Percent Increase/Decrease (Finding the Percent)

Although it's not at all hard to increase or decrease a number by a percent (review the Percent Increase/Decrease lesson if this doesn't sound familiar), it can be tricky to find the percent a number was increase or decreased by.

Essentially, what you have to remember is that any time a number is increased or decrease by a percent, that percent is based on the original number, not the new number.  (So, if you buy a \$40 shirt, at a 25% discount, for \$30, the discount of \$10 is 25% of the original price, \$40, not the discounted price of \$30). 

So, how can you figure out the percent change when you're given the before and after numbers?

Example: A pair of shoes is on sale for \$15.  The original price was \$40.  What percent discount is that?

We like to use proportions that follow this format: $\dfrac{change}{original}=\dfrac{\text{%}}{100}$

The "change" is the difference between the original and new price.  "Original" is the original number, the number that your new price is based upon.  "%" is the percent you are trying to find.  "100" always represents "whole" in percents.  

$$\eqalign{\text{change}&=40-15\\&=25\\\text{original}&=40\\\text{%}&=x\\\text{100}&=100}$$

The proportion for this problem would be: $\dfrac{25}{40}=\dfrac{x}{100}$

Essentially, we know that the change (in this case, the discount) is a portion of the original price, so we're trying to find out, exactly what percent of the original price it is!

We put an $x$ in for the "%" because that's what we're trying to solve for.  Now, just cross multiply to solve the proportion for $x$.

$$\eqalign{\dfrac{25}{40}&=\dfrac{x}{100} && \text{Set up proportion}\\ x\times40&=25\times100 &&\text{Cross multiply}\\40x&=2500 &&\text{Solve for variable}\\x&=62.5&&\text{The discount was 62.5%}}$$

This set up can also be used to find percent increases. The term change can be used to capture any kind of change -- either up or down. 

Example: A test prep company promises to increase students' scores by at least 50%.  A student's math SAT score rose from 320 to 400.  Did the student achieve a 50% increase?

We're going to use our "change over original" proportion again: $\dfrac{change}{original}=\dfrac{\text{%}}{100}$

$$\eqalign{\text{change}&=400-320\\&=80\\\text{original}&=320\\\text{%}&=x\\\text{100}&=100}$$

The proportion for this problem would be: $\dfrac{80}{320}=\dfrac{x}{100}$

Now, just cross multiply to solve the proportion for $x$.

$$\eqalign{\dfrac{80}{320}&=\dfrac{x}{100} && \text{Set up proportion}\\ x\times320&=80\times100 &&\text{Cross multiply}\\320x&=8000 &&\text{Solve for variable}\\x&=25&&\text{The score increase was 25% -not 50%.}}$$

Overall, tests like to ask students to find the percent of increases and decreases -- and a lot of students get tripped up.  The most common error that students make, is they find the percent of the final number rather than the original number. If you remember our proportion, you'll always get these problems correct! $\dfrac{change}{original}=\dfrac{\text{%}}{100}$