Digital SAT · Math · Exam Prep · NYC Test Prep
Digital SAT Practice Test 6 — Module 2 Math: Full Walkthrough of All 27 Questions with Answers & Explanations
Every question on Digital SAT Practice Test #6, 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.
Looking for the answers, explanations, and full walkthrough of Digital SAT Practice Test 6 — Math Module 2? You are in the right place. This is a complete, question-by-question walkthrough of SAT Test #6, Math Module 2 — every one of the 27 questions transcribed verbatim, with the official College Board answer for each item and a plain-English worked solution showing exactly which SAT-Math tool each question wants and how to apply it. Theory refreshers on every skill category are included, plus a free PDF download of the entire test for offline studying.
Module 2 is the second Math module on the Digital SAT, and it is adaptive: the specific set of 27 questions you see depends on how you performed on Module 1. Students who do well on Module 1 are routed into a harder Module 2 with access to the full 200–800 scoring range; students who do not are routed into an easier Module 2. The official Practice Test 6 form reproduced here is an excellent way to practice making accurate decisions under the second-module time limit.
Whether you are a student preparing for an upcoming SAT, a parent looking for answer explanations, or a teacher building a review packet, use this walkthrough actively: solve first, then open each answer to compare your setup and arithmetic with the full solution.
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 | B |
| 2 | A |
| 3 | B |
| 4 | A |
| 5 | B |
| 6 | 6 |
| 7 | 10 |
| 8 | B |
| 9 | A |
| 10 | A |
| 11 | D |
| 12 | C |
| 13 | 774 |
| 14 | 5 |
| 15 | B |
| 16 | A |
| 17 | D |
| 18 | B |
| 19 | A |
| 20 | 7/24 |
| 21 | 1677 |
| 22 | B |
| 23 | A |
| 24 | D |
| 25 | A |
| 26 | A |
| 27 | −28 |
Theory refresher
Use these compact rules to review the skills that appear in this module before you return to the question-by-question walkthrough.
Linear Equations in 1 Var
Undo operations in reverse order, or use equivalent equations. Dividing both sides of an equation by the same nonzero number preserves its solution.
Linear Equations in 2 Vars
A linear relationship has a constant rate of change. From a table, find the slope first, then use y = mx + b to find the intercept.
Linear Systems
Substitution is efficient when one variable is already isolated; elimination is efficient when terms cancel after adding or subtracting. Parallel distinct lines have no solution.
Linear Functions
In y = mx + b, m is the rate of change and b is the value when x = 0. Perpendicular nonvertical lines have slopes that are negative reciprocals.
Ratios/Rates/Proportions
Ratio parts are equal-sized pieces. If a quantity represents 6 parts, divide by 6 to find 1 part; scale up only after identifying the requested number of parts.
Percentages
Convert a percent to a decimal before multiplying: 20% = 0.20. “Of” signals multiplication.
Probability
For equally likely outcomes, P(event) = favorable outcomes / total outcomes. Keep the denominator as the total possible selections.
One-Variable Data
To find a median, order the values. With an even number of values, average the two central values.
Two-Variable Data
If every y-coordinate is multiplied by a constant, both the slope and y-intercept of a line of best fit are multiplied by that constant.
Equivalent Expressions
Use exponent rules to rewrite radicals and powers, and factor only terms shared by every term in an expression.
Nonlinear Equations in 1 Var
For a factored equation equal to zero, set each factor equal to zero. In a geometric context, reject roots that do not make sense, such as a negative width.
Nonlinear Functions
Exponential expressions can be rewritten by applying exponent rules. For radical functions, use a known zero and a sign condition to constrain the constants.
Area & Volume
A cylinder’s volume is V = Bh, where B is the base area. Solve for the missing quantity using the units provided.
Lines/Angles/Triangles
Vertical angles are congruent. Mark this relationship immediately rather than adding to 180°; supplementary angles are a different relationship.
Right Triangles & Trig
For a right triangle, a² + b² = c², with c opposite the right angle. Square radical side lengths carefully before simplifying.
Circles
Every point on a circle is one radius from the center. Two radii with a 90° included angle form a right isosceles triangle, so use the Pythagorean theorem for the chord.
Full walkthrough: all 27 questions
Each answer is hidden so that you can genuinely attempt the problem first.
Question 1 · Linear Equations in 1 Var
The function f is defined by f(x) = 8x. For what value of x does f(x) = 72?
- A) 8
- B) 9
- C) 64
- D) 80
Correct answer: B) 9
Quick approach. Replace f(x) with 8x, then divide by 8.
f(x) = 72 8x = 72 x = 72 ÷ 8 x = 9
Question 2 · Lines/Angles/Triangles
In the figure, two lines intersect at a point. Angle 1 and angle 2 are vertical angles. The measure of angle 1 is 72°. What is the measure of angle 2?
See figure in the original College Board PDF above.
- A) 72°
- B) 108°
- C) 144°
- D) 288°
Correct answer: A) 72°
Quick approach. Vertical angles have equal measures.
Vertical angles are congruent. m∠2 = m∠1 m∠2 = 72°
Question 3 · Probability
On a street with 7 houses, 2 houses are blue. If a house from this street is selected at random, what is the probability of selecting a house that is blue?
- A) 1/7
- B) 2/7
- C) 5/7
- D) 7/7
Correct answer: B) 2/7
Quick approach. Use blue houses as the favorable outcomes and all houses as the total.
probability = favorable outcomes / total outcomes
= 2 / 7
= 2/7
Question 4 · Linear Functions
The graph of function f is shown, where y = f(x). Which of the following describes function f?
See figure in the original College Board PDF above.
- A) Increasing linear
- B) Decreasing linear
- C) Increasing exponential
- D) Decreasing exponential
Correct answer: A) Increasing linear
Quick approach. Classify the graph by its shape and its left-to-right direction.
The graph is a straight line, so it is linear. It rises from left to right, so it is increasing. Therefore, f is increasing linear.
Question 5 · Linear Functions
The graph of the function f is shown, where y = f(x). What is the y-intercept of the graph?
See figure in the original College Board PDF above.
- A) (0, −1)
- B) (0, −4)
- C) (0, 1)
- D) (0, 4)
Correct answer: B) (0, −4)
Quick approach. Read the point where the line crosses the y-axis.
At the y-intercept, x = 0. From the graph, the line crosses the y-axis at y = −4. y-intercept = (0, −4)
Question 6 · Linear Systems
The solution to the given system of equations is (x, y). What is the value of y?
x = 8
x + 3y = 26
Student-produced response — no answer choices; enter a value in the Bluebook grid.
Correct answer: 6
Quick approach. Substitute the known value of x into the second equation.
x + 3y = 26 8 + 3y = 26 3y = 18 y = 6
Question 7 · Percentages
The amount of Hanna’s bill for a food order was $50. Hanna gave a tip of 20% of the amount of the bill. What is the amount, in dollars, of the tip Hanna gave?
Student-produced response — no answer choices; enter a value in the Bluebook grid.
Correct answer: 10
Quick approach. Convert 20% to 0.20 and multiply by the bill.
20% of $50 = 0.20 × 50
= 10
Question 8 · Equivalent Expressions
Which expression is equivalent to 5x⁵ − 6x⁴ + 8x³?
- A) x⁴(5x − 6)
- B) x³(5x² − 6x + 8)
- C) 8x³(5x² − 6x + 1)
- D) 6x⁵(−6x⁴ + 8x³ + 1)
Correct answer: B) x³(5x² − 6x + 8)
Quick approach. Factor out the greatest common factor, x³.
5x⁵ − 6x⁴ + 8x³ = x³(5x²) − x³(6x) + x³(8) = x³(5x² − 6x + 8)
Question 9 · Ratios/Rates/Proportions
The ratio of the length of line segment XY to the length of line segment ZV is 6 to 1. If the length of line segment XY is 102 inches, what is the length, in inches, of line segment ZV?
- A) 17
- B) 96
- C) 102
- D) 612
Correct answer: A) 17
Quick approach. The 102-inch segment represents 6 equal ratio parts.
XY : ZV = 6 : 1 102 : ZV = 6 : 1 ZV = 102 ÷ 6 ZV = 17
Question 10 · Linear Equations in 1 Var
7(2x − 3) = 63
Which equation has the same solution as the given equation?
- A) 2x − 3 = 9
- B) 2x − 3 = 56
- C) 2x − 21 = 63
- D) 2x − 21 = 70
Correct answer: A) 2x − 3 = 9
Quick approach. Divide both sides by the common factor 7.
7(2x − 3) = 63 Divide both sides by 7: 2x − 3 = 9
Question 11 · Linear Functions
The function f defined by f(t) = 14t + 9 gives the estimated length, in inches, of a vine plant t months after Tavon purchased it. Which of the following is the best interpretation of 9 in this context?
- A) Tavon will keep the vine plant for 9 months.
- B) The vine plant is expected to grow 9 inches each month.
- C) The vine plant is expected to grow to a maximum length of 9 inches.
- D) The estimated length of the vine plant was 9 inches when Tavon purchased it.
Correct answer: D) The estimated length of the vine plant was 9 inches when Tavon purchased it.
Quick approach. Evaluate at t = 0 to interpret the constant term.
f(t) = 14t + 9 At purchase, t = 0: f(0) = 14(0) + 9 f(0) = 9 So 9 is the starting length in inches.
Question 12 · Nonlinear Equations in 1 Var
(x + 2)(x − 5)(x + 9) = 0
What is a positive solution to the given equation?
- A) 3
- B) 4
- C) 5
- D) 18
Correct answer: C) 5
Quick approach. Set each factor equal to zero and keep the positive root.
(x + 2)(x − 5)(x + 9) = 0 x + 2 = 0 → x = −2 x − 5 = 0 → x = 5 x + 9 = 0 → x = −9 The positive solution is 5.
Question 13 · Ratios/Rates/Proportions
Brian saves 2/5 of the $215 he earns each week from his job. If Brian continues to save at this rate, how much money, in dollars, will Brian save in 9 weeks?
Student-produced response — no answer choices; enter a value in the Bluebook grid.
Correct answer: 774
Quick approach. First find the weekly savings, then multiply by 9 weeks.
weekly savings = (2/5)(215)
= 86
9-week savings = 9(86)
= 774
Question 14 · Nonlinear Equations in 1 Var
A rectangle has an area of 155 square inches. The length of the rectangle is 4 inches less than 7 times the width of the rectangle. What is the width of the rectangle, in inches?
Student-produced response — no answer choices; enter a value in the Bluebook grid.
Correct answer: 5
Quick approach. Translate the length statement, then use the area formula and reject the negative root.
Let w = width. Then length = 7w − 4. area = length × width 155 = (7w − 4)w 155 = 7w² − 4w 0 = 7w² − 4w − 155 0 = (7w + 31)(w − 5) w = −31/7 or w = 5 A width must be positive, so w = 5.
Question 15 · One-Variable Data
4, 10, 18, 4, 4, 5, 6, 5
What is the median of the data set shown?
- A) 4
- B) 5
- C) 7
- D) 14
Correct answer: B) 5
Quick approach. Order the data; the two middle values are both 5.
Order the values:
4, 4, 4, 5, 5, 6, 10, 18
There are 8 values, so use the middle two:
median = (5 + 5) / 2
= 5
Question 16 · Area & Volume
A right circular cylinder has a volume of 432 cubic centimeters. The area of the base of the cylinder is 24 square centimeters. What is the height, in centimeters, of the cylinder?
- A) 18
- B) 24
- C) 216
- D) 10,368
Correct answer: A) 18
Quick approach. Use volume = base area × height.
V = Bh 432 = 24h h = 432 ÷ 24 h = 18
Question 17 · Nonlinear Equations in 1 Var
x² = −841
How many distinct real solutions does the given equation have?
- A) Exactly one
- B) Exactly two
- C) Infinitely many
- D) Zero
Correct answer: D) Zero
Quick approach. A real number squared cannot be negative.
For every real number x: x² ≥ 0 But the equation requires x² = −841, which is negative. This is impossible for real x. Number of distinct real solutions = 0
Question 18 · Linear Functions
Line k is defined by y = 7x + 1/8. Line j is perpendicular to line k in the xy-plane. What is the slope of line j?
- A) −8
- B) −1/7
- C) 1/8
- D) 7
Correct answer: B) −1/7
Quick approach. Take the negative reciprocal of the slope 7.
slope of line k = 7 Perpendicular slopes are negative reciprocals. slope of line j = −1/7
Question 19 · Linear Equations in 2 Vars
The table shows the linear relationship between the number of cars, c, on a commuter train and the maximum number of passengers and crew, p, that the train can carry. Which equation represents the linear relationship between c and p?
| Number of cars | Maximum number of passengers and crew |
|---|---|
| 3 | 174 |
| 5 | 284 |
| 10 | 559 |
- A) 55c − p = −9
- B) 55c − p = 9
- C) 55p − c = −9
- D) 55p − c = 9
Correct answer: A) 55c − p = −9
Quick approach. Find the slope from two rows, then solve for the intercept.
slope = (284 − 174) / (5 − 3)
= 110 / 2
= 55
p = 55c + b
174 = 55(3) + b
174 = 165 + b
b = 9
p = 55c + 9
55c − p = −9
Question 20 · Equivalent Expressions
If 48c = (∛4)7, what is the value of c?
Student-produced response — no answer choices; enter a value in the Bluebook grid.
Correct answer: 7/24 (or .2916 or .2917)
Quick approach. Rewrite the cube root as an exponent, then equate exponents.
4⁸ᶜ = (4¹ᐟ³)⁷ 4⁸ᶜ = 4⁷ᐟ³ 8c = 7/3 c = (7/3) ÷ 8 c = 7/24 c ≈ 0.2917
Question 21 · Linear Systems
(x − 2) − 4(y + 7) = 117
(x − 2) + 4(y + 7) = 442
The solution to the given system of equations is (x, y). What is the value of 6(x − 2)?
Student-produced response — no answer choices; enter a value in the Bluebook grid.
Correct answer: 1677
Quick approach. Add the equations so the y-expression cancels, then multiply the result requested.
(x − 2) − 4(y + 7) = 117
(x − 2) + 4(y + 7) = 442
--------------------------------
2(x − 2) = 559
x − 2 = 559/2
6(x − 2) = 6(559/2)
= 3(559)
= 1677
Question 22 · Right Triangles & Trig
In triangle ABC, angle B is a right angle. The length of side AB is 10√37 and the length of side BC is 24√37. What is the length of side AC?
- A) 14√37
- B) 26√37
- C) 34√37
- D) √(34 · 37)
Correct answer: B) 26√37
Quick approach. Use the Pythagorean theorem because AC is opposite the right angle.
AC² = AB² + BC² AC² = (10√37)² + (24√37)² AC² = 100(37) + 576(37) AC² = 676(37) AC = √(676 · 37) AC = 26√37
Question 23 · Nonlinear Functions
The function f is defined by the given equation. f(x) = (1.84)x/4. The equation can be rewritten as f(x) = (1 + p/100)x, where p is a constant. Which of the following is closest to the value of p?
- A) 16
- B) 21
- C) 46
- D) 96
Correct answer: A) 16
Quick approach. Rewrite the exponent so the growth factor is raised to x, then convert that factor to a percent.
(1.84)ˣᐟ⁴ = [(1.84)¹ᐟ⁴]ˣ
So 1 + p/100 = (1.84)¹ᐟ⁴
≈ 1.1646
p/100 ≈ 0.1646
p ≈ 16.46
The closest choice is 16.
Question 24 · Nonlinear Functions
The function f is defined by f(x) = a√(x + b), where a and b are constants. In the xy-plane, the graph of y = f(x) passes through the point (−24, 0), and f(24) < 0. Which of the following must be true?
- A) f(0) = 24
- B) f(0) = −24
- C) a > b
- D) a < b
Correct answer: D) a < b
Quick approach. The zero at x = −24 fixes b, and the negative output at x = 24 fixes the sign of a.
f(−24) = 0 a√(−24 + b) = 0 Because f(24) < 0, a cannot be 0. So √(b − 24) = 0, giving b = 24. f(24) = a√(48) < 0 Since √48 is positive, a < 0. Thus a < 0 < 24 = b, so a < b.
Question 25 · Circles
In the xy-plane, a circle has center C with coordinates (h, k). Points A and B lie on the circle. Point A has coordinates (h + 1, k + √102), and ∠ACB is a right angle. What is the length of AB?
- A) √206
- B) 2√102
- C) 103√2
- D) 103√3
Correct answer: A) √206
Quick approach. First find the radius from the center to A; then use the right isosceles triangle formed by the two radii.
CA² = [(h + 1) − h]² + [(k + √102) − k]² CA² = 1² + (√102)² CA² = 103 CA = CB = √103, and ∠ACB = 90°. AB² = (√103)² + (√103)² AB² = 206 AB = √206
Question 26 · Two-Variable Data
The scatterplot shows the relationship between two variables, x and y, for data set E. A line of best fit is shown. Data set F is created by multiplying the y-coordinate of each data point from data set E by 3.9. Which of the following could be an equation of a line of best fit for data set F?
See figure in the original College Board PDF above.
- A) y = 46.8 + 5.9x
- B) y = 46.8 + 1.5x
- C) y = 12 + 5.9x
- D) y = 12 + 1.5x
Correct answer: A) y = 46.8 + 5.9x
Quick approach. Multiplying every y-value by 3.9 multiplies both the intercept and slope by 3.9.
For data set E, the displayed line of best fit is approximately: y = 12 + 1.5x Multiply every y-value by 3.9: y = 3.9(12 + 1.5x) y = 46.8 + 5.85x y ≈ 46.8 + 5.9x
Question 27 · Linear Systems
48x − 64y = 48y + 24
ry = 1/8 − 12x
In the given system of equations, r is a constant. If the system has no solution, what is the value of r?
Student-produced response — no answer choices; enter a value in the Bluebook grid.
Correct answer: −28
Quick approach. Put both equations in slope-intercept form; parallel distinct lines have no solution.
48x − 64y = 48y + 24 48x − 112y = 24 6x − 14y = 3 y = (3/7)x − 3/14 ry = 1/8 − 12x y = (−12/r)x + 1/(8r) No solution means equal slopes, different intercepts: −12/r = 3/7 −84 = 3r r = −28
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Practice Test 6 Math Module 2 FAQ
How many questions are in Digital SAT Practice Test 6 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 linear equations and systems, functions, ratios and percentages, data, nonlinear equations, probability, geometry, circles, and right triangles.
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.