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In a previous section, Percentages, ratios, proportions, decimals, fractions, we talked about how to use percentages as decimals. In this section, we will be expanding on this skill. We will discover how to find out how much a number will change when we apply a percentage to it, especially by using algebra.

You have undoubtedly heard the phrase “$50$ percent increase.” What does this mean, exactly? Perhaps our first thought is that a $50%$ increase is our original number multiplied by $0.5$. However, this is not quite right. We have to pay very close attention to the words **increase** and **decrease**.

When finding a percent increase, we must first recognize that in order for the number to increase, the number has to be **at least $100%$**. So, to find this increase we will actually **add our percentage of increase to a base of $100%$.** Let’s use the idea of a $50%$ increase in the example below.

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

Adding our $50%$ to a base of $100%$ the actual value we will multiply by our number is $150%$ or $1.5$:

$20∗1.5=30$

Finding the percent decrease of a number **will also use a base percentage of $100%$**. Instead of adding our percentage, we will simply subtract it from the starting $100%$. Let’s do a $30%$ decrease.

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

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Subtract our $30%$ from the initial $100%$:

$100%−30%=70%$

Now, find the new value by multiplying the decimal form of the percentage by the value of $x$:

$(0.7)∗12=8.4$

To help solidify this idea in your mind, take a close 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 $100%=20$

In algebra, it is common for you to be asked to find the percent change given one number and the value it is changed to. Let’s jump straight into an example.

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?

We must set up an equation that represents this change. What we know is that the starting number is $7$, which is multiplied by some percentage to result in $17.50$. So how would that equation look?

$7∗x=17.50$

It is as simple as that! We just need to find $x$, which is $715 $ or $2.5$.

Now, this is a percent change of $250%$, but what is the **percent increase**?

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$%increase=150%$

There is also a formula: $%change=(old)[(new)−(old)] $

Let’s do the example above but use the formula instead. We should get the same answer.

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$% increase =(17.50−7.00)/7.00=10.50/7.00=1.5=150% $

This will also work if the new value is lower than the old value. In this case, you will always get a **negative** $%$ change value, which is the same as a decrease.

This form of percent change will require a bit more technique. We will actually be relating two different numbers and percent changes. Here is an example.

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

What do you think we should do here? We definitely need to make an algebraic equation, so let’s start with that. $50%$ of $16$ can be written as $(0.5)∗16$ and $20%$ of some number can be written as $(0.2)∗x$. Altogether, we get:

$(0.5)∗16=(0.2)x$

Now solve for $x$!

(spoiler)

$x=8/(0.2)=40$

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