Percent change

In a previous section, Percentages, ratios, proportions, decimals, fractions, you learned how to write percentages as decimals. In this section, you’ll build on that skill by finding how much a number changes when you apply a percentage to it, including how to set up and solve percent-change problems using algebra.

Percent increase and decrease

You’ve probably heard the phrase “$50$ percent increase.” What does that mean? A common first thought is to multiply the original number by $0.5$, but that would give you half of the original number, not an increase.

The key is to pay attention to the words increase and decrease.

Percent increase

For a number to increase, you start with the original amount (that’s $100\%$) and then add the percent increase on top of it. In other words, you add the percent increase to a base of $100\%$, then multiply.

Let’s use a $50\%$ increase.

Given that $x=20$, what is the value of $x$ after a $50\%$ increase?

Add $50\%$ to the base $100\%$:

• $100\%+50\%=150\%=1.5$

Now multiply:

$$20\ast 1.5=30$$

Percent decrease

A percent decrease also starts from a base of $100\%$. This time, you subtract the percent decrease from $100\%$, then multiply.

Let’s do a $30\%$ decrease.

Given that $x=12$, what is the value of $x$ after a $30\%$ decrease?

To make the pattern clearer, look at these percent changes of $10$.

• $10$ decreased by $100\%=0$
• $10$ decreased by $50\%=5$
• $100\%$ of $10=10$
• $10$ increased by $50\%=15$
• $10$ increased by $100\%=20$

$\%$ change in algebra

In algebra, you’ll often be given an original value and a new value, and you’ll be asked for the percent change.

When Jack started his job, he earned $\$7.00$ an hour. Now, he earns $\$17.50$ an hour. What is the $\%$ change in his wage?

One way to model this is with an equation: the original value ($7$) is multiplied by some factor $x$ to produce the new value ($17.50$).

$$7\ast x=17.50$$

Solve for $x$:

• $x=\frac{17.50}{7}=2.5=\frac{15}{7}$

This means the new wage is $2.5$ times the old wage, or $250\%$ of the original. The question asks for the percent increase, which is how much was added beyond the original $100\%$.

There is also a formula:

There is also a formula: $\%\text{change}=\frac{(\text{new})-(\text{old})}{(\text{old})}\times 100$

Let’s do the same example using the formula.

This method also works when the new value is less than the old value. In that case, the result will be a negative $\%$ change, which indicates a decrease.

$\%$ of another number

In this type of problem, you relate two different “percent of” expressions and solve for the unknown.

$50\%$ of $16$ is $20\%$ of what number?

Write each “percent of” as a decimal times a number:

• $50\%$ of $16$ is $(0.5)\ast 16$
• $20\%$ of $x$ is $(0.2)\ast x$

Set them equal:

$$(0.5)\ast 16=(0.2)x$$

Now solve for $x$.