SAT Math Practice Questions: 12 Problems Solved Step by Step
Twelve original digital SAT math questions in the same mix as the real test, including two student-produced responses, solved step by step.
The SAT Math section has 44 questions in 70 minutes, split into two 35-minute modules. It is adaptive: how you do on the first module decides whether the second one is harder or easier. Most questions are multiple choice, and some ask you to enter your own answer. Below are 12 original SAT-style questions in the same mix of topics as the real test, each solved step by step.
What's on SAT Math
The College Board's official overview divides the section into four areas:
| Area | Questions on the test | Questions here |
|---|---|---|
| Algebra | 13 to 15 | 1 to 4 |
| Advanced Math | 13 to 15 | 5 to 8 |
| Problem-Solving and Data Analysis | 5 to 7 | 9, 10 |
| Geometry and Trigonometry | 5 to 7 | 11, 12 |
Algebra and Advanced Math together are about two-thirds of the section, so that is where practice pays off most. About 30% of the questions are set in a real-world context, and questions from all four areas appear in both modules.
Algebra
1.If \(3(2x - 5) = 4x + 7\), what is the value of \(x\)?
- A4
- B\(\tfrac{8}{5}\)
- C11
- D22
Show solution
\[\begin{gathered}6x - 15 = 4x + 7 \\ \Rightarrow\; 2x = 22 \\ \Rightarrow\; x = 11\end{gathered}\]Common trap: Stopping at \(2x = 22\) and choosing 22. The SAT often lists the value of an intermediate step as a wrong choice.
Answer: C (11)
2.A line passes through the points \((-1, 4)\) and \((3, -8)\). Which equation represents the line?
- A\(y = -3x + 1\)
- B\(y = 3x + 7\)
- C\(y = -\tfrac{1}{3}x + \tfrac{11}{3}\)
- D\(y = -3x - 7\)
Show solution
\[\begin{gathered}m = \frac{-8 - 4}{3 - (-1)} = \frac{-12}{4} = -3 \\ 4 = -3(-1) + b \\ \Rightarrow\; b = 1\end{gathered}\]Common trap: Getting the slope right but the intercept wrong, as in \(y = -3x - 7\). Check your final equation with both points: \(-3(3) + 1 = -8\).
Answer: A (\(y = -3x + 1\))
3.If \(2x + 3y = 12\) and \(x - y = 1\), what is the value of \(x + y\)? (Enter your answer.)
Your answer: ________
Show solution
From the second equation, \(x = y + 1\). Substitute into the first:
\[\begin{gathered}2(y + 1) + 3y = 12 \\ \Rightarrow\; 5y = 10 \\ \Rightarrow\; y = 2, \; x = 3\end{gathered}\]So \(x + y = 5\).
Common trap: Entering 3 or 2, the value of one variable. The question asks for \(x + y\). On student-produced response questions, there are no choices to remind you what was asked.
Answer: 5
4.A gym charges a $45 membership fee plus $12 per class. Maya has $165 to spend. What is the greatest number of classes she can take?
- A13
- B3
- C14
- D10
Show solution
\[\begin{gathered}45 + 12c \le 165 \\ \Rightarrow\; 12c \le 120 \\ \Rightarrow\; c \le 10\end{gathered}\]Common trap: Dividing \(165 \div 12\) and rounding to 13 or 14. The membership fee comes out of her budget first.
Answer: D (10)
Advanced Math
5.What is the sum of the solutions of \(x^2 - 8x + 15 = 0\)? (Enter your answer.)
Your answer: ________
Show solution
\[\begin{gathered}(x - 3)(x - 5) = 0 \\ \Rightarrow\; x = 3 \text{ or } x = 5 \\ 3 + 5 = 8\end{gathered}\]Shortcut: for \(x^2 + bx + c = 0\), the solutions always add up to \(-b\), here \(-(-8) = 8\).
Common trap: Entering \(-8\), the coefficient itself. The sum of the solutions is the opposite of \(b\) when the \(x^2\) coefficient is 1.
Answer: 8
6.The population of a town is modeled by \(P(t) = 500(1.04)^t\), where \(t\) is the number of years since 2020. What does 1.04 tell you?
- AThe population decreases by 4% each year
- BThe population grows by 4% each year
- CThe population grows by 104% each year
- DThe population grows by 1.04 people each year
Show solution
In \(a(1 + r)^t\), the growth factor \(1 + r = 1.04\), so \(r = 0.04\): the population is multiplied by 1.04 each year, which is a 4% increase.
Common trap: Reading 1.04 as 104% growth. A factor of 1.04 keeps 100% and adds 4%; a 104% increase would be a factor of 2.04.
Answer: B (The population grows by 4% each year)
7.For \(x \ne -3\), which expression is equivalent to \(\dfrac{x^2 - 9}{x + 3}\)?
- A\(x + 3\)
- B\(x - 3\)
- C\(x - 9\)
- D\(x^2 - 3\)
Show solution
Factor the numerator as a difference of squares, then cancel.
\[\frac{(x - 3)(x + 3)}{x + 3} = x - 3\]Common trap: Canceling term by term, as in \(\tfrac{x^2}{x} - \tfrac{9}{3}\). You can only cancel factors, never pieces of a sum.
Answer: B (\(x - 3\))
8.What is the minimum value of \(f(x) = 2(x - 3)^2 + 5\)?
- A3
- B2
- C5
- D\(-5\)
Show solution
A square is never negative, so \(2(x - 3)^2 \ge 0\). The smallest value happens when \(x = 3\):
\[f(3) = 2(0)^2 + 5 = 5\]Common trap: Choosing 3, the \(x\)-value where the minimum happens. The question asks for the minimum value of the function, which is the \(y\)-coordinate of the vertex.
Answer: C (5)
Problem-Solving and Data Analysis
9.The data set 4, 7, 7, 9, 13 is changed by replacing 13 with 23. Which statement is true?
- AThe mean increases by 2 and the median stays the same
- BThe mean and the median both increase by 2
- CThe mean increases by 10 and the median stays the same
- DThe median increases by 2 and the mean stays the same
Show solution
The total goes up by 10, and there are 5 values, so the mean goes up by \(10 \div 5 = 2\). The middle value is still 7, so the median does not change.
Common trap: Thinking the mean rises by the full 10. The change is spread across all five values. The median ignores how big the largest value is.
Answer: A (The mean increases by 2 and the median stays the same)
10.A runner covers 3 kilometers in 15 minutes. At this rate, what is her speed in kilometers per hour?
- A0.2
- B12
- C45
- D5
Show solution
15 minutes is \(\tfrac{1}{4}\) of an hour.
\[\frac{3 \text{ km}}{0.25 \text{ h}} = 12 \text{ km/h}\]Common trap: Dividing \(3 \div 15 = 0.2\), which is kilometers per minute. Convert the time to hours before dividing.
Answer: B (12)
Geometry and Trigonometry
11.A right triangle has legs of length 7 and 24. What is the length of the hypotenuse?
- A31
- B17
- C\(\sqrt{527}\)
- D25
Show solution
\[\begin{gathered}c = \sqrt{7^2 + 24^2} \\ = \sqrt{49 + 576} = \sqrt{625} = 25\end{gathered}\]Common trap: Adding the legs (31) or subtracting the squares, \(\sqrt{576 - 49} = \sqrt{527}\). The hypotenuse is the square root of the sum of the squares.
Answer: D (25)
12.A circle has an area of \(49\pi\). What is its circumference?
- A\(14\pi\)
- B\(7\pi\)
- C\(49\pi\)
- D\(98\pi\)
Show solution
\[\begin{gathered}\pi r^2 = 49\pi \\ \Rightarrow\; r = 7 \\ C = 2\pi r = 14\pi\end{gathered}\]Common trap: Stopping at \(r = 7\) and choosing \(7\pi\). Circumference is \(2\pi r\), so double the radius.
Answer: A (\(14\pi\))
Answer key
Three habits that raise SAT Math scores
- Read the last line first. Questions like 3 and 8 ask for something other than \(x\). Knowing the target before you start saves you from the most common wrong answer.
- Use the built-in graphing calculator to check, not to think. Graphing both sides of an equation and finding where they meet is a quick way to confirm an algebra answer.
- Bank the first module. A strong first module leads to the harder second module, which is where the higher scores are. Go carefully on module 1 rather than fast.
To review the skills behind these questions, see our lessons on slope, systems of equations, factoring, exponent rules, and percent word problems. For full-length practice, the College Board's free Bluebook app has official practice tests.
Teacher's note: The SAT rewards students who notice structure before they calculate. Question 7 looks like long division until you see a difference of squares; question 5 looks like it needs the quadratic formula until you see it factors. Before you start computing, give each question five seconds to ask, "Is there a shortcut built into this?" On this test, there usually is.
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