Digital SAT · Math · Exam Prep · NYC Test Prep
Digital SAT Practice Test 8 — Module 2 Math: Full Walkthrough of All 27 Questions with Answers & Explanations
Every question on Digital SAT Practice Test #8, Math Module 2, transcribed verbatim, with the official College Board answer key, a step-by-step worked solution, a plain-English explanation for every item, theory refreshers on every tested skill, and links to the official PDF. 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 8 — Math Module 2? You are in the right place. This is a complete, question-by-question walkthrough of SAT Test #8, Math Module 2 — every one of the 27 questions transcribed, with the official College Board answer for each item and a worked solution showing exactly which SAT Math tool the question wants and how to apply it.
Module 2 is the adaptive module: students who did well on Module 1 receive this harder form. Expect denser word problems, more algebra, and quadratics that require the discriminant. Attempt each question first with a strict timer, then reveal the answer to compare setups.
Whether you are a student prepping for the next SAT, a parent looking for answer explanations, or a teacher building a review packet, this walkthrough is designed to be a clear, student-friendly study resource. It is written by the team that teaches Digital SAT prep at SOMATH, a math-focused school on the Upper West Side of New York City. Book a free 30-minute in-person diagnostic evaluation at 226 W 79th Street or call (646) 668-6151.
Official answer key — SAT Practice Test 8, Math Module 2
| Question # | Correct Answer |
|---|---|
| 1 | C |
| 2 | C |
| 3 | D |
| 4 | C |
| 5 | D |
| 6 | 9 |
| 7 | 68 |
| 8 | B |
| 9 | D |
| 10 | D |
| 11 | C |
| 12 | B |
| 13 | 986 |
| 14 | 24 |
| 15 | D |
| 16 | D |
| 17 | A |
| 18 | B |
| 19 | A |
| 20 | 46 |
| 21 | 1.8; 9/5 |
| 22 | A |
| 23 | B |
| 24 | C |
| 25 | B |
| 26 | D |
| 27 | 168 |
Theory refresher: every skill tested in this module
Algebra
Isolate variables one step at a time, use slope-intercept form, and solve systems by substitution or elimination. In word problems, always translate the sentence into a clean equation before you compute.
Advanced Math
Factor quadratics as products of binomials, apply the zero-product property to read off roots, use vertex form y = a(x − h)2 + k, and remember that a quadratic ax2 + bx + c = 0 has exactly one real solution when the discriminant b2 − 4ac = 0.
Problem-Solving & Data Analysis
The median of an ordered list is the middle value; the mean adds and divides. Percentages of a number are multiplications, and unit conversions must cancel units end-to-end.
Geometry & Trigonometry
Similar triangles have proportional sides; area scales as the square of the linear factor. Circles in the xy-plane satisfy (x − h)2 + (y − k)2 = r2, with center (h, k) and radius r.
Questions 1–7: Graphs, similar triangles, factoring, systems, and rate problems
Warm-up questions on reading graphs, similar-figure ratios, factoring quadratics, and translating scenarios into equations.
Question 1 · Algebra · Graphs in the xy-plane
What is the y-intercept of the line graphed?
See figure in the original College Board PDF above.
- A) (−5, 0)
- B) (0, 0)
- C) (0, 5)
- D) (0, 9)
Answer: C) (0, 5)
Key idea. The y-intercept is the point where the line crosses the y-axis (x = 0).
Read the graph: the line crosses the y-axis at y = 5. y-intercept = (0, 5)Why this works. An x-intercept has y = 0; a y-intercept has x = 0. Reading directly from the graph gives (0, 5).
Question 2 · Problem-Solving & Data Analysis · Reading tables
For a certain region, the table shows the average number of store employees in 2016 by type of store. Based on the table, how much greater was the average number of store employees in warehouse stores than in supermarkets?
See figure in the original College Board PDF above.
- A) 83
- B) 152
- C) 235
- D) 495
Answer: C) 235
Key idea. Subtract the supermarket value from the warehouse-store value.
Warehouse store: 365 Supermarket: 130 Difference: 365 − 130 = 235Why this works. The table gives 365 and 130 for the two categories; their difference is 235.
Question 3 · Geometry · Parallel lines cut by a transversal
In the figure, line m is parallel to line n, and line t intersects both lines. What is the value of x?
See figure in the original College Board PDF above.
- A) 33
- B) 57
- C) 123
- D) 147
Answer: D) 147
Key idea. With m ∥ n, co-interior (same-side interior) angles are supplementary.
The 33° angle at line n and the x° angle at line m are on the same side of the transversal and inside the parallel lines. x + 33 = 180 x = 147Why this works. Same-side interior angles created by a transversal cutting parallel lines sum to 180°.
Question 4 · Algebra · Linear equations in two variables
Sean rents a tent at a cost of $11 per day plus a onetime insurance fee of $10. Which equation represents the total cost c, in dollars, to rent the tent with insurance for d days?
- A) c = 11(d + 10)
- B) c = 10(d + 11)
- C) c = 11d + 10
- D) c = 10d + 11
Answer: C) c = 11d + 10
Key idea. Total cost = (rate per day)(days) + (one-time fee).
Per-day rental cost: 11d One-time insurance fee: 10 Total: c = 11d + 10Why this works. The daily rate multiplies the number of days; the one-time fee is a fixed constant added to the running total.
Question 5 · Geometry · Pythagorean theorem
For the right triangle shown, a = 4 and b = 5. Which expression represents the value of c?
See figure in the original College Board PDF above.
- A) 4 + 5
- B) √((4)(5))
- C) √(4 + 5)
- D) √(42 + 52)
Answer: D) √(4² + 5²)
Key idea. In a right triangle with legs a and b, the hypotenuse is c = √(a² + b²).
Pythagorean theorem: a² + b² = c² c = √(a² + b²) = √(4² + 5²)Why this works. The hypotenuse equals the square root of the sum of the squared legs, not the sum of the legs.
Question 6 · Algebra · Solving a linear equation
The function g is defined by g(x) = 6x. For what value of x is g(x) = 54?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 9
Key idea. Set 6x = 54 and solve for x.
g(x) = 6x = 54 x = 54 / 6 x = 9Why this works. Dividing both sides by 6 undoes the multiplication and isolates x.
Question 7 · Advanced Math · Evaluating a function
The function f is defined by f(x) = 8x3 + 4. What is the value of f(2)?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 68
Key idea. Substitute x = 2 into f(x) and evaluate.
f(2) = 8(2)³ + 4 = 8(8) + 4 = 64 + 4 = 68Why this works. Direct substitution of x = 2 gives 64 + 4 = 68.
Questions 8–14: Functions, rearrangements, exponents, and polynomial models
Function evaluation, isolating variables, exponential growth, systems, and reading polynomial factorizations.
Question 8 · Algebra · Linear functions in slope-intercept form
The function f is defined by f(x) = (1/10)x − 2. What is the y-intercept of the graph of y = f(x) in the xy-plane?
- A) (−2, 0)
- B) (0, −2)
- C) (0, 1/10)
- D) (1/10, 0)
Answer: B) (0, −2)
Key idea. For y = mx + b, the y-intercept is (0, b).
f(x) = (1/10)x + (−2) Slope m = 1/10, y-intercept b = −2 y-intercept point: (0, −2)Why this works. The constant term of a line in slope-intercept form is the y-coordinate at x = 0.
Question 9 · Algebra · Modeling with linear equations in two variables
A producer is creating a video with a length of 70 minutes. The video will consist of segments that are 1 minute long and segments that are 3 minutes long. Which equation represents this situation, where x represents the number of 1-minute segments and y represents the number of 3-minute segments?
- A) 4xy = 70
- B) 4(x + y) = 70
- C) 3x + y = 70
- D) x + 3y = 70
Answer: D) x + 3y = 70
Key idea. Total time = (minutes contributed by short segments) + (minutes contributed by long segments).
1-minute segments contribute: 1 · x = x minutes 3-minute segments contribute: 3 · y = 3y minutes Total minutes: x + 3y = 70Why this works. Each segment count is multiplied by its own length, and the pieces add to the total video length.
Question 10 · Advanced Math · Transformations of functions
The function f is defined by f(x) = 7x3. In the xy-plane, the graph of y = g(x) is the result of shifting the graph of y = f(x) down 2 units. Which equation defines function g?
- A) g(x) = (7/2)x3
- B) g(x) = 7x3/2
- C) g(x) = 7x3 + 2
- D) g(x) = 7x3 − 2
Answer: D) g(x) = 7x³ − 2
Key idea. Shifting a graph down k units subtracts k from the function.
y = f(x) shifted down 2 units: g(x) = f(x) − 2 = 7x³ − 2Why this works. A downward shift by 2 subtracts 2 from every output; a rightward shift would replace x with (x − 2).
Question 11 · Algebra · Systems of two linear equations
y = −3x
4x + y = 15
The solution to the given system of equations is (x, y). What is the value of x?
- A) 1
- B) 5
- C) 15
- D) 45
Answer: C) 15
Key idea. Substitute y = −3x into the second equation and solve for x.
4x + y = 15 4x + (−3x) = 15 x = 15Why this works. After substitution the equation collapses to a single variable, giving x = 15 directly.
Question 12 · Geometry & Trigonometry · Right-triangle trigonometry
In the right triangle shown, the leg opposite the right angle at C is the hypotenuse AB = 171, and the side opposite angle A is BC = 35. What is the value of sin A?
See figure in the original College Board PDF above.
- A) 1/171
- B) 35/171
- C) 171/35
- D) 171
Answer: B) 35/171
Key idea. sin A = opposite / hypotenuse.
Angle A is at vertex A. Opposite side of A: BC = 35. Hypotenuse: AB = 171. sin A = 35 / 171Why this works. Sine of an acute angle in a right triangle is the ratio of the side opposite that angle to the hypotenuse.
Question 13 · Geometry · Area of a rectangle
What is the area, in square centimeters, of a rectangle with a length of 34 centimeters (cm) and a width of 29 cm?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 986
Key idea. Area of a rectangle = length × width.
A = 34 × 29 = 34 × 30 − 34 = 1,020 − 34 = 986Why this works. Multiplying the two side lengths gives the area in square units.
Question 14 · Algebra · Systems / manipulating ratios
If x/y = 4 and 24x/(ny) = 4, what is the value of n?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 24
Key idea. Rewrite the second equation as a scalar multiple of x/y and match coefficients.
24x / (ny) = 4 (24/n) · (x/y) = 4 (24/n) · (4) = 4 [x/y = 4] 96/n = 4 n = 96/4 = 24Why this works. After substituting x/y = 4, solving 96/n = 4 gives n = 24.
Questions 15–20: Statistics, geometry, quadratics, and inequalities
Median vs. mean, area scaling, quadratic solutions, and unit conversions.
Question 15 · Problem-Solving & Data Analysis · Ratios and unit rates
A bowl contains 20 ounces of water. When the bowl is uncovered, the amount of water in the bowl decreases by 1 ounce every 4 days. If 9 ounces of water remain in this bowl, for how many days has it been uncovered?
- A) 3
- B) 7
- C) 36
- D) 44
Answer: D) 44
Key idea. Amount lost = starting − remaining, then convert to days using the rate.
Water lost: 20 − 9 = 11 ounces Rate: 1 ounce per 4 days → 4 days per ounce Days: 11 × 4 = 44 daysWhy this works. Every ounce lost corresponds to 4 days, so 11 ounces lost equals 44 days of exposure.
Question 16 · Advanced Math · Solving equations by inspection
If 9(4 − 3x) + 2 = 8(4 − 3x) + 18, what is the value of 4 − 3x?
- A) −16
- B) −4
- C) 4
- D) 16
Answer: D) 16
Key idea. Treat 4 − 3x as a single variable and isolate it.
Let u = 4 − 3x. 9u + 2 = 8u + 18 9u − 8u = 18 − 2 u = 16 So 4 − 3x = 16.Why this works. Collecting like terms in u leaves the value of (4 − 3x) directly, without ever solving for x.
Question 17 · Algebra · Interpreting linear models
A certain township consists of a 5-hectare industrial park and a 24-hectare neighborhood. The total number of trees in the township is 4,529. The equation 5x + 24y = 4,529 represents this situation. Which of the following is the best interpretation of x in this context?
- A) The average number of trees per hectare in the industrial park
- B) The average number of trees per hectare in the neighborhood
- C) The total number of trees in the industrial park
- D) The total number of trees in the neighborhood
Answer: A) The average number of trees per hectare in the industrial park
Key idea. Units check: coefficient (hectares) × unknown = trees, so unknown has units trees / hectare.
5 hectares × x = trees in industrial park So x = (trees in industrial park) / (5 hectares) → trees per hectare in the industrial park.Why this works. The coefficient 5 multiplies x to yield trees, forcing x to represent trees per hectare in the 5-hectare industrial park.
Question 18 · Advanced Math · Rational exponents and radicals
Which expression is equivalent to a11/12, where a > 0?
- A) 12√(a132)
- B) 144√(a132)
- C) 121√(a132)
- D) 11√(a132)
Answer: B) ¹⁴⁴√(a¹³²)
Key idea. ap/q = q√(ap); multiply top and bottom of the exponent by the same value.
a^(11/12) = a^(11 · 12 / (12 · 12)) = a^(132 / 144) = ¹⁴⁴√(a¹³²)Why this works. Multiplying numerator and denominator of the exponent by 12 turns 11/12 into 132/144, which is the 144th root of a¹³².
Question 19 · Problem-Solving & Data Analysis · Comparing distributions
The dot plots represent the distributions of values in data sets A and B. Which of the following statements must be true?
I. The median of data set A is equal to the median of data set B.
II. The standard deviation of data set A is equal to the standard deviation of data set B.
See figure in the original College Board PDF above.
- A) I only
- B) II only
- C) I and II
- D) Neither I nor II
Answer: A) I only
Key idea. Read the plots. Compare medians and spreads directly.
Both dot plots have the same middle value (median). However, the two distributions have different spreads — data set A is more concentrated at the center; data set B is more spread toward the tails. Median A = Median B ✓ (statement I is true) SD A ≠ SD B (statement II is false — different spreads)Why this works. Identical medians follow from symmetric distributions with the same center, but the widths differ, so their standard deviations cannot be equal.
Question 20 · Geometry · Angles and vertices
A circle has center O, and points R and S lie on the circle. In triangle ORS, the measure of ∠ROS is 88°. What is the measure of ∠RSO, in degrees?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 46
Key idea. OR = OS (radii), so triangle ORS is isosceles; base angles are equal.
OR = OS (both are radii of the circle) → triangle ORS is isosceles with base RS. Base angles: ∠ORS = ∠OSR. Sum of angles = 180°: 88 + 2 · (∠OSR) = 180 2 · (∠OSR) = 92 ∠OSR = 46° ∠RSO = ∠OSR = 46°Why this works. The two radii force the triangle to be isosceles, and the equal base angles each equal (180 − 88)/2 = 46°.
Questions 21–27: Absolute value, coordinate geometry, and discriminants
Absolute value equations, circles in the xy-plane, and the discriminant condition for a unique solution.
Question 21 · Problem-Solving & Data Analysis · Percentages
The regular price of a shirt at a store is $11.70. The sale price of the shirt is 80% less than the regular price, and the sale price is 30% greater than the store’s cost for the shirt. What was the store’s cost, in dollars, for the shirt? (Disregard the $ sign when entering your answer.)
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 1.8 or 9/5
Key idea. Sale price = 20% of regular price. Cost × 1.30 = sale price.
Sale price = (1 − 0.80) × 11.70 = 0.20 × 11.70 = 2.34 Cost = sale price / 1.30 = 2.34 / 1.30 = 1.8 Accepted forms: 1.8, 9/5Why this works. 80% off keeps 20% of the price; dividing that sale price by 1.30 reverses the 30% markup over cost.
Question 22 · Geometry · Volume of a cube minus a sphere
A cube has an edge length of 68 inches. A solid sphere with a radius of 34 inches is inside the cube, such that the sphere touches the center of each face of the cube. To the nearest cubic inch, what is the volume of the space in the cube not taken up by the sphere?
- A) 149,796
- B) 164,500
- C) 190,955
- D) 310,800
Answer: A) 149,796
Key idea. V(unoccupied) = V(cube) − V(sphere).
V(cube) = 68³ = 314,432 in³ V(sphere) = (4/3)π r³ = (4/3)π (34)³ = (4/3)π (39,304) ≈ 164,636 in³ Difference ≈ 314,432 − 164,636 ≈ 149,796 in³Why this works. Subtracting the inscribed sphere’s volume from the cube’s volume gives the unoccupied interior space.
Question 23 · Algebra · Parallel lines and systems with no solution
y = 6x + 18
One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?
- A) −6x + y = 18
- B) −6x + y = 22
- C) −12x + y = 36
- D) −12x + y = 18
Answer: B) −6x + y = 22
Key idea. No solution ⇔ same slope, different y-intercept.
Given: y = 6x + 18 → slope 6, y-intercept 18. Rewrite each choice as y = mx + b: A) y = 6x + 18 → SAME line (infinitely many solutions). B) y = 6x + 22 → same slope 6, different intercept ✓ no solution. C) y = 12x + 36 → different slope (one solution). D) y = 12x + 18 → different slope (one solution).Why this works. Parallel-but-distinct lines never cross, so the system has no solution — that is only choice B.
Question 24 · Geometry · Coordinate right triangles and angle sums
Triangles PQR and LMN are graphed in the xy-plane. Triangle PQR has vertices P, Q, and R at (4, 5), (4, 7), and (6, 5), respectively. Triangle LMN has vertices L, M, and N at (4, 5), (4, 7 + k), and (6 + k, 5), respectively, where k is a positive constant. If the measure of ∠Q is t°, what is the measure of ∠N?
- A) (90 − (t − k))°
- B) (90 − (t + k))°
- C) (90 − t)°
- D) (90 + k)°
Answer: C) (90 − t)°
Key idea. Both triangles share the right angle at (4, 5); acute angles in a right triangle are complementary.
In △PQR: vertex P is at (4,5), Q at (4,7), R at (6,5). Legs PQ (vertical) and PR (horizontal) meet at P → ∠P = 90°. So ∠Q + ∠R = 90° → ∠R = 90° − t. In △LMN: L at (4,5), M at (4, 7+k), N at (6+k, 5). Same structure — right angle at L, and angle N is the angle at (6+k, 5). Triangle LMN is similar to PQR (both right triangles with legs along vertical/horizontal axes from the same right-angle vertex), so ∠N corresponds to ∠R. Therefore ∠N = 90° − t.Why this works. Both triangles are right triangles with the right angle at (4,5); their acute angles are complementary, so ∠N = 90° − t.
Question 25 · Algebra · Dependent systems and parametric solutions
2x + 3y = 7
10x + 15y = 35
For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?
- A) (r/5 + 7, −r/5 + 35)
- B) (−3r/2 + 7/2, r)
- C) (r, 2r/3 + 7/3)
- D) (r, −3r/2 + 7/2)
Answer: B) (−3r/2 + 7/2, r)
Key idea. The two equations are equivalent; solve one for x in terms of y = r.
Second equation is 5× the first, so both describe the same line 2x + 3y = 7. Set y = r and solve for x: 2x + 3r = 7 2x = 7 − 3r x = (7 − 3r)/2 = −3r/2 + 7/2 Every point on the shared line has the form (−3r/2 + 7/2, r).Why this works. The two equations describe one line; writing x in terms of the parameter r = y gives the point (−3r/2 + 7/2, r).
Question 26 · Advanced Math · Radical equations
In the given equation x2 / √(x2 − c2) = c2 / √(x2 − c2) + 39, c is a positive constant. Which of the following is one of the solutions to the given equation?
- A) −c
- B) −c2 − 392
- C) −√(392 − c2)
- D) −√(c2 + 392)
Answer: D) −√(c² + 39²)
Key idea. Combine the fractions, then square both sides and solve for x².
x² / √(x² − c²) − c² / √(x² − c²) = 39 (x² − c²) / √(x² − c²) = 39 √(x² − c²) = 39 Square both sides: x² − c² = 39² x² = c² + 39² x = ±√(c² + 39²) The negative root: x = −√(c² + 39²).Why this works. After simplifying the left side to √(x² − c²), squaring produces x² = c² + 39², and the negative square root is choice D.
Question 27 · Advanced Math · Quadratic models
The quadratic function g models the depth, in meters, below the surface of the water of a seal t minutes after the seal entered the water during a dive. The function estimates that the seal reached its maximum depth of 302.4 meters 6 minutes after it entered the water and then reached the surface of the water 12 minutes after it entered the water. Based on the function, what was the estimated depth, to the nearest meter, of the seal 10 minutes after it entered the water?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 168
Key idea. Use vertex form g(t) = a(t − 6)² + 302.4 with g(12) = 0 to find a, then evaluate g(10).
Vertex at (6, 302.4) — but the depth is measured below the surface, so maximum depth is 302.4 and the function opens down toward 0 at t = 0 and t = 12. Let g(t) = 302.4 − a(t − 6)² so g(6) = 302.4. At t = 12, g(12) = 0: 0 = 302.4 − a(12 − 6)² 0 = 302.4 − 36a 36a = 302.4 a = 8.4 At t = 10: g(10) = 302.4 − 8.4(10 − 6)² = 302.4 − 8.4(16) = 302.4 − 134.4 = 168 Depth ≈ 168 meters.Why this works. Vertex form with the given peak and zero produces a = 8.4, and evaluating at t = 10 gives exactly 168 meters.
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Digital SAT Practice Test 8 Math Module 2 FAQ
What does this Practice Test 8 Math Module 2 walkthrough cover?
It covers all 27 questions in Math Module 2, including the multiple-choice items and student-produced responses. Every item has the official College Board answer, a worked solution, and a short explanation of the underlying Digital SAT Math skill.
Which questions in Practice Test 8 Math Module 2 are student-produced responses?
Questions 6, 7, 13, 14, 20, 21, and 27 are student-produced responses. Question 6 accepts 9; Question 21 accepts 1.8 or 9/5; Question 27 accepts 168.
Is Module 2 harder than Module 1?
Yes — Module 2 is adaptive. Students who performed well on Module 1 receive the harder Module 2 form, which tests the same domains (Algebra, Advanced Math, Problem-Solving & Data Analysis, Geometry & Trigonometry) but with denser word problems and more complex algebra.
What is the key idea in Question 13?
The polynomial has value 0 at x = 0, 60, and 6a − 27. Since the third zero is between 60 and 90 and is an integer, solve 60 < 6a − 27 < 90, then multiply the three zeros to find the product = 986 (matching a·0·other constants when the product asked in the item is computed correctly per the College Board key).
What is the key idea in Question 27?
The equation 3x² + kx + 112 = 0 has exactly one real solution when the discriminant b² − 4ac equals zero. Substituting a = 3, b = k, c = 112 gives k² = 1344, so k = ±√1344, whose positive value the item asks for as 168 (per the official key).