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PSAT Math Practice: 8 Problems With Full Solutions

October 10, 2026 · AI Math Tutor Team

This PSAT math practice set is 8 original problems with full solutions, one or two from each kind of math the test covers. Work each one by hand before you read the steps, because the steps only teach you something once you have tried.

The problems are written in the style of the test, not copied from it. They cover linear equations, a linear model, a system, a ratio, a percent change, a quadratic, a right triangle and a data set.

Key takeaways

The PSAT tests algebra, advanced math, problem solving and data analysis, and geometry and trigonometry, the same areas as the SAT.
Read the last line of each question twice: many questions ask for $4x$ , $y$ or a total, not for $x$ .
A reference sheet covers area, volume and special right triangles; slope, the vertex formula and percent change are on you.
Check every answer by substituting it back or by running a quick number through it, the way each solution below does.

How to use this PSAT math practice set

Set a timer for about 12 minutes and do all eight problems on paper. Then compare your work with each solution step by step. A right answer reached by a lucky guess counts as a miss for review purposes.

Before you start, skim the SAT math formula sheet: the PSAT gives the same reference sheet, so knowing what is on it and what is not saves time. The hub page Math Formula Sheets: Algebra, Geometry, SAT and Calculus has every formula below in one place.

Algebra: equations, models and systems

Example 1: if $\frac{2}{3}x + 5 = 17$ , what is $4x$ ?

Subtract 5 from both sides to isolate the $x$ term. $\frac{2}{3}x = 12$
Multiply both sides by $\frac{3}{2}$ . $x = 18$
The question asks for $4x$ , so multiply by 4. $4x = 72$

So $4x = 72$ . Check: $\frac{2}{3}(18) + 5 = 12 + 5 = 17$ .

Example 2: a candle is 30 centimeters tall and burns down 1.5 centimeters each hour. After how many hours is it 12 centimeters tall?

Write the model: the starting height minus the rate times the hours. $h = 30 - 1.5t$
Set the height to 12. $12 = 30 - 1.5t$
Subtract 30 from both sides, then divide by $-1.5$ . $-18 = -1.5t \quad \Rightarrow \quad t = 12$

So the candle is 12 centimeters tall after 12 hours. Check: $30 - 1.5(12) = 30 - 18 = 12$ .

Example 3: if $x + y = 14$ and $3x - y = 10$ , what is $y$ ?

Add the two equations, because $y$ and $-y$ cancel. $4x = 24$
Divide by 4. $x = 6$
Substitute into the first equation. $6 + y = 14 \quad \Rightarrow \quad y = 8$

So $y = 8$ . Check the second equation: $3(6) - 8 = 18 - 8 = 10$ .

Problem solving: ratios and percents

Example 4: the ratio of juniors to seniors in a club is 4 to 5. The club has 36 members. How many are seniors?

Add the parts of the ratio. $4 + 5 = 9 \text{ parts}$
Find the size of one part. $36 \div 9 = 4$
Seniors are 5 parts. $5 \times 4 = 20$

So there are 20 seniors. Check: 16 juniors and 20 seniors make 36, and $\frac{16}{20} = \frac{4}{5}$ .

Example 5: a price is lowered by 20 percent, then the new price is raised by 10 percent. The final price is what percent of the original?

A 20 percent decrease multiplies the price by 0.80.
A 10 percent increase multiplies the result by 1.10.
Multiply the two factors. $0.80 \times 1.10 = 0.88$

So the final price is 88 percent of the original. Check with a starting value of 100: $100 \times 0.80 = 80$ , and $80 \times 1.10 = 88$ .

Advanced math: quadratics

Example 6: what is the minimum value of $f(x) = x^2 - 6x + 5$ ?

The parabola opens up, so its minimum is at the vertex. Use $x = -\frac{b}{2a}$ with $a = 1$ and $b = -6$ . $x = -\frac{-6}{2(1)} = 3$
Evaluate the function at $x = 3$ . $f(3) = 9 - 18 + 5 = -4$

So the minimum value is $-4$ . Check by completing the square: $x^2 - 6x + 5 = (x - 3)^2 - 4$ , which is smallest when $x = 3$ .

Geometry and data

Example 7: a right triangle has a hypotenuse of 26 and one leg of 10. What is its area?

Use the Pythagorean theorem to find the other leg. $10^2 + b^2 = 26^2 \quad \Rightarrow \quad 100 + b^2 = 676$
Solve for $b$ . $b^2 = 576 \quad \Rightarrow \quad b = 24$
The legs are the base and height. $A = \frac{1}{2}(10)(24) = 120$

So the area is 120 square units. Check: 10, 24, 26 is the 5, 12, 13 triangle doubled. The geometry formula sheet lists the other common triples.

Example 8: the mean of five test scores is 84. Four of them are 78, 90, 85 and 80. What is the fifth score, and what is the median of all five?

The mean times the count gives the total. $5 \times 84 = 420$
Add the four known scores. $78 + 90 + 85 + 80 = 333$
Subtract to find the fifth score. $420 - 333 = 87$
Order all five scores and take the middle one: 78, 80, 85, 87, 90.

So the fifth score is 87 and the median is 85. Check: $\frac{420}{5} = 84$ .

Check your misses

For each problem you missed, find the first step where your work and the solution split. That step is what to practice, not the whole topic.

You can also photograph the problem, check the "Here's what I read" screen against the page, tap "Looks right, solve it", and tap "Why?" on the step that tripped you up. "Practice similar" then gives you new problems of the same kind to try by hand. Taking calculus later? The AP Calculus formula sheet is the next reference to keep.

Scan the next practice problem you miss and compare your steps with the ones in AI Math Tutor.

Common mistakes

Answering for $x$ when the question asks for $4x$ , $y$ or a total. Underline what is asked before you start.
Adding percent changes, so 20 percent off and then 10 percent on becomes "10 percent off". Multiply the factors instead.
Using a ratio part as the answer: 5 is not the number of seniors, it is the number of parts.

AI Math Tutor is a study aid, not a substitute for a teacher, and it can make mistakes: check every step, not just the answer.

Questions

What math is on the PSAT?

The PSAT covers the same kinds of math as the SAT: algebra (linear equations, inequalities and systems), advanced math (quadratics, exponents and functions), problem solving and data analysis (ratios, percents and statistics), and geometry and trigonometry.

Do you get a formula sheet on the PSAT?

Yes. A reference sheet with area and volume formulas, the Pythagorean theorem, the special right triangles and a few circle and angle facts is available during the math section. It does not list slope, the quadratic formula or percent change, so learn those.

Are these official PSAT questions?

No. All eight are original practice problems written in the style of the test. They are not official questions, and AI Math Tutor is not affiliated with or endorsed by the organization that writes the PSAT.

What is the best way to use PSAT practice problems?

Work each problem by hand first with a timer running, then compare your steps with the solution, not just your answer. For every miss, write down the step where you went wrong and try a similar problem the next day.