A percent problem with money finds a percent of an original amount, then adds it (tax, tip, markup) or subtracts it (discount). Write the percent as a decimal and multiply, or multiply by 1 plus or minus the rate. Commission is a percent of sales, and simple interest is I = Prt.
Key facts:
- Tax, tips and markups increase an amount: multiply by \(1+r\). Discounts decrease it: multiply by \(1-r\).
- The percent is taken of the original amount, which is 100%.
- Two successive discounts multiply, so 20% off and then 10% off equals 28% off, not 30%.
- Commission equals the rate times sales, and simple interest is \(I=Prt\) with \(r\) written as a decimal.
- Follow the order of the story and round money to the nearest cent at the end.
How do you find sales tax and tips?
Sales tax and tips are a percent of an amount. Find that percent of the amount, then add it. For a $24.00 meal with 8% sales tax, the tax is \(0.08\times 24=1.92\) and the total is \(24+1.92=25.92\) dollars. A shortcut gives the total in one step: multiply by \(1.08\).
To increase an amount by \(r\%\), multiply by \(1+\frac{r}{100}\). To decrease it by \(r\%\), multiply by \(1-\frac{r}{100}\). The percent is always taken of the original amount, which is 100%.
| Situation | Change | Multiplier |
|---|---|---|
| 8% sales tax | increase 8% | \(1.08\) |
| 20% tip | increase 20% | \(1.20\) |
| 35% markup | increase 35% | \(1.35\) |
| 25% discount | decrease 25% | \(0.75\) |
How do markups and discounts work?
A store buys an item at a wholesale price and adds a markup to set the selling price. If a shop pays $40 for a skateboard deck and marks it up 35%, the markup is \(0.35\times 40=14\) and the selling price is \(40+14=54\) dollars. A discount (or markdown) lowers a price. A $60 jacket that is 25% off saves \(0.25\times 60=15\) dollars, so it sells for $45.
Two discounts do not add. A 20% discount followed by a 10% discount multiplies: \(0.80\times 0.90=0.72\), so the item costs 72% of its price, a 28% discount, not 30%.
What is the method for multistep percent problems?
- Name the original amount. It is 100%.
- Write each percent as a decimal, such as \(6\%=0.06\).
- Multiply the rate by the amount, or use the multiplier \(1+r\) or \(1-r\).
- Follow the order of the story, for example discount first and tax second.
- Round money to the nearest cent.
Worked example. Sneakers cost $80, are 25% off, and then 6% sales tax is added. Sale price: \(80\times 0.75=60\). Tax: \(60\times 0.06=3.60\). Total: \(60+3.60=63.60\) dollars.
How do commission, fees and simple interest work?
Commission is a percent of sales. Sofia earns 6% on $2,500 of sales, so she makes \(0.06\times 2500=150\) dollars. A fee based on a percent works the same way. Simple interest uses \(I=Prt\), where \(P\) is the principal (starting amount), \(r\) is the yearly rate as a decimal, and \(t\) is the time in years. For $800 at 3% for 4 years, \(I=800\times 0.03\times 4=96\) dollars. The interest is $24 every year, so it grows in equal steps.
Percent Problems: Tax, Tips, Markups and Discounts: frequently asked questions
How do you calculate sales tax?
Change the tax rate to a decimal and multiply by the price. A 7% tax on $28 is \(0.07\times 28=1.96\), so the total is $29.96. You can also multiply the price by \(1.07\) to get the total directly.
Is 20% off and then 10% off the same as 30% off?
No. The second discount applies to the reduced price. On a $100 item, 20% off gives $80, and 10% off $80 gives $72, which is a 28% discount.
What is the difference between a markup and a discount?
A markup raises a price: a store adds it to the amount it paid. A discount lowers a price: a store subtracts it from the regular price. Both are a percent of the original amount.
How do you find the original price from a sale price?
Find what percent of the original the sale price is, then divide. If a 30% discount leaves $21, then $21 is 70% of the original, so the original is \(21: 0.70=30\) dollars.
What does each letter in I = Prt mean?
The letter \(I\) is the interest earned, \(P\) is the principal (the starting amount), \(r\) is the annual interest rate written as a decimal, and \(t\) is the time in years. Simple interest is always figured on the original principal.
- Percent Problems: Tax, Tips, Markups and Discounts worksheet (20 problems)
- Percent Problems: Tax, Tips, Markups and Discounts quiz (10 questions)
- Percent Problems: Tax, Tips, Markups and Discounts test (45 minutes)
- Previous lesson: Equations of Proportional Relationships
- Next lesson: Percent Increase, Percent Decrease and Percent Error
See every chapter on the Grade 7 math page.