SAT Practice Test 1 Math: Free Questions, Answer Key & Scoring Guide

SAT Practice Test 1 Math Section — Free Questions, Answers & Scoring Guide

Preparing for the SAT? This guide covers official SAT Practice Test #1 Math questions from both the No‑Calculator and Calculator sections, written with clean LaTeX math notation, plus a full answer key and a score conversion guide. It's built for students, tutors, and parents who want a fast, accurate way to practice SAT Math.

Math Test – No Calculator

25 Minutes, 20 Questions

Calculator use is not permitted on this section. All variables and expressions represent real numbers unless otherwise indicated, and figures (where present) are drawn to scale unless stated otherwise.

1. If $\dfrac{x-1}{3} = k$ and $k = 3$, what is the value of $x$?

A) 2 B) 4 C) 9 D) 10

2. For $i = \sqrt{-1}$, what is the sum $(7 + 3i) + (-8 + 9i)$?

A) $-1 + 12i$ B) $-1 - 6i$ C) $15 + 12i$ D) $15 - 6i$

3. On Saturday afternoon, Armand sent $m$ text messages each hour for 5 hours, and Tyrone sent $p$ text messages each hour for 4 hours. Which of the following represents the total number of messages sent by Armand and Tyrone on Saturday afternoon?

A) $9mp$ B) $20mp$ C) $5m + 4p$ D) $4m + 5p$

4. Kathy is a repair technician for a phone company. Each week, she receives a batch of phones that need repairs. The number of phones that she has left to fix at the end of each day can be estimated with the equation $P = 108 - 23d$, where $P$ is the number of phones left and $d$ is the number of days she has worked that week. What is the meaning of the value 108 in this equation?

A) Kathy will complete the repairs within 108 days. B) Kathy starts each week with 108 phones to fix. C) Kathy repairs phones at a rate of 108 per hour. D) Kathy repairs phones at a rate of 108 per day.

5. Which of the following is equivalent to the expression below?

A) $4x^2y^2$ B) $8xy^2 - 6y^2$ C) $2x^2y + 2xy^2$ D) $2x^2y + 8xy^2 - 6y^2$

6. A pediatrician uses the model below to estimate the height $h$ of a boy, in inches, in terms of the boy's age $a$, in years, between the ages of 2 and 5.

Based on the model, what is the estimated increase, in inches, of a boy's height each year?

A) 3 B) 5.7 C) 9.5 D) 14.3

7. The formula below gives the monthly payment $m$ needed to pay off a loan of $P$ dollars at $r$ percent annual interest over $N$ months.

Which of the following gives $P$ in terms of $m$, $r$, and $N$?

A) $P = \dfrac{\left(\dfrac{r}{1{,}200}\right)\left(1 + \dfrac{r}{1{,}200}\right)^N}{\left(1 + \dfrac{r}{1{,}200}\right)^N - 1}, m$

B) $P = \dfrac{\left(1 + \dfrac{r}{1{,}200}\right)^N - 1}{\left(\dfrac{r}{1{,}200}\right)\left(1 + \dfrac{r}{1{,}200}\right)^N}, m$

C) $P = \left(\dfrac{r}{1{,}200}\right) m$

D) $P = \left(\dfrac{1{,}200}{r}\right) m$

8. If $\dfrac{a}{b} = 2$, what is the value of $\dfrac{4b}{a}$?

A) 0 B) 1 C) 2 D) 4

9. What is the solution $(x, y)$ to the system of equations below?

A) $(-5, -2)$ B) $(3, -8)$ C) $(4, -6)$ D) $(9, -6)$

10. For the function $g(x) = ax^2 + 24$, where $a$ is a constant, $g(4) = 8$. What is the value of $g(-4)$?

A) 8 B) 0 C) $-1$ D) $-8$

11. In the equations below, $b$ and $c$ represent the price per pound, in dollars, of beef and chicken, respectively, $x$ weeks after July 1 during last summer.

What was the price per pound of beef when it was equal to the price per pound of chicken?

A) $2.60 B) $2.85 C) $2.95 D) $3.35

12. A line in the $xy$-plane passes through the origin and has a slope of $\dfrac{1}{7}$. Which of the following points lies on the line?

A) $(0, 7)$ B) $(1, 7)$ C) $(7, 7)$ D) $(14, 2)$

13. If $x > 3$, which of the following is equivalent to

A) $\dfrac{2x+5}{x^2+5x+6}$

B) $\dfrac{x^2+5x+6}{2x+5}$

C) $2x+5$

D) $x^2+5x+6$

14. If $3x - y = 12$, what is the value of $\dfrac{8^x}{2^y}$?

A) $2^{12}$ B) $4^4$ C) $8^2$ D) The value cannot be determined from the information given.

15. If $(ax + 2)(bx + 7) = 15x^2 + cx + 14$ for all values of $x$, and $a + b = 8$, what are the two possible values for $c$?

A) 3 and 5 B) 6 and 35 C) 10 and 21 D) 31 and 41

Questions 16–20 are student-produced response ("grid-in") questions. Solve each problem and enter your own numeric answer — there are no answer choices.

16. If $t > 0$ and $t^2 - 4 = 0$, what is the value of $t$?

18. According to the system of equations below, what is the value of $x$?

19. In a right triangle, one angle measures $x°$, where $\sin x° = \dfrac{4}{5}$. What is $\cos(90° - x°)$?

20. If $a = 5\sqrt{2}$ and $2a = \sqrt{2x}$, what is the value of $x$?

Math Test – Calculator

55 Minutes, 38 Questions

Calculator use is permitted on this section. All variables and expressions represent real numbers unless otherwise indicated.

2. If $y = kx$, where $k$ is a constant, and $y = 24$ when $x = 6$, what is the value of $y$ when $x = 5$?

A) 6 B) 15 C) 20 D) 23

4. If $16 + 4x$ is 10 more than 14, what is the value of $8x$?

A) 2 B) 6 C) 16 D) 80

6. Given the conversions below:

A hospital stores one type of medicine in 2‑decagram containers. Based on the information given above, how many 1‑milligram doses are there in one 2‑decagram container?

A) 0.002 B) 200 C) 2,000 D) 20,000

8. For what value of $n$ is $|n - 1| + 1$ equal to $0$?

A) 0 B) 1 C) 2 D) There is no such value of $n$.

Questions 9 and 10 refer to the following information:

The speed of a sound wave in air depends on the air temperature. The formula above shows the relationship between $a$, the speed of a sound wave, in feet per second, and $t$, the air temperature, in degrees Fahrenheit (°F).

9. Which of the following expresses the air temperature in terms of the speed of a sound wave?

A) $t = \dfrac{a - 1{,}052}{1.08}$

B) $t = \dfrac{a + 1{,}052}{1.08}$

C) $t = \dfrac{1{,}052 - a}{1.08}$

D) $t = \dfrac{1.08}{a + 1{,}052}$

10. At which of the following air temperatures will the speed of a sound wave be closest to 1,000 feet per second?

A) $-46°F$ B) $-48°F$ C) $-49°F$ D) $-50°F$

11. Which of the following numbers is NOT a solution of the inequality $3x - 5 \ge 4x - 3$?

A) $-1$ B) $-2$ C) $-3$ D) $-5$

13. A group of tenth-grade students responded to a survey that asked which math course they were currently enrolled in. The survey data were broken down as shown in the table below.

Which of the following categories accounts for approximately 19 percent of all the survey respondents?

A) Females taking Geometry B) Females taking Algebra II C) Males taking Geometry D) Males taking Algebra I

14. The table below lists the lengths, to the nearest inch, of a random sample of 21 brown bullhead fish. The outlier measurement of 24 inches is an error.

Of the mean, median, and range of the values listed, which will change the most if the 24-inch measurement is removed from the data?

A) Mean B) Median C) Range D) They will all change by the same amount.

18. In the $xy$-plane, if $(0, 0)$ is a solution to the system of inequalities below, which of the following relationships between $a$ and $b$ must be true?

A) $a > b$ B) $b > a$ C) $|a| > |b|$ D) $a = -b$

19. A food truck sells salads for $6.50 each and drinks for$2.00 each. The food truck's revenue from selling a total of 209 salads and drinks in one day was $836.50. How many salads were sold that day?

A) 77 B) 93 C) 99 D) 105

20. Alma bought a laptop computer at a store that gave a 20 percent discount off its original price. The total amount she paid to the cashier was $p$ dollars, including an 8 percent sales tax on the discounted price. Which of the following represents the original price of the computer in terms of $p$?

A) $0.88p$

B) $\dfrac{p}{0.88}$

C) $(0.8)(1.08)p$

D) $\dfrac{p}{(0.8)(1.08)}$

21. The data below were produced by a sleep researcher studying the number of dreams people recall when asked to record their dreams for one week. Group X consisted of 100 people who observed early bedtimes, and Group Y consisted of 100 people who observed later bedtimes.

If a person is chosen at random from those who recalled at least 1 dream, what is the probability that the person belonged to Group Y?

A) $\dfrac{68}{100}$

B) $\dfrac{79}{100}$

C) $\dfrac{79}{164}$

D) $\dfrac{164}{200}$

Questions 22 and 23 refer to the following information:

Annual Budgets for Different Programs in Kansas, 2007 to 2010 (in thousands of dollars)

22. Which of the following best approximates the average rate of change in the annual budget for agriculture/natural resources in Kansas from 2008 to 2010?

A) $50,000,000 per year B) $65,000,000 per year C) $75,000,000 per year D) $130,000,000 per year

23. Of the following, which program's ratio of its 2007 budget to its 2010 budget is closest to the human resources program's ratio of its 2007 budget to its 2010 budget?

A) Agriculture/natural resources B) Education C) Highways and transportation D) Public safety

24. Which of the following is an equation of a circle in the $xy$-plane with center $(0, 4)$ and a radius with endpoint $\left(\dfrac{4}{3}, 5\right)$?

A) $x^2 + (y-4)^2 = \dfrac{25}{9}$

B) $x^2 + (y+4)^2 = \dfrac{25}{9}$

C) $x^2 + (y-4)^2 = \dfrac{5}{3}$

D) $x^2 + (y+4)^2 = \dfrac{3}{5}$

25. The equation below expresses the approximate height $h$, in meters, of a ball $t$ seconds after it is launched vertically upward from the ground with an initial velocity of 25 meters per second.

After approximately how many seconds will the ball hit the ground?

A) 3.5 B) 4.0 C) 4.5 D) 5.0

26. Katarina is a botanist studying the production of pears by two types of pear trees. She noticed that Type A trees produced 20 percent more pears than Type B trees did. Based on Katarina's observation, if the Type A trees produced 144 pears, how many pears did the Type B trees produce?

A) 115 B) 120 C) 124 D) 173

27. A square field measures 10 meters by 10 meters. Ten students each mark off a randomly selected region of the field; each region is square and has side lengths of 1 meter, and no two regions overlap. The students count the earthworms contained in the soil to a depth of 5 centimeters beneath the ground's surface in each region. The results are shown below.

Which of the following is a reasonable approximation of the number of earthworms to a depth of 5 centimeters beneath the ground's surface in the entire field?

A) 150 B) 1,500 C) 15,000 D) 150,000

29. For a polynomial $p(x)$, the value of $p(3)$ is $-2$. Which of the following must be true about $p(x)$?

A) $x - 5$ is a factor of $p(x)$. B) $x - 2$ is a factor of $p(x)$. C) $x + 2$ is a factor of $p(x)$. D) The remainder when $p(x)$ is divided by $x - 3$ is $-2$.

Questions 31–38 are student-produced response ("grid-in") questions. Solve each problem and enter your own numeric answer — there are no answer choices.

31. Wyatt can husk at least 12 dozen ears of corn per hour and at most 18 dozen ears of corn per hour. Based on this information, what is a possible amount of time, in hours, that it could take Wyatt to husk 72 dozen ears of corn?

32. The posted weight limit for a covered wooden bridge in Pennsylvania is 6,000 pounds. A delivery truck that is carrying $x$ identical boxes, each weighing 14 pounds, will pass over the bridge. If the combined weight of the empty delivery truck and its driver is 4,500 pounds, what is the maximum possible value for $x$ that will keep the combined weight of the truck, driver, and boxes below the bridge's posted weight limit?

34. A local television station sells time slots for programs in 30-minute intervals. If the station operates 24 hours per day, every day of the week, what is the total number of 30-minute time slots the station can sell for Tuesday and Wednesday?

36. For what value of $x$ is the function $h$ below undefined?

Questions 37 and 38 refer to the following information:

Jessica opened a bank account that earns 2 percent interest compounded annually. Her initial deposit was $100, and she uses the expression$100(x)^t$to find the value of the account after$t$ years.

37. What is the value of $x$ in the expression?

38. Jessica's friend Tyshaun found an account that earns 2.5 percent interest compounded annually. Tyshaun made an initial deposit of $100 into this account at the same time Jessica made a deposit of$100 into her account. After 10 years, how much more money will Tyshaun's initial deposit have earned than Jessica's initial deposit? (Round your answer to the nearest cent and ignore the dollar sign when gridding your response.)

Answer Key

Math Test – No Calculator

Math Test – Calculator

Note: A few answers above (e.g., No‑Calculator Q17; Calculator Q1, 3, 5, 7, 12, 15, 16, 17, 28, 30, 33, 35) correspond to figure‑based questions that were omitted from this transcription, but are listed here so the answer key stays complete and numbered exactly as in the original test.

How SAT Math Scoring Works

The SAT Math section is scored on a 200–800 scale, combining raw scores from the No‑Calculator and Calculator sections into a single Math Section Raw Score (0–58 correct answers), which is then converted using the College Board's official conversion table.

Steps to score your practice test:

1. Count your correct answers on the No‑Calculator section (out of 20).

2. Count your correct answers on the Calculator section (out of 38).

3. Add them together for your Math Section Raw Score (0–58).

4. Use the official raw‑score‑to‑scaled‑score conversion table to convert that raw score into your Math Section Score (200–800).

5. Add your Math Section Score to your Evidence‑Based Reading and Writing score (200–800) to get your Total SAT Score (400–1600).

Your math raw score also splits into three official subscores (each rated 1–15) that appear on your score report:

• Heart of Algebra

• Problem Solving and Data Analysis

• Passport to Advanced Math

These subscores help pinpoint exactly which math skill areas need more review before your next SAT attempt.

Frequently Asked Questions

Is a calculator allowed on the entire SAT Math section? No. SAT Math is split into two parts: a 25‑minute No‑Calculator section (20 questions) and a 55‑minute Calculator section (38 questions).

How many grid-in (student-produced response) questions are there? 5 appear in the No‑Calculator section (Q16–20) and 8 appear in the Calculator section (Q31–38), for 13 total.

What score range does SAT Math cover? The Math Section Score ranges from 200 to 800 and combines with the Evidence‑Based Reading and Writing score (200–800) for a Total SAT Score between 400 and 1600.