Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Percent problems show up all over everyday life: a 20% tip, a 30% off sale, 6% sales tax, simple interest on savings, a store's markup or a measurement that is off by some percent. In seventh grade students solve these as proportional relationships, recognizing that a percent is a rate per 100 and that the same percent always scales with the base amount.
What makes these problems multistep is that one percent change often feeds into another, or the question asks for something other than the percent itself. A jacket is marked down 25% and then taxed at 6%, so the tax is taken on the sale price, not the original. A price that goes up 20% and then down 20% does not return to where it started. Percent increase and decrease always compare the change to the original amount, and percent error compares the error to the actual value. Students learn to write each step clearly, and many learn to use a single multiplier such as 0.75 or 1.06 as a shortcut.
After a discount, tax is charged on the sale price. Students who tax the original price get an answer that is a little too high, so ask them to say which amount each percent applies to.
A $200 item raised 20% becomes $240, and 20% off $240 is $48, giving $192. The two percents are taken of different amounts, so they do not cancel.
Percent increase from $50 to $65 is 15 ÷ 50 = 30%, not 15 ÷ 65. The original value is always the base for percent change.
Interest at 4% per year for 3 years is 3 times one year's interest. Writing I = P × r × t with every value labeled keeps the time factor in.
A jacket costs $80. It is on sale for 25% off, and then 6% sales tax is added. What is the final price?
Answer: The final price is $63.60.
Percent bar models (a strip marked 0% to 100% above and $0 to the full price below) make the base of each percent visible, which is the root of most errors here. Move from the bar to multipliers once students can explain why 25% off is the same as paying 75%.
Real menus, receipts and store flyers make good sources for problems, and comparing two offers ('$15 off' versus '20% off') pushes students to reason rather than follow one rule. Tests often combine two steps in one item, so practice should too.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 7.20 (also accepted: 7.2, $7.20)
18% of $40 is 0.18 × 40 = 7.20, so the tip is $7.20.
Answer: A) 30%
The increase is 65 - 50 = 15. Compare it to the original: 15 ÷ 50 = 0.30, which is 30%.
Answer: 72 (also accepted: $72)
I = P × r × t = 600 × 0.04 × 3 = 72. The interest is $72.
Answer: C) $42
A 75% markup adds 0.75 × 24 = 18 dollars, so the price is 24 + 18 = $42. In one step, 24 × 1.75 = 42.
Answer: 20% (also accepted: 20)
The error is 50 - 40 = 10. Compare it to the actual value: 10 ÷ 50 = 0.2, which is a 20% error.
Answer: A) $150
5% of $3,000 is 0.05 × 3,000 = $150.
Answer: B) $192
After the markdown the price is 0.8 × 200 = $160. Raising $160 by 20% gives 1.2 × 160 = $192. The second 20% is taken of a smaller amount.
A full lesson with slides, activities and an exit ticket on multistep ratio and percent problems, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.RP.A.3 with an answer key, ready in about a minute.
Make a worksheet →Turn multistep ratio and percent problems into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Simple interest, tax, markups, markdowns, tips, commissions, fees, percent increase and decrease, and percent error, usually with more than one step.
Yes, once they understand why they work. A multiplier combines 'find the percent' and 'add or subtract it' into one step and makes chains of percent changes much easier.
In sixth grade (6.RP.A.3) students find a percent of a quantity and solve for the whole. Seventh grade adds multistep situations and percent change.