Percent questions appear throughout the PSAT/NMSQT Math section, and while the underlying idea — a percent is just a fraction out of 100 — is simple, the College Board finds many ways to make percent problems genuinely challenging. Successive discounts, percent increase versus percent decrease, working backward from a final price to an original price, and percents expressed algebraically in terms of a variable are all common variations. Each one rewards a careful, methodical setup far more than it rewards speed.
One of the most common traps on the PSAT is assuming that two percent changes combine by simple addition or subtraction. A 25% discount followed by a 20% discount does not equal a 45% discount — each discount applies to a new, already-reduced price, so the two multipliers (0.75 and 0.80) must be multiplied together, not their percentages added. The same logic applies to percent increases: increasing a quantity by 20% and then decreasing a related quantity by 20% does not cancel out, because the two percent changes are being applied to different amounts. Recognizing when a problem requires multiplying successive decimal multipliers — rather than adding or subtracting percentages directly — is often the single biggest factor in getting these questions right.
Below are 13 PSAT/NMSQT-style practice problems covering successive percent changes, percent increase and decrease, mixtures, algebraic percent expressions, and setting up systems of equations from percent-based word problems. Work through each problem, select your answer, and check whether you got it right — every problem includes a complete, step-by-step explanation. Once you've worked through these, keep building your skills with thousands more official-style questions in our free PSAT/NMSQT Math QBank.
There are n candies in a jar. If one candy is removed, what percent of the candies are left in terms of n?
After removing one candy, there are n − 1 candies left out of the original n. As a fraction of the original amount, that's (n − 1)/n.
To convert any fraction to a percent, multiply by 100: 100 × (n − 1)/n, written as 100((n−1)/n)%.
The correct answer is D.
The price of a cellphone was discounted by 25% and then discounted an additional 20%, to become $348. What was the original price of the cellphone before it was discounted twice?
Two successive discounts do not add together — each one applies to the price after the previous discount. A 25% discount leaves 75% of the price (a multiplier of 0.75), and a further 20% discount leaves 80% of that (a multiplier of 0.80).
Let P be the original price: P × 0.75 × 0.80 = 348
P × 0.6 = 348 → P = 348 / 0.6 = 580
The original price was $580.00. The correct answer is A.
A chemist mixes a 40% acid solution and a 30% acid solution. How many liters of the 40% solution must be added to produce 50 liters of a solution that is 36% acid?
Let x be the liters of the 40% solution used. Since the total mixture is 50 liters, the amount of 30% solution is 50 − x.
The total amount of pure acid from both solutions must equal the acid in the final 36% mixture:
0.40x + 0.30(50 − x) = 0.36(50)
0.4x + 15 − 0.3x = 18 → 0.1x = 3 → x = 30
30 liters of the 40% solution are needed. The correct answer is C.
Victor invests part of his $5,000 in a savings account that pays 4.5% annual simple interest. He invests the rest in bonds that pay 8% annual simple interest. Let s be the amount invested in savings and r be the amount invested in bonds. Victor's total income in one year from these investments is $305.50. Which of the following systems of equations represents this relationship?
The total amount invested is $5,000, split between the two accounts: s + r = 5,000. This rules out choices A and B, which incorrectly set the interest expression equal to 5,000.
The total income is $305.50, earned as 4.5% interest on the savings amount plus 8% interest on the bonds amount: 0.045s + 0.08r = 305.50.
Choice D swaps the interest rates onto the wrong variables (pairing 8% with s, the savings account, instead of r, the bonds).
The correct system is s + r = 5,000 and 0.045s + 0.08r = 305.50. The correct answer is C.
A sporting goods store added 50% profit cost and 8% tax to the price of a backpack, which then became $129.60. What was the price of the backpack before adding profit and tax?
Just like successive discounts, successive increases apply one after the other, not by adding percentages. A 50% profit markup means multiplying by 1.50, and the 8% tax that follows means multiplying that result by 1.08.
Let P be the original price: P × 1.50 × 1.08 = 129.60
P × 1.62 = 129.60 → P = 129.60 / 1.62 = 80
The backpack originally cost $80. The correct answer is B.
There are 800 students in a school and 45% of the students are male. If 30% of the male students and 25% of the female students play varsity sports, how many students play varsity sports?
Find the number of male and female students. Male: 45% × 800 = 360. Female: the remaining 55%, or 800 − 360 = 440.
Find how many of each group play varsity sports. Male athletes: 30% × 360 = 108. Female athletes: 25% × 440 = 110.
Add the two groups together: 108 + 110 = 218.
The correct answer is C.
Keep the momentum going.
Explore 2,500+ Free PSAT/NMSQT Math ProblemsWhich of the following is equivalent to 0.03% of 4?
Convert 0.03% to a decimal first. A percent is divided by 100, so 0.03% = 0.03/100 = 0.0003.
"Of 4" means multiply: 0.0003 × 4 = 0.0012.
A common mistake is converting 0.03% to 0.003 (moving the decimal only one place instead of two) — always divide the percent by 100, not just by 10.
The correct answer is C.
1/400 is equivalent to which of the following percents?
Convert the fraction to a decimal first: 1/400 = 0.0025.
To express a decimal as a percent, multiply by 100 (which just shifts the decimal point two places right): 0.0025 × 100 = 0.25%.
The correct answer is A.
The quantities x and y are positive. If x is decreased by 20 percent and y is increased by 20 percent, then the product of x and y is
A decrease of 20% means multiplying x by 0.80. An increase of 20% means multiplying y by 1.20. The new product is:
(0.80x)(1.20y) = 0.96xy
A multiplier of 0.96 means the product is now 96% of its original value — a decrease of 100% − 96% = 4%. Notice the two 20% changes do not cancel out to leave the product unchanged.
The correct answer is B.
By what percent is 4.5 × 10⁵ greater than 9 × 10⁴?
Rewrite both numbers in standard form to compare them directly: 4.5 × 10⁵ = 450,000 and 9 × 10⁴ = 90,000.
Percent greater is found using: (new − old)/old × 100.
(450,000 − 90,000)/90,000 × 100 = 360,000/90,000 × 100 = 4 × 100 = 400%
A common trap is finding that 450,000 is 5 times as large as 90,000 and mistakenly answering 500% — being 5 times as large means a 400% increase (since the original 100% is already included in "5 times"). The correct answer is B.
The temperature increased from 60°F to 72°F. What is the percent increase in temperature?
Percent increase is the change divided by the original value, times 100. The change here is 72 − 60 = 12 degrees.
12/60 × 100 = 20%
The correct answer is C. A common mistake is dividing the change by the new value (12/72) instead of the original value, which gives an incorrect result.
This year's enrollment in Mesa School District is 6,000, which is 20 percent higher than last year's. What was last year's enrollment in Mesa School District?
Let E be last year's enrollment. This year's enrollment is 20% higher, meaning it's 120% of last year's value: 1.20 × E = 6,000.
E = 6,000 / 1.20 = 5,000
A common mistake is subtracting 20% of 6,000 directly (6,000 − 1,200 = 4,800), but that assumes the 20% applies to this year's value rather than last year's — the base of a percent increase is always the original (earlier) amount.
The correct answer is C.
If 125% of x is 80 and x is n% of 400, what is the value of n?
First find x from "125% of x is 80": 1.25x = 80 → x = 64.
Now use "x is n% of 400" to find n: 64 = (n/100) × 400.
64 = 4n → n = 16
The correct answer is C.
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