Which questions are student-produced responses in Test 7 Math Module 2?

Questions 6, 7, 13, 14, 20, 21, and 27 are student-produced response questions in this module.

What skills appear in Practice Test 7 Math Module 2?

This module includes linear equations and systems, functions, ratios and percentages, data, nonlinear equations, probability, geometry, circles, and right triangles.

Why are the answers hidden in this walkthrough?

The reveal format lets you attempt each question first, then compare your reasoning with a complete worked solution.

Digital SAT · Math · Exam Prep · NYC Test Prep

Digital SAT Practice Test 7 — Module 2 Math: Full Walkthrough of All 27 Questions with Answers & Explanations

Every question on Digital SAT Practice Test #7, Math Module 2, transcribed verbatim, with the official College Board answer key, a step-by-step worked solution, a plain-English explanation for every choice, theory refreshers on every tested skill, and a free PDF download. Built by SOMATH, the math school on the Upper West Side of Manhattan.

2026-08-12 · By the SOMATH team · 226 W 79th St, UWS · (646) 668-6151

Looking for the answers, explanations, and full walkthrough of Digital SAT Practice Test 7 — Math Module 2? You are in the right place. This complete walkthrough includes every one of the 27 questions from SAT Test #7, the official answer for each item, and a plain-English worked solution that identifies the SAT Math skill in play.

Math Module 2 is the second Math module on the Digital SAT, and it is adaptive: the specific set of 27 questions depends on how you performed on Module 1. Practice each question under a focused 30-minute review block, then use the reveal to check both your setup and final answer.

Whether you are a student preparing for an upcoming SAT, a parent looking for answer explanations, or a teacher building a review packet, solve first and then open each answer to compare your reasoning with the worked steps.

Practice Test 7 PDFOpen the original College Board test alongside this walkthrough.

Download the PDF

Official answer key

The table below uses the College Board scoring key. Student-produced responses are shown in an accepted exact form.

Question #	Correct Answer
1	A
2	B
3	C
4	A
5	B
6	2850
7	11/4 or 2.75
8	C
9	C
10	D
11	D
12	D
13	4.41 or 441/100
14	153
15	C
16	A
17	A
18	D
19	B
20	120
21	1660
22	B
23	C
24	B
25	C
26	B
27	14

How to use this walkthrough: Work a question without opening the answer first. Then compare your setup—not only your final answer—with the solution. Module 2 includes multiple-choice and grid-in questions, so practice choosing and entering answers carefully.

Theory refresher

Use these compact rules to review the skills that appear in this module before returning to the question-by-question walkthrough.

Algebra

Use substitution or elimination in systems, keep equivalent operations balanced, and test every table row against all inequalities.

Advanced Math

For quadratics, identify vertex form or use the discriminant. When dividing powers with like bases, subtract exponents; a negative exponent moves its factor to the other side of a fraction.

Problem-Solving and Data Analysis

Use a unit rate for “per,” convert percentages to decimals before multiplying, and estimate a line’s slope as rise over run. Squared units require squared conversion factors.

Geometry and Trigonometry

Use complementary and supplementary angle relationships deliberately. For circles, A = πr²; for a right triangle, tan θ = opposite/adjacent and a² + b² = c².

Full walkthrough: all 27 questions

Each answer is hidden so that you can genuinely attempt the problem first.

Algebra

Questions 6, 8, 11, 16, 18, 22, and 23 use linear relationships, equations, inequalities, and systems.

Question 6 · Linear Functions

f(x) = 45x + 600
The function f gives the monthly fee f(x), in dollars, a facility charges to keep x crates in storage. What is the monthly fee, in dollars, the facility charges to keep 50 crates in storage?

Student-produced response — no answer choices; enter a value in the Bluebook grid.

Correct answer: 2850

Quick approach. Evaluate the fee function at 50 crates.

f(50) = 45(50) + 600
      = 2,250 + 600
      = 2,850

Intuition. The expression combines a per-crate charge of $45 with a fixed $600 fee.

Question 8 · Linear Equations in 1 Var

If 8x = 6, what is the value of 72x?

A) 3
B) 15
C) 54
D) 57

Correct answer: C) 54

Quick approach. Multiply both sides of the known equation by 9.

8x = 6
9(8x) = 9(6)
72x = 54

Intuition. Since 72 is nine times 8, the matching product on the other side must also be multiplied by nine.

Question 11 · Linear Equations in 2 Vars

At a state fair, attendees can win tokens that are worth a different number of points depending on the shape. One attendee won S square tokens and C circle tokens worth a total of 1,120 points. The equation 80S + 90C = 1,120 represents this situation. How many more points is a circle token worth than a square token?

A) 950
B) 90
C) 80
D) 10

Correct answer: D) 10

Quick approach. Read each coefficient as the point value of one token and compare them.

square token: 80 points
circle token: 90 points

difference = 90 − 80 = 10

Intuition. The variables count tokens; their coefficients are the points contributed by a single token.

Question 16 · Linear Inequalities

y < x
x < 22
For which of the following tables are all the values of x and their corresponding values of y solutions to the given system of inequalities?

A)
x y
19 18
20 19
21 20
B)
x y
19 20
20 21
21 22
C)
x y
23 22
24 23
25 24
D)
x y
23 24
24 25
25 26

x	y
19	20
20	21
21	22

x	y
23	22
24	23
25	24

x	y
23	24
24	25
25	26

Correct answer: A)

x	y
19	18
20	19
21	20

Quick approach. Check both inequalities against every row of each choice.

For choice A:
18 < 19 and 19 < 22
19 < 20 and 20 < 22
20 < 21 and 21 < 22

Every row works.

Intuition. A valid row must put y below x and also keep x below 22; only choice A does both throughout.

Question 18 · Linear Systems

3y = 4x + 17
−3y = 9x − 23
The solution to the given system of equations is (x, y). What is the value of 39x?

A) −18
B) −6
C) 6
D) 18

Correct answer: D) 18

Quick approach. Add the equations to eliminate y, then scale the resulting value of x.

3y = 4x + 17
−3y = 9x − 23
----------------
0 = 13x − 6
13x = 6
39x = 3(13x) = 18

Intuition. The opposite y-terms are designed to cancel, so there is no need to solve for y.

Question 22 · Linear Functions

x	f(x)
−4	0
−19/5	1
−18/5	2

For the linear function f, the table shows three values of x and their corresponding values of f(x). If h(x) = f(x) − 13, which equation defines h?

A) h(x) = 5x − 4
B) h(x) = 5x + 7
C) h(x) = 5x + 9
D) h(x) = 5x + 20

Correct answer: B) h(x) = 5x + 7

Quick approach. Find the linear rule for f from the table, then subtract 13.

slope = (1 − 0) / [−19/5 − (−4)] = 5
f(x) = 5x + 20

h(x) = f(x) − 13
     = (5x + 20) − 13
     = 5x + 7

Intuition. The transformation h(x) = f(x) − 13 shifts every output of f down by 13, leaving the slope unchanged.

Question 23 · Linear Functions

The linear function g is defined by g(x) = b − 15x, where b is a constant. If g(c + 7) = c/4, where c is a constant, which of the following expressions represents the value of b?

A) 15c/4
B) 19c/4 + 7
C) 61c/4 + 105
D) 15c + 105

Correct answer: C) 61c/4 + 105

Quick approach. Substitute c + 7 into g and solve the resulting equation for b.

c/4 = b − 15(c + 7)
c/4 = b − 15c − 105
b = c/4 + 15c + 105
b = 61c/4 + 105

Intuition. The given function value supplies both the input, c + 7, and output, c/4; substitute them in their proper places.

Advanced Math

Questions 3, 7, 9, 10, 15, 17, 19, 20, and 27 focus on nonlinear functions, equivalent expressions, and quadratics.

Question 3 · Nonlinear Functions

The graph of the quadratic function y = f(x) is shown. What is the vertex of the graph?

See figure in the original College Board PDF above.

A) (0, −2)
B) (0, −3)
C) (0, 2)
D) (0, 3)

Correct answer: C) (0, 2)

Quick approach. Read the parabola’s lowest point directly from the graph.

The vertex is the turning point of a parabola.

From the graph, the minimum occurs at:
(0, 2)

Intuition. Because this parabola opens upward, its vertex is its minimum, not simply any point where it crosses an axis.

Question 7 · Nonlinear Functions

The function f is defined by f(x) = 5(1/4 − x)² + 11/4. What is the value of f(1/4)?

Student-produced response — no answer choices; enter a value in the Bluebook grid.

Correct answer: 11/4 or 2.75

Quick approach. Substitute x = 1/4; the squared difference becomes zero.

f(1/4) = 5(1/4 − 1/4)² + 11/4
       = 5(0)² + 11/4
       = 11/4
       = 2.75

Intuition. In vertex form, the squared term vanishes at x = 1/4, leaving the vertical shift.

Question 9 · Equivalent Expressions

Which expression is equivalent to 23x³ + 2x² + 9x?

A) 23x(x² + 2x + 9)
B) 9x(23x³ + 2x² + 1)
C) x(23x² + 2x + 9)
D) 34(x³ + x² + x)

Correct answer: C) x(23x² + 2x + 9)

Quick approach. Factor out the common factor x from every term.

23x³ + 2x² + 9x
= x(23x² + 2x + 9)

Intuition. Each term contains one x, but no larger numerical factor is shared by all three terms.

Question 10 · Equivalent Expressions

Which expression is equivalent to (9x³ + 5x + 7) + (6x³ + 5x² − 5)?

A) 15x⁶ + 5x² − 5x − 35
B) 15x³ + 10x² + 2
C) 15x⁶ + 5x² + 5x + 2
D) 15x³ + 5x² + 5x + 2

Correct answer: D) 15x³ + 5x² + 5x + 2

Quick approach. Combine only like terms after removing the parentheses.

(9x³ + 5x + 7) + (6x³ + 5x² − 5)
= 15x³ + 5x² + 5x + 2

Intuition. Adding polynomials changes coefficients, not exponents; x³ terms combine with x³ terms only.

Question 15 · Nonlinear Functions

An investment account was opened with an initial value of $890. The value of the account doubled every 10 years. Which equation represents the value of the account M(t), in dollars, t years after the account was opened?

A) M(t) = 890(1/2)^t/10
B) M(t) = 890(1/10)^t/2
C) M(t) = 890(2)^t/10
D) M(t) = 890(10)^t/2

Correct answer: C) M(t) = 890(2)^t/10

Quick approach. Use 890 as the initial value, factor 2 for doubling, and t/10 for the number of decades.

number of 10-year periods = t/10

M(t) = 890(2)^t/10

Intuition. An exponent of t/10 makes the exponent increase by one exactly every 10 years, which doubles the account each time.

Question 17 · Equivalent Expressions

Which expression is equivalent to h¹⁵q⁷ / h⁵q²¹, where h > 0 and q > 0?

A) h¹⁰/q¹⁴
B) h³/q³
C) h¹⁰q¹⁴
D) h³q³

Correct answer: A) h¹⁰/q¹⁴

Quick approach. Subtract exponents when dividing like bases, then rewrite the negative exponent.

h¹⁵q⁷ / (h⁵q²¹)
= h¹⁵⁻⁵q⁷⁻²¹
= h¹⁰q⁻¹⁴
= h¹⁰ / q¹⁴

Intuition. A negative exponent moves its factor across the fraction bar; h stays in the numerator because its exponent is positive.

Question 19 · Nonlinear Functions

h(t) = −16t² + b
The function h estimates an object’s height, in feet, above the ground t seconds after the object is dropped, where b is a constant. The function estimates that the object is 3,364 feet above the ground when it is dropped at t = 0. Approximately how many seconds after being dropped does the function estimate the object will hit the ground?

A) 7.25
B) 14.50
C) 105.13
D) 210.25

Correct answer: B) 14.50

Quick approach. Use the starting height to find b, set height to zero, and take the positive square root.

h(0) = b = 3,364
0 = −16t² + 3,364
16t² = 3,364
t² = 210.25
t = √210.25 = 14.50

Intuition. Time after a drop is positive, so the negative square-root solution is not meaningful.

Question 20 · Nonlinear Equations in 1 Var

2x² − 8x − 7 = 0
One solution to the given equation can be written as (8 − √k)/4, where k is a constant. What is the value of k?

Student-produced response — no answer choices; enter a value in the Bluebook grid.

Correct answer: 120

Quick approach. Use the quadratic formula and match the discriminant under the radical.

a = 2, b = −8, c = −7

x = [8 ± √((−8)² − 4(2)(−7))] / 4
  = [8 ± √(64 + 56)] / 4
  = [8 ± √120] / 4

k = 120

Intuition. The k is the discriminant after simplification, because that is exactly the expression inside the square root.

Question 27 · Nonlinear Equations in 1 Var

y = −2.5
y = x² + 8x + k
In the given system of equations, k is a positive integer constant. The system has no real solutions. What is the least possible value of k?

Student-produced response — no answer choices; enter a value in the Bluebook grid.

Correct answer: 14

Quick approach. Set the equations equal and require the resulting quadratic to have a negative discriminant.

−2.5 = x² + 8x + k
0 = x² + 8x + k + 2.5

discriminant < 0:
8² − 4(1)(k + 2.5) < 0
64 − 4k − 10 < 0
54 < 4k
13.5 < k

least positive integer k = 14

Intuition. No real intersection means the horizontal line never meets the parabola, which corresponds to a negative discriminant.

Problem-Solving and Data Analysis

Questions 1, 2, 4, 5, 12, and 26 ask you to reason with quantities, percentages, probability, models, and units.

Question 1 · Ratios/Rates/Proportions

A total of 165 people contributed to a charity event as either a donor or a volunteer. 130 people contributed as a donor. How many people contributed as a volunteer?

A) 35
B) 130
C) 165
D) 330

Correct answer: A) 35

Quick approach. Subtract the donor count from the total number of contributors.

total = donors + volunteers
165 = 130 + volunteers
volunteers = 165 − 130
volunteers = 35

Intuition. The total has exactly two nonoverlapping groups, so the amount not accounted for by donors must be volunteers.

Question 2 · Percentages

There are 250 trees in a park. Of these trees, 6% are birch trees. How many birch trees are in the park?

A) 6
B) 15
C) 75
D) 244

Correct answer: B) 15

Quick approach. Convert 6% to 0.06 and multiply by the total.

birch trees = 6% of 250
             = 0.06(250)
             = 15

Intuition. A percent is a part out of 100; 6% of 250 is six hundredths of the park’s trees.

Question 4 · Ratios/Rates/Proportions

The number of raccoons in a 131-square-mile area is estimated to be 2,358. What is the estimated population density, in raccoons per square mile, of this area?

A) 18
B) 131
C) 149
D) 2,376

Correct answer: A) 18

Quick approach. Divide the number of raccoons by the number of square miles.

population density = raccoons / square miles
                   = 2,358 / 131
                   = 18 raccoons per square mile

Intuition. “Per square mile” calls for a unit rate, so square miles belong in the denominator.

Question 5 · Probability

−11, −9, 26
A data set of three numbers is shown. If a number from this data set is selected at random, what is the probability of selecting a positive number?

A) 0
B) 1/3
C) 2/3
D) 1

Correct answer: B) 1/3

Quick approach. Count the positive entries and divide by all possible selections.

positive values: 26
total values: 3

P(positive) = 1 / 3

Intuition. Only 26 is greater than zero; the two negative values do not qualify.

Question 12 · Two-Variable Data

In the given scatterplot, a line of best fit for the data is shown. Which of the following is closest to the slope of the line of best fit shown?

See figure in the original College Board PDF above.

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

Correct answer: D) 2

Quick approach. Use two convenient approximate points on the trend line to estimate rise over run.

Approximate points: (1, 3.3) and (7, 14.5)

slope ≈ (14.5 − 3.3) / (7 − 1)
      = 11.2 / 6
      ≈ 1.87

Closest choice: 2

Intuition. A best-fit line need not pass exactly through data points, so an estimate from visible points is enough.

Question 26 · Ratios/Rates/Proportions

The speed of a vehicle is increasing at a rate of 7.3 meters per second squared. What is this rate, in miles per minute squared, rounded to the nearest tenth? (Use 1 mile = 1,609 meters.)

A) 0.3
B) 16.3
C) 195.8
D) 220.4

Correct answer: B) 16.3

Quick approach. Convert meters to miles and seconds squared to minutes squared.

7.3 m/s² × (1 mile / 1,609 m) × (60² s² / 1 min²)
= (7.3 × 3,600) / 1,609 mi/min²
≈ 16.3 mi/min²

Intuition. Because the time unit is squared, changing seconds to minutes requires multiplying by 60², not merely 60.

Geometry and Trigonometry

Questions 13, 14, 21, 24, and 25 use circle formulas, angle relationships, trig ratios, and geometric structure.

Question 13 · Area & Volume

A circle has a radius of 2.1 inches. The area of the circle is bπ square inches, where b is a constant. What is the value of b?

Student-produced response — no answer choices; enter a value in the Bluebook grid.

Correct answer: 4.41 or 441/100

Quick approach. Apply the circle-area formula and isolate the coefficient of π.

A = πr²
bπ = π(2.1)²
b = (2.1)²
b = 4.41

Intuition. The π appears on both sides, so b is simply the radius squared.

Question 14 · Lines/Angles/Triangles

In triangle XYZ, angle Y is a right angle, point P lies on XZ, and point Q lies on YZ such that PQ is parallel to XY. If the measure of angle XZY is 63°, what is the measure, in degrees, of angle XPQ?

Student-produced response — no answer choices; enter a value in the Bluebook grid.

Correct answer: 153

Quick approach. Use complementary angles in the right triangle, then a supplementary pair at P.

∠X = 180° − 90° − 63° = 27°

Because PQ ∥ XY, ∠ZPQ = 27°.
∠XPQ = 180° − 27°
∠XPQ = 153°

Intuition. The smaller triangle has matching angles because its horizontal side is parallel to XY; the requested angle is the straight-line supplement.

Question 21 · Lines/Angles/Triangles

A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of the acute angles is (9x − 560)°. The sum of the measures of one of the acute angles and three of the obtuse angles is (−18x + w)°. What is the value of w?

Student-produced response — no answer choices; enter a value in the Bluebook grid.

Correct answer: 1660

Quick approach. Express an obtuse angle as the supplement of the acute angle, then match the given sum.

acute = 9x − 560
obtuse = 180 − (9x − 560) = −9x + 740

acute + 3(obtuse)
= (9x − 560) + 3(−9x + 740)
= −18x + 1,660

Therefore w = 1,660.

Intuition. With parallel lines, all acute angles match and all obtuse angles match; adjacent acute and obtuse angles total 180°.

Question 24 · Right Triangles & Trig

In triangle XYZ, angle Z is a right angle and the length of YZ is 24 units. If tan X = 12/35, what is the perimeter, in units, of triangle XYZ?

A) 188
B) 168
C) 84
D) 71

Correct answer: B) 168

Quick approach. Use tangent to find the other leg, then the Pythagorean theorem for the hypotenuse.

tan X = opposite / adjacent = YZ / XZ
12/35 = 24 / XZ
XZ = 70

XY = √(24² + 70²) = 74
perimeter = 24 + 70 + 74 = 168

Intuition. The ratio 12:35 scales to 24:70, creating the familiar 24-70-74 right triangle.

Question 25 · Circles

x² + 14x + y² = 6y + 109
In the xy-plane, the graph of the given equation is a circle. What is the length of the circle’s radius?

A) √109
B) √149
C) √167
D) √341

Correct answer: C) √167

Quick approach. Complete the square in x and y to put the equation in circle form.

x² + 14x + y² − 6y = 109
(x + 7)² + (y − 3)² = 109 + 49 + 9
(x + 7)² + (y − 3)² = 167

r = √167

Intuition. In standard circle form, the constant on the right is r², so its positive square root is the radius.

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Practice Test 7 Math Module 2 FAQ

How many questions are in Digital SAT Practice Test 7 Math Module 2?

This module has 27 Math questions, including multiple-choice and student-produced response questions.

Which questions are student-produced responses in this module?

Questions 6, 7, 13, 14, 20, 21, and 27 require you to enter your own answer in the Bluebook grid.

What skills appear in this module?

You will see algebra, advanced math, problem-solving and data analysis, and geometry and trigonometry.

Can I use Desmos on SAT Math Module 2?

Yes. The digital SAT provides the embedded Desmos graphing calculator throughout Math, although many questions here are quickest with direct algebra or arithmetic.

Why are the answers hidden?

Use the reveal only after making a real attempt. Comparing your setup with a full solution is more useful than simply checking a final answer.

Digital SAT Practice Test 7 — Module 2 Math: Full Walkthrough of All 27 Questions with Answers & Explanations

Official answer key

Theory refresher

Algebra

Advanced Math

Problem-Solving and Data Analysis

Geometry and Trigonometry

Full walkthrough: all 27 questions

Algebra

Question 6 · Linear Functions

Question 8 · Linear Equations in 1 Var

Question 11 · Linear Equations in 2 Vars

Question 16 · Linear Inequalities

Question 18 · Linear Systems

Question 22 · Linear Functions

Question 23 · Linear Functions

Advanced Math

Question 3 · Nonlinear Functions

Question 7 · Nonlinear Functions

Question 9 · Equivalent Expressions

Question 10 · Equivalent Expressions

Question 15 · Nonlinear Functions

Question 17 · Equivalent Expressions

Question 19 · Nonlinear Functions

Question 20 · Nonlinear Equations in 1 Var

Question 27 · Nonlinear Equations in 1 Var

Problem-Solving and Data Analysis

Question 1 · Ratios/Rates/Proportions

Question 2 · Percentages

Question 4 · Ratios/Rates/Proportions

Question 5 · Probability

Question 12 · Two-Variable Data

Question 26 · Ratios/Rates/Proportions

Geometry and Trigonometry

Question 13 · Area & Volume

Question 14 · Lines/Angles/Triangles

Question 21 · Lines/Angles/Triangles

Question 24 · Right Triangles & Trig

Question 25 · Circles

Preparing for the Digital SAT at SOMATH

Turn practice-test review into a score plan.

Practice Test 7 Math Module 2 FAQ

How many questions are in Digital SAT Practice Test 7 Math Module 2?

Which questions are student-produced responses in this module?

What skills appear in this module?

Can I use Desmos on SAT Math Module 2?

Why are the answers hidden?

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