Official Exam Prep
PSAT Math
20 Essential Problems
Concept review · Key formulas · Worked examples · Full exam-style questions across all tested units
20Questions
4Units
25Minutes
Heart of Algebra
Unit 1 — Concept Review
Linear Equations, Inequalities & Systems
Linear equations form the backbone of algebra. A linear equation in one variable has exactly one solution (unless it's an identity or contradiction). Systems of two linear equations can be solved by substitution or elimination. Inequalities follow the same rules, except multiplying/dividing by a negative flips the inequality sign.
Slope-Intercept
y = mx + b
Point-Slope
y − y₁ = m(x − x₁)
Slope Formula
m = (y₂ − y₁) / (x₂ − x₁)
No Solution / Infinite
Same slope → no sol. (≠b)
Same line → ∞ sol.
Same line → ∞ sol.
✏️ Worked Example
If $3x - 7 = 2(x + 4)$, find $x$.
$3x - 7 = 2x + 8 \Rightarrow x = 15$
Answer: x = 15
01
If $\dfrac{2x + 5}{3} = \dfrac{x - 1}{2}$, what is the value of $x$?
✅ Explanation
Cross-multiply: $2(2x+5) = 3(x-1)$$4x + 10 = 3x - 3$
$x = -13$
Correct Answer: A
02
A system of equations is shown below:
$3x + 2y = 16$
$5x - 2y = 0$
What is the value of $x + y$?
$3x + 2y = 16$
$5x - 2y = 0$
What is the value of $x + y$?
✅ Explanation
Add equations: $8x = 16 \Rightarrow x = 2$Substitute: $5(2) - 2y = 0 \Rightarrow y = 5$
$x + y = 2 + 5 = 7$
Correct Answer: C
03
Which of the following is the solution set of $-3(2x - 4) > 6x - 12$?
✅ Explanation
$-6x + 12 > 6x - 12$$24 > 12x$
$x < 2$
Correct Answer: B
04
A rental car company charges a flat fee of $\$30$ plus $\$0.25$ per mile driven. A customer's total bill was $\$67.50$. How many miles did the customer drive?
✅ Explanation
Let $m$ = miles. $30 + 0.25m = 67.50$$0.25m = 37.50$
$m = 150$
Correct Answer: C
05
How many solutions does the equation $|2x - 6| = 3x + 1$ have?
✅ Explanation
Case 1: $2x - 6 = 3x + 1 \Rightarrow x = -7$. Check: $|{-20}| = -20$? No (RHS negative). Reject.Case 2: $-(2x-6) = 3x+1 \Rightarrow -2x+6=3x+1 \Rightarrow x=1$. Check: $|{-4}|=4=4$. ✓
Only 1 valid solution.
Correct Answer: B
Problem Solving & Data Analysis
Unit 2 — Concept Review
Ratios, Percentages, Statistics & Probability
This unit tests real-world quantitative reasoning. Master ratios/proportions, percent change, weighted averages, and interpreting data from tables/graphs. Probability = favorable outcomes ÷ total outcomes. Key statistics: mean, median, mode, range, standard deviation direction.
Percent Change
% Change = (New−Old)/Old × 100
Probability
P(A) = favorable / total
Mean
Mean = Σxᵢ / n
Unit Rate
Rate = Quantity / Time
✏️ Worked Example
A shirt costs $\$40$. After a 25% discount, what is the sale price?
Sale price $= 40 \times (1 - 0.25) = 40 \times 0.75 = \$30$
Answer: $30
06
A car travels 180 miles in 3 hours. At the same rate, how many miles will it travel in 5 hours?
✅ Explanation
Rate = 180 ÷ 3 = 60 mphDistance = 60 × 5 = 300 miles
Correct Answer: C
07
The five numbers in a data set are: 4, 7, 12, $x$, and 20. If the mean of the data set is 11, what is the median?
✅ Explanation
Mean = $\frac{4+7+12+x+20}{5} = 11$$43 + x = 55 \Rightarrow x = 12$
Sorted: 4, 7, 12, 12, 20 → Median = 12
Correct Answer: D
08
A store's revenue increased from $\$12,000$ to $\$15,600$ in one year. What was the percent increase in revenue?
✅ Explanation
$\%$ Change $= \frac{15600-12000}{12000} \times 100 = \frac{3600}{12000} \times 100 = 30\%$Correct Answer: B
09
A bag contains 4 red, 5 blue, and 6 green marbles. If two marbles are drawn at random without replacement, what is the probability that both marbles are blue?
✅ Explanation
Total marbles = 15. $P(\text{both blue}) = \frac{5}{15} \times \frac{4}{14} = \frac{20}{210} = \frac{2}{21}$Correct Answer: B
10
A line of best fit for a data set has the equation $y = 2.5x + 14$. Based on this equation, what is the predicted value of $y$ when $x = 8$?
✅ Explanation
$y = 2.5(8) + 14 = 20 + 14 = 34$Correct Answer: B
Passport to Advanced Math
Unit 3 — Concept Review
Quadratics, Polynomials, Functions & Exponentials
Advanced math covers nonlinear equations. Factoring, the quadratic formula, function notation, and exponential growth/decay are essential. Know how to manipulate complex expressions and understand function transformations: $f(x+h)$ shifts left by $h$; $f(x)+k$ shifts up by $k$.
Quadratic Formula
x = (−b ± √(b²−4ac)) / 2a
Vertex Form
y = a(x − h)² + k
Exponential Growth
y = A · (1 + r)ᵗ
Discriminant
D = b²−4ac
D>0: 2 roots, D=0: 1 root
D>0: 2 roots, D=0: 1 root
✏️ Worked Example
Factor $x^2 - 5x + 6$.
Find two numbers that multiply to 6 and add to −5: that's −2 and −3.
$(x - 2)(x - 3)$, so $x = 2$ or $x = 3$
11
What are the solutions to $x^2 - 3x - 10 = 0$?
✅ Explanation
Factor: $(x-5)(x+2) = 0$$x = 5$ or $x = -2$
Correct Answer: B
12
If $f(x) = 3x^2 - 2x + 1$, what is $f(-2)$?
✅ Explanation
$f(-2) = 3(-2)^2 - 2(-2) + 1 = 3(4) + 4 + 1 = 12 + 4 + 1 = 17$Correct Answer: C
13
A bacteria culture starts with 200 cells and doubles every 3 hours. Which expression gives the number of cells after $t$ hours?
✅ Explanation
Every 3 hours it doubles. In $t$ hours there are $t/3$ doubling periods.Cells $= 200 \cdot 2^{t/3}$
Correct Answer: B
14
The polynomial $p(x) = x^3 - 2x^2 - 5x + 6$ has a factor of $(x - 3)$. Which of the following shows the complete factored form of $p(x)$?
✅ Explanation
Divide: $p(x) \div (x-3) = x^2 + x - 2$Factor: $(x+2)(x-1)$
So $p(x) = (x-3)(x+2)(x-1)$
Correct Answer: D
15
Which expression is equivalent to $\dfrac{x^2 - 9}{x^2 + x - 6}$ for all valid values of $x$?
✅ Explanation
Numerator: $(x-3)(x+3)$Denominator: $(x+3)(x-2)$
Cancel $(x+3)$: $\dfrac{x-3}{x-2}$
Correct Answer: A
Additional Topics — Geometry & Trigonometry
Unit 4 — Concept Review
Geometry, Circles & Trigonometry
Geometry on the PSAT covers area, perimeter, volume, coordinate geometry, and properties of triangles, circles, and angles. Trigonometry focuses on SOHCAHTOA in right triangles. Remember: sum of angles in a triangle = 180°; opposite angles in a parallelogram are equal.
Circle
A = πr² · C = 2πr
Pythagorean
a² + b² = c²
SOHCAHTOA
sin = opp/hyp
cos = adj/hyp
tan = opp/adj
cos = adj/hyp
tan = opp/adj
Special Triangles
30-60-90: 1·√3·2
45-45-90: 1·1·√2
45-45-90: 1·1·√2
✏️ Worked Example
A right triangle has legs of length 5 and 12. What is the hypotenuse?
$c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
Answer: 13
16
A rectangle has a perimeter of 56 cm. If the length is 4 cm more than twice the width, what is the area of the rectangle?
✅ Explanation
Let width $= w$, length $= 2w + 4$.Perimeter: $2(w + 2w + 4) = 56 \Rightarrow 6w + 8 = 56 \Rightarrow w = 8$ cm
Length $= 2(8) + 4 = 20$ cm
Area $= 8 \times 20 = 160$ cm²
Correct Answer: A
17
A circle has a circumference of $18\pi$. A sector of the circle has a central angle of 120°. What is the area of that sector?
✅ Explanation
$C = 2\pi r = 18\pi \Rightarrow r = 9$Area of circle $= \pi(9)^2 = 81\pi$
Sector area $= \frac{120}{360} \times 81\pi = \frac{1}{3} \times 81\pi = 27\pi$
Correct Answer: B
18
In a right triangle, the angle $\theta$ has $\sin\theta = \dfrac{5}{13}$. What is the value of $\cos\theta$?
✅ Explanation
opp = 5, hyp = 13 → adj $= \sqrt{13^2 - 5^2} = \sqrt{144} = 12$$\cos\theta = \frac{12}{13}$
Correct Answer: B
19
A cylinder has a radius of 4 cm and a height of 9 cm. A cone with the same base and height is removed from the cylinder. What is the remaining volume? (Use $\pi \approx 3.14$)
✅ Explanation
Cylinder: $\pi r^2 h = \pi(16)(9) = 144\pi$Cone: $\frac{1}{3}\pi r^2 h = \frac{1}{3}\pi(16)(9) = 48\pi$
Remaining: $144\pi - 48\pi = 96\pi$ cm³
Correct Answer: A
20
A circle in the coordinate plane has the equation $(x-3)^2 + (y+5)^2 = 49$. Which of the following points lies on the circle?
✅ Explanation
Center $(3, -5)$, radius $= 7$. Test each point:A: $(10-3)^2 + (-5+5)^2 = 49+0 = 49$ ✓
B: $0 + 49 = 49$ ✓ — but let's check: $(3-3)^2+(2+5)^2=0+49=49$ ✓
The question asks which lies on the circle. (A): $(7)^2+(0)^2=49$ ✓
Correct Answer: A
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📖 Answer Key & Explanations
Full solutions for all 20 questions — ideal for review and self-study