Percent Increase/Decrease (Discounts/Mark Ups)
Finding a percent of a number means finding a part or portion of a number. Increasing/decreasing a number by percent means finding a percent of a number and adding or subtracting that amount from the original number. We deal with percent increases (mark-ups, tips, sales tax) and decreases (sales, discounts) all the time. Calculating percent increases and decreases mostly involves finding a percent (if you need to review that, review the Finding a Percent lesson) and then adding it to, or subtracting it from, your original number.
Example: Find the sales price if a shirt's original price of $60 is discounted by 15%
First you need to find 15% of 60.
We like to use proportions that follow this format: $\dfrac{part}{whole}=\dfrac{\text{%}}{100}$ (again, review the Finding a Percent lesson if you need to practice this step)
$$\eqalign{\dfrac{x}{60}&=\dfrac{15}{100} && \text{Set up proportion}\\ x\times100&=15\times60 &&\text{Cross multiply}\\100x&=900 &&\text{Solve for variable}\\x&=9}$$
So, the discount is \$9.
Because this is a discount (percent decrease), we'll subtract the discount from the original price: $60-9=51$
The final, discounted price is \$51.
The process is the same with percent increases, except once you find the percent amount, you add it to the original.
Example: Miriam can sew 20 shirts a week at her job. Her new boss demans that everyone increase production by 25%. How many shirts will she be expected to make each week?
First you need to find 25% of 20.
$$\eqalign{\dfrac{x}{20}&=\dfrac{25}{100} && \text{Set up proportion}\\ x\times100&=25\times20 &&\text{Cross multiply}\\100x&=500 &&\text{Solve for variable}\\x&=5}$$
So, the improvement demanded by Miriam's boss is 5 shirts.
Because this is a percent increase), in order to find the total number of shirts that Miriam will have to prodice, we'll add the increae to the number of shirts she produced: $20+5=25$
Miriam needs to produce 25 shirts a week.
The only tricks to finding percent increases and decreases are finding the percents correctly, and remembering to add or subtract before giving your final answer.
There is a way to find percent increases and decreases in one step. That step involves understanding that a whole equals 100%. So, if a Miriam must INCREASE her shirt production by 25% she must do everything she is doing now (100%) plus another 25%. So, if you multiply her current production by 125%, you'll get the total number of shirts she needs to produce for the new boss:
$$\eqalign{\dfrac{x}{20}&=\dfrac{125}{100} && \text{Set up proportion}\\ x\times100&=125\times20 &&\text{Cross multiply}\\100x&=2500 &&\text{Solve for variable}\\x&=25}$$
Likewise, when finding a discount, you recognize that you will not be paying 100% of the price, but rather less. So, if you are buying a shirt at a 15% discount, you are actually only paying 85% of the cost.
$$\eqalign{\dfrac{x}{60}&=\dfrac{85}{100} && \text{Set up proportion}\\ x\times100&=85\times60 &&\text{Cross multiply}\\100x&=5100&&\text{Solve for variable}\\x&=51}$$
Although people who add and subtract percents quickly in their heads think of this as a one-step alternate method, technically, you're just doing your adding/subtracting first (to the percents rather than to the final numbers). So, unless you feel more comfortable with this method, the method above works just as well and may be less confusing for some people. If you feel comfortable with both methods, feel free to use the one that is most efficient for any given problem!
Practice Problems:
Percent Increase/Decrease
- A store reduces its stock of 150 comic books by 20%. How many comic books do they keep in stock after the reduction?
- A shirt that sells for \$39.95 is on sale for 15% off. How much is the discount?
- A shirt that sells for \$39.95 is on sale for 15% off. How much does the discounted shirt cost?
- Ms. Chin usually gives her students tests with 75 questions. The week before winter break she decides to be kind and take 12% of the questions off of the test. How many questions were on the test that the kids took before winter break?
- Ms. Chin usually gives her students tests with 75 questions. The week before winter break she decides to be kind and take 12% of the questions off of the test. How many questions did she remove from the test?
- Sunny decided that the students weren't working hard enough. He decided to increase everyone's work by 35%. If Imani had 40 math problems to do, how many total math questions did Sunny make her do on the day she was not working hard enough?
Answer Key: