $5\%$ to $7\%$.
- The increase in percentage points is $7\% - 5\% = 2$ percentage points.
- The percent increase in the rate itself is:
$$
\frac{7\% - 5\%}{5\%} \times 100\%
= \frac{2}{5} \times 100\%
= 40\%
$$
So you can say:
- “The rate increased by $2$ percentage points.”
- Or: “The rate increased by $40\%$.”
They describe two different comparisons, so it is important to distinguish them when reading problems.
Common Real-World Percent Calculations
Many everyday situations are straightforward applications of the ideas above. A few patterns:
- Discounts: usually a percent decrease from the original price.
If an item costing \$80 has a $25\%$ discount:
$$
\text{sale price} = 80 \times (1 - 0.25) = 80 \times 0.75 = 60
$$
- Sales tax or tips: typically a percent increase added to a base amount.
For a \$50 meal with $18\%$ tip:
$$
\text{tip} = 0.18 \times 50 = 9, \quad \text{total} = 50 + 9 = 59
$$
or directly:
$$
\text{total} = 50 \times 1.18 = 59
$$
- Percent of a mixture or group: using the “what percent is this of that?” formula.
Example: In a group of 40 students, 10 are left-handed:
$$
\frac{10}{40} \times 100\% = 25\%
$$
Whenever you use percentages, the key is to be clear about:
- What is the “whole” (the base).
- Whether you are finding a part, a percent, or the original whole.
- Whether changes are single or successive, and whether you need relative change (percent) or simple difference (percentage points).