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MathProblem-Solving and Data AnalysisHard
A store raises the price of a jacket by 20%. Later, it lowers the new price by 20%. The final price is what percent of the original price?
A80%
B96%Correct
C100%The trap
D104%
Check with $100: $100→then$120→then$96
Turn each change into a multiplier.
Up 20% means × 1.20. Down 20% means × 0.80.
Apply them in order.
1.20 × 0.80 = 0.96
Read the answer.
The final price is 96% of the original. That’s B.
Why not 100%?
The 20% cut is taken from the higher price, so it removes more than the 20% increase added.
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Reading & Writing · Module 1
27 questions32 min
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Reading & Writing · Module 2
27 questions32 min
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22 questions35 min
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Math · Module 2
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The Daily, plus 36 practice questions every day: 12 easy, 12 medium and 12 hard, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too. No card needed.
Pro is $20 a month or $200 a year: unlimited practice with no daily limit, plus every full-length test and Challenge test. Peak is $29 a month or $280 a year: everything in Pro, plus the full Learn course and Peak Plan. Cancel anytime from your account.
98 questions across four modules, in the digital SAT’s order: Reading & Writing Module 1 and Module 2 (27 questions each), a 10-minute break, then Math Module 1 and Module 2 (22 questions each). On official timing that’s 2 hr 14 min of testing. You get estimated section scores and a total at the end.
Yes. In the standard full-length tests, Module 2 of each section is the harder version if you get 60% or more of Module 1 right, and the easier version if you don’t. Challenge tests use the hardest material throughout.
Official timing, extended 1.5× time, or untimed, on every test, including the free ones. Extended time allows 48 minutes for each Reading & Writing module and about 52 for each Math module. Untimed removes the clock so you can focus on accuracy.
A quick run with three hearts. Each right answer builds your streak: the questions get harder as it grows, and your points multiply (×2 from three in a row, ×3 from six). A miss costs a heart and drops you back to easy. Best is your longest streak.
Every day at midnight. You get a fresh 12 easy, 12 medium and 12 hard.
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Your next point starts today.
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Percentages · Lesson 3
Successive percent changes
When one percent change follows another, the second percent is taken of the new amount, not the original. So you can't add the percents. Multiply the multipliers instead.
A 20% increase followed by a 20% decrease: . The result is 96% of the original, a 4% decrease, not a return to the start. With a start of 100: up 20% gives 120. Then 20% of 120 is 24, so down 20% gives 96.
Two 5% increases: , a 10.25% increase overall, not 10%.
The order of the changes doesn't matter, because you can multiply in any order: is also 0.96. The same change repeated times gives a multiplier of , the idea behind exponential growth.
To read the overall change, compare the product with 1. Above 1 is an increase (1.1025 means up 10.25%). Below 1 is a decrease (0.96 means down 4%).
Worked example Easy
A jacket originally priced at 200 dollars is marked down 25%. Later, the store takes an additional 10% off the marked-down price.
What is the final price of the jacket, in dollars?
- Answer
How to solve it
- First markdown: dollars.
- The extra 10% is taken off the marked-down price: dollars.
- In one step: .
Why each choice is right or wrong
- A. Incorrect. This adds the discounts to get 35% off the original: . The 10% comes off the 150 dollar price, so it removes only 15 dollars, not 20.
- B. Correct. , and then .
- C. Incorrect. 150 dollars is the price after the first markdown only. The additional 10% still comes off.
- D. Incorrect. This applies only the 10% discount: . The 25% markdown comes first.
Worked example Medium
A quantity is increased by 30%, and then the result is decreased by 30%. The final value is what percent of the original value?
- Answer
How to solve it
- The multipliers are 1.3 for the increase and 0.7 for the decrease.
- Multiply: .
- The final value is 91% of the original. With a start of 100: up 30% gives 130, and 30% of 130 is 39, so down 30% gives 91.
Why each choice is right or wrong
- A. Incorrect. 9% is the size of the overall drop, . The question asks what percent of the original the final value is.
- B. Incorrect. 70% comes from applying only the 30% decrease to the original and ignoring the increase. The decrease is taken of 130% of the original.
- C. Correct. , so the final value is 91% of the original.
- D. Incorrect. 100% assumes the two changes cancel. The 30% decrease is taken of the larger amount, so it removes , more than the 30 that was added.
Trap Adding the percents
25% off and then 10% off is not 35% off, because the 10% is taken of a smaller price. Write each change as a multiplier and multiply: , which is 32.5% off.
Trap Expecting an equal rise and fall to cancel
An increase and a decrease by the same percent always leave you below where you started, in either order: the multipliers give , which is less than 1 for any nonzero . A choice that says the value is unchanged is a trap.
Desmos When Desmos is faster
For a chain of known changes, type the product of the multipliers, such as 1.3*0.7, and Desmos shows 0.91.
Desmos is faster when the percent is unknown. If a quantity goes up and then down and ends at 91% of the start, graph and . Zoom out until you see where they cross and click the intersection with positive : , so . The curve is symmetric, so ignore the matching negative intersection.
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