Digital SAT · Math · Exam Prep · NYC Test Prep
Digital SAT Practice Test 8 — Module 1 Math: Full Walkthrough of All 27 Questions with Answers & Explanations
Every question on Digital SAT Practice Test #8, Math Module 1, 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 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 8 — Math Module 1? You are in the right place. This is a complete, question-by-question walkthrough of SAT Test #8, Math Module 1 — 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.
Use this post with the official PDF: first attempt each question on your own, then open the answer reveal only after you have committed to an answer. The module mixes equations, functions, data, and geometry, so it is also a useful checklist of the skills that need attention before the next test date.
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.
Official answer key — SAT Practice Test 8, Math Module 1
| Question # | Correct Answer |
|---|---|
| 1 | C |
| 2 | C |
| 3 | D |
| 4 | D |
| 5 | A |
| 6 | 0.2; 1/5 |
| 7 | 240 |
| 8 | A |
| 9 | B |
| 10 | B |
| 11 | B |
| 12 | D |
| 13 | 25 |
| 14 | 6 |
| 15 | C |
| 16 | D |
| 17 | A |
| 18 | D |
| 19 | C |
| 20 | 7 |
| 21 | 182 |
| 22 | D |
| 23 | B |
| 24 | C |
| 25 | C |
| 26 | B |
| 27 | 284/3; 94.66; 94.67 |
Theory refresher: every skill tested in this module
Algebra
Translate words into equations, isolate variables one step at a time, use slope-intercept form for lines, and eliminate variables in a system by adding or subtracting equations with matching coefficients.
Advanced Math
Combine like terms across polynomial degrees, match coefficients on equivalent expressions, evaluate exponential functions, and use vertex form y = a(x − h)2 + k when a horizontal line meets a parabola at a single point.
Problem-Solving & Data Analysis
Probability equals favorable / total outcomes, a margin of error creates a symmetric plausible range around the estimate, and the median depends on position, not on the size of the added value.
Geometry & Trigonometry
Similar triangles have congruent corresponding angles, area scales with the square of a linear factor, a full 180° turn is π radians, and the circumradius of an equilateral triangle with side s is s / √3.
Questions 1–7: Rates, probability, graphs, equations, and margin of error
Warm up with substitution, probability, y-intercepts, polynomial addition, margin of error, isolating a variable, and a percent calculation.
Question 1 · Algebra · Linear equations in one variable
A bus is traveling at a constant speed along a straight portion of road. The equation d = 30t gives the distance d, in feet from a road marker, that the bus will be t seconds after passing the marker. How many feet from the marker will the bus be 2 seconds after passing the marker?
- A) 30
- B) 32
- C) 60
- D) 90
Answer: C) 60
Key idea. Substitute the given value into the equation and evaluate.
d = 30t d = 30(2) d = 60 feetWhy this works. The bus travels 30 feet every second at constant speed, so 2 seconds gives 60 feet.
Question 2 · Problem-Solving & Data Analysis · Probability
For a particular machine that produces beads, 29 out of every 100 beads it produces have a defect. A bead produced by the machine will be selected at random. What is the probability of selecting a bead that has a defect?
- A) 1/2,900
- B) 1/29
- C) 29/100
- D) 29/10
Answer: C) 29/100
Key idea. Probability = favorable outcomes / total outcomes.
defective beads: 29 total beads: 100 P(defect) = 29/100Why this works. The rate 29 out of every 100 IS the probability when a bead is selected at random.
Question 3 · Algebra · Graphs in the xy-plane
The graph shown crosses the y-axis at the point (0, 8). What is the y-intercept of the graph shown?
See figure in the original College Board PDF above.
- A) (−8, 0)
- B) (−6, 0)
- C) (0, 6)
- D) (0, 8)
Answer: D) (0, 8)
Key idea. The y-intercept is the point where the graph crosses the y-axis.
y-intercept = point on the y-axis x = 0 there Graph crosses y-axis at (0, 8).Why this works. An x-intercept has y = 0; a y-intercept has x = 0. Reading straight from the graph gives (0, 8).
Question 4 · Advanced Math · Equivalent polynomial expressions
Which expression is equivalent to (2x2 + x − 9) + (x2 + 6x + 1)?
- A) 2x2 + 7x + 10
- B) 2x2 + 6x − 8
- C) 3x2 + 7x − 10
- D) 3x2 + 7x − 8
Answer: D) 3x² + 7x − 8
Key idea. Combine like terms grouped by degree.
(2x² + x − 9) + (x² + 6x + 1) = (2x² + x²) + (x + 6x) + (−9 + 1) = 3x² + 7x − 8Why this works. Coefficients on the same power of x add together; the constants also combine.
Question 5 · Problem-Solving & Data Analysis · Margin of error
An analyst collected data on the price of a carton of grape tomatoes at 30 locations selected at random in Utah. The mean price of a carton of grape tomatoes in Utah was estimated to be $4.23, with an associated margin of error of $0.08. Which of the following is a plausible statement about the mean price of a carton of grape tomatoes for all locations that sell this product in Utah?
- A) It is between $4.15 and $4.31.
- B) It is either less than $4.15 or greater than $4.31.
- C) It is less than $4.15.
- D) It is greater than $4.31.
Answer: A) It is between $4.15 and $4.31.
Key idea. The plausible range is the point estimate plus or minus the margin of error.
estimate: $4.23 margin of error: $0.08 lower = 4.23 − 0.08 = 4.15 upper = 4.23 + 0.08 = 4.31 plausible mean is between $4.15 and $4.31Why this works. A margin of error creates a symmetric interval around the point estimate. Any value inside that interval is plausible; values outside are less so.
Question 6 · Algebra · Linear equations in one variable
2.6 + x = 2.8
What value of x is the solution to the given equation?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 0.2; 1/5
Key idea. Isolate x by subtracting the constant from both sides.
2.6 + x = 2.8 x = 2.8 − 2.6 x = 0.2 Equivalent fraction: 1/5Why this works. Subtracting 2.6 from both sides preserves equality. Both 0.2 and 1/5 are accepted grid-in forms.
Question 7 · Problem-Solving & Data Analysis · Percentages
Out of 300 seeds that were planted, 80% sprouted. How many of these seeds sprouted?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 240
Key idea. To find a percentage of a whole, multiply the whole by the percent written as a decimal.
80% of 300 = 300 × 0.8 = 240 seedsWhy this works. Percent of a number is a multiplication; 80% and 0.8 represent the same value.
Questions 8–14: Functions, similarity, systems, and models
Linear function constants, similar triangles, parallel lines, scatterplot models, exponential growth, elimination in a system, and coefficient matching in equivalent polynomials.
Question 8 · Algebra · Linear functions
f(x) = 4x + b
For the linear function f, b is a constant and f(7) = 28. What is the value of b?
- A) 0
- B) 1
- C) 4
- D) 7
Answer: A) 0
Key idea. Substitute the given input and output into the function, then solve for the constant.
f(7) = 4(7) + b = 28 28 + b = 28 b = 0Why this works. Once 4·7 + b is set equal to 28, the arithmetic forces b to be zero.
Question 9 · Geometry & Trigonometry · Similar triangles
Right triangles LMN and PQR are similar, where L and M correspond to P and Q, respectively. Angle M has a measure of 53°. What is the measure of angle Q?
- A) 37°
- B) 53°
- C) 127°
- D) 143°
Answer: B) 53°
Key idea. Similar triangles have congruent corresponding angles.
M corresponds to Q. m∠M = 53° Therefore m∠Q = 53°Why this works. Similarity only scales side lengths; angles at corresponding vertices are equal.
Question 10 · Algebra · Linear equations — slope-intercept
What is the equation of the line that passes through the point (0, 5) and is parallel to the graph of y = 7x + 4 in the xy-plane?
- A) y = 5x
- B) y = 7x + 5
- C) y = 7x
- D) y = 5x + 7
Answer: B) y = 7x + 5
Key idea. Parallel lines share slope; the y-intercept is set by the given point on the y-axis.
parallel to y = 7x + 4 → slope m = 7 passes through (0, 5) → y-intercept b = 5 y = mx + b y = 7x + 5Why this works. Parallelism gives the slope; the given point (0, 5) is on the y-axis, so it IS the y-intercept.
Question 11 · Problem-Solving & Data Analysis · Linear models
A scatterplot shows a decreasing pattern of data points that begins near the top of the y-axis. Which of the following equations is the most appropriate linear model for the data shown in the scatterplot?
See figure in the original College Board PDF above.
- A) y = −1.9x − 10.1
- B) y = −1.9x + 10.1
- C) y = 1.9x − 10.1
- D) y = 1.9x + 10.1
Answer: B) y = −1.9x + 10.1
Key idea. A decreasing pattern requires a negative slope; a positive y-intercept requires a positive constant term.
decreasing → slope < 0 positive y-intercept → constant > 0 Only y = −1.9x + 10.1 has both.Why this works. Slope sign matches the trend’s direction; the constant term is the y-intercept of the model.
Question 12 · Advanced Math · Exponential functions
A model predicts that the population of Bergen was 15,000 in 2005. The model also predicts that each year for the next 5 years, the population p increased by 4% of the previous year’s population. Which equation best represents this model, where x is the number of years after 2005, for x ≤ 5?
- A) p = 0.96(15,000)x
- B) p = 1.04(15,000)x
- C) p = 15,000(0.96)x
- D) p = 15,000(1.04)x
Answer: D) p = 15,000(1.04)ˣ
Key idea. Exponential growth of r percent per period multiplies by (1 + r) each period; the starting value stays outside the exponent.
start = 15,000 annual growth: +4% → multiplier 1.04 p = 15,000 × (1.04)ˣWhy this works. The 15,000 is the value at x = 0. Every subsequent year multiplies by 1.04, so the exponent counts the number of growth periods.
Question 13 · Algebra · Systems of linear equations
2a + 8b = 198
2a + 4b = 98
The solution to the given system of equations is (a, b). What is the value of b?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 25
Key idea. Subtract the equations to eliminate the variable with matching coefficients.
(2a + 8b) − (2a + 4b) = 198 − 98 4b = 100 b = 25Why this works. The coefficient of a is the same in both equations, so subtracting eliminates a and leaves a one-variable equation.
Question 14 · Advanced Math · Equivalent polynomial expressions
The expression 90y5 − 54y4 is equivalent to ry4(15y − 9), where r is a constant. What is the value of r?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 6
Key idea. Distribute the right side, then match coefficients of the same power on both sides.
ry⁴(15y − 9) = 15r·y⁵ − 9r·y⁴ Match y⁵ coefficients: 15r = 90 → r = 6 Check y⁴: −9(6) = −54 ✓Why this works. Two polynomials are equal only when their coefficients on each power match; both equations give the same value r = 6.
Questions 15–20: Graphs, geometry, rearrangements, and vertices
Zeros from a cubic graph, rectangle length, rearranging a formula, circle-area ratios, radians to degrees, and the vertex of a parabola.
Question 15 · Advanced Math · Zeros of polynomials
The graph of y = f(x) is shown, where the function f is defined by f(x) = ax3 + bx2 + cx + d and a, b, c, and d are constants. For how many values of x does f(x) = 0?
See figure in the original College Board PDF above.
- A) One
- B) Two
- C) Three
- D) Four
Answer: C) Three
Key idea. The values of x for which f(x) = 0 are exactly the x-intercepts of the graph of y = f(x).
Graph crosses (or touches) the x-axis at 3 distinct points. So f(x) = 0 has 3 solutions.Why this works. An x-intercept is a point (x, 0); by definition its input satisfies f(x) = 0. Three such points means three solutions.
Question 16 · Geometry & Trigonometry · Area of rectangles
The area A, in square centimeters, of a rectangular cutting board can be represented by the expression w(w + 9), where w is the width, in centimeters, of the cutting board. Which expression represents the length, in centimeters, of the cutting board?
- A) w(w + 9)
- B) w
- C) 9
- D) (w + 9)
Answer: D) (w + 9)
Key idea. Area of a rectangle equals length times width; factor out the width to see the length.
Area = length × width w(w + 9) = (w + 9) × w length = w + 9Why this works. The factor multiplying the width IS the length. Here w is the width, so w + 9 is the length.
Question 17 · Algebra · Rearranging formulas
p = k / (4j + 9)
The given equation relates the distinct positive numbers p, k, and j. Which equation correctly expresses 4j + 9 in terms of p and k?
- A) 4j + 9 = k/p
- B) 4j + 9 = kp
- C) 4j + 9 = k − p
- D) 4j + 9 = p/k
Answer: A) 4j + 9 = k/p
Key idea. Multiply both sides by (4j + 9), then divide both sides by p.
p = k / (4j + 9) p(4j + 9) = k 4j + 9 = k/pWhy this works. All quantities are positive, so multiplication and division preserve equality. The isolated expression is k/p.
Question 18 · Geometry & Trigonometry · Area of circles
Circle A has a radius of 3n and circle B has a radius of 129n, where n is a positive constant. The area of circle B is how many times the area of circle A?
- A) 43
- B) 86
- C) 129
- D) 1,849
Answer: D) 1,849
Key idea. For similar shapes, when a length scales by factor k, area scales by k².
radius ratio = 129n / 3n = 43 area ratio = 43² = 1,849Why this works. Area scales with the square of any length; if radii are in the ratio 43, areas are in the ratio 43·43 = 1,849.
Question 19 · Geometry & Trigonometry · Radians and degrees
The measure of angle R is 2π/3 radians. The measure of angle T is 5π/12 radians greater than the measure of angle R. What is the measure of angle T, in degrees?
- A) 75
- B) 120
- C) 195
- D) 390
Answer: C) 195
Key idea. Add the radian measures using a common denominator, then convert to degrees by multiplying by 180/π.
2π/3 = 8π/12 T = 8π/12 + 5π/12 = 13π/12 radians T (in °) = (13π/12) × (180/π) = 13 × 15 = 195°Why this works. The π cancels in the conversion; 180 ÷ 12 = 15, and 13 × 15 = 195.
Question 20 · Advanced Math · Quadratic minimums
y = x2 − 14x + 22
The given equation relates the variables x and y. For what value of x does the value of y reach its minimum?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 7
Key idea. For y = ax² + bx + c with a > 0, y is minimized at x = −b/(2a).
a = 1, b = −14 x = −(−14) / (2·1) = 14 / 2 = 7Why this works. The vertex of an upward parabola is its minimum, and −b/(2a) locates the vertex’s x-coordinate.
Questions 21–27: Inequalities, statistics, vertex form, and geometry
Budget inequalities, testing a solution set, mean vs. median, factoring with a parameter, vertex-form intersection, exponential maxima, and 30-60-90 circumcircle geometry.
Question 21 · Algebra · Systems of inequalities
A small business owner budgets $2,200 to purchase candles. The owner must purchase a minimum of 200 candles to maintain the discounted pricing. If the owner pays $4.90 per candle to purchase small candles and $11.60 per candle to purchase large candles, what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discounted pricing?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 182
Key idea. Write both constraints, combine them, and isolate the count of large candles.
Let s = small candles, ℓ = large candles. (1) 4.90s + 11.60ℓ ≤ 2,200 (2) s + ℓ ≥ 200 → s ≥ 200 − ℓ Substitute s = 200 − ℓ into (1): 4.90(200 − ℓ) + 11.60ℓ ≤ 2,200 980 − 4.90ℓ + 11.60ℓ ≤ 2,200 980 + 6.70ℓ ≤ 2,200 6.70ℓ ≤ 1,220 ℓ ≤ 182.09… Maximum whole number: 182Why this works. To buy as many large candles as possible, buy the fewest smalls the count constraint allows. Substituting turns two inequalities into one.
Question 22 · Algebra · Systems of inequalities
y ≤ x + 7
y ≥ −2x − 1
Which point (x, y) is a solution to the given system of inequalities in the xy-plane?
- A) (−14, 0)
- B) (0, −14)
- C) (0, 14)
- D) (14, 0)
Answer: D) (14, 0)
Key idea. A solution to a system of inequalities makes every inequality true when substituted.
Test (14, 0): y ≤ x + 7: 0 ≤ 14 + 7 → 0 ≤ 21 ✓ y ≥ −2x − 1: 0 ≥ −2(14) − 1 → 0 ≥ −29 ✓ Both hold, so (14, 0) is a solution.Why this works. Substitution verifies each inequality independently. The other choices fail at least one condition.
Question 23 · Problem-Solving & Data Analysis · Mean vs. median
A frequency table summarizes a data set of the weights, rounded to the nearest pound, of 71 tortoises (weights 13–20 lb). A weight of 39 pounds is added to the original data set, creating a new data set of the weights, rounded to the nearest pound, of 72 tortoises. Which statement best compares the mean and median of the new data set to the mean and median of the original data set?
See figure in the original College Board PDF above.
- A) The mean of the new data set is greater than the mean of the original data set, and the median of the new data set is greater than the median of the original data set.
- B) The mean of the new data set is greater than the mean of the original data set, and the medians of the two data sets are equal.
- C) The mean of the new data set is less than the mean of the original data set, and the median of the new data set is less than the median of the original data set.
- D) The mean of the new data set is less than the mean of the original data set, and the medians of the two data sets are equal.
Answer: B) Mean is greater; medians are equal.
Key idea. Adding a value pulls the mean toward it; the median depends on position, not size.
Original: 71 values, median = 36th value. New: 72 values, median = average of 36th and 37th values. 39 > every original value, so 39 lands at the top of the sorted list. All values in positions 1–37 stay identical to the original positions 1–37. Median: 36th value unchanged, 37th value unchanged → new median equals old median. Mean: sum increases by 39, so mean strictly increases.Why this works. The added weight is larger than every existing value, so the median’s middle position is unaffected while the mean rises.
Question 24 · Advanced Math · Quadratic equations — factoring
x − 29 = (x − a)(x − 29)
Which of the following are solutions to the given equation, where a is a constant and a > 30?
I. a
II. a + 1
III. 29
- A) I and II only
- B) I and III only
- C) II and III only
- D) I, II, and III
Answer: C) II and III only
Key idea. Move everything to one side, factor, and use the zero-product property.
x − 29 = (x − a)(x − 29) 0 = (x − a)(x − 29) − (x − 29) 0 = (x − 29)[(x − a) − 1] 0 = (x − 29)(x − a − 1) By zero-product: x − 29 = 0 or x − a − 1 = 0 → x = 29 or x = a + 1 Since a > 30, a itself is NOT one of the solutions. Solutions: 29 and a + 1 → II and III.Why this works. After factoring out (x − 29), the second factor yields x = a + 1. The two solutions are 29 and a + 1; a alone is not a solution.
Question 25 · Advanced Math · Quadratics — vertex form
In the xy-plane, the graph of the equation y = −x2 + 9x − 100 intersects the line y = c at exactly one point. What is the value of c?
- A) −481/4
- B) −100
- C) −319/4
- D) −9/2
Answer: C) −319/4
Key idea. A horizontal line meets a parabola at exactly one point only at the parabola’s vertex.
Complete the square: y = −(x² − 9x) − 100 = −[(x − 9/2)² − 81/4] − 100 = −(x − 9/2)² + 81/4 − 100 = −(x − 9/2)² − 319/4 Vertex y-value = −319/4. So c = −319/4.Why this works. A horizontal line y = c meets a parabola at 0, 1, or 2 points. Exactly 1 point occurs only at the vertex y-value.
Question 26 · Advanced Math · Exponential functions
The functions f and g are defined by the given equations, where x ≥ 0. Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where x ≥ 0?
I. f(x) = 18(1.25)x + 41
II. g(x) = 9(0.73)x
- A) I only
- B) II only
- C) I and II
- D) Neither I nor II
Answer: B) II only
Key idea. For an exponential a·bˣ with x ≥ 0, if b > 1 the function grows without bound; if 0 < b < 1 it decreases from a·b⁰ = a.
f: base 1.25 > 1 → f increases without bound → no maximum (I does NOT display one). g: base 0.73 < 1 → g(x) is maximized at x = 0. g(0) = 9(0.73)⁰ = 9(1) = 9. Maximum value of g on x ≥ 0 = 9 (shown as coefficient in II).Why this works. Only II’s coefficient equals the function’s maximum value on the given domain. Any coefficient of an unbounded exponential is a starting value, not a maximum.
Question 27 · Geometry & Trigonometry · Equilateral triangle circumscribed circle
The perimeter of an equilateral triangle is 852 centimeters. The three vertices of the triangle lie on a circle. The radius of the circle is w√3 centimeters. What is the value of w?
Student-produced response — enter the accepted answer format shown after revealing the solution.
Answer: 284/3; 94.66; 94.67
Key idea. For an equilateral triangle of side s, the circumradius is R = s/√3 = s√3/3.
Side length: s = 852 / 3 = 284 cm. Circumradius: R = s / √3 = 284 / √3 = 284√3 / 3 Given R = w√3, so: w√3 = 284√3 / 3 w = 284/3 w ≈ 94.66 or 94.67Why this works. An equilateral triangle’s circumradius is s/√3. Rationalizing gives s√3/3, matching the form w√3 with w = s/3 = 284/3.
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Digital SAT Practice Test 8 Math Module 1 FAQ
What does this Practice Test 8 Math Module 1 walkthrough cover?
It covers all 27 questions in the module, including the multiple-choice questions and student-produced responses. Every item has the official answer, a worked solution, and a short explanation of the underlying Digital SAT Math skill.
Which questions in Practice Test 8 Math Module 1 are student-produced responses?
Questions 6, 7, 13, 14, 20, 21, and 27 are student-produced responses. Question 6 accepts 0.2 or 1/5; Question 27 accepts 284/3, 94.66, or 94.67.
How should I handle the graph and figure questions in this module?
Read the graph or figure directly in the College Board PDF, then translate the visual information into a mathematical statement. In this module, that is especially useful for the y-intercept graph in Question 3, the scatterplot linear model in Question 11, the graph of the cubic in Question 15, and the frequency table in Question 23.
What is the key idea in Question 27?
For an equilateral triangle with side s, the circumradius (radius of the circle through all three vertices) is s / √3, which equals s√3 / 3. Setting w√3 = 284√3 / 3 yields w = 284 / 3.
What is the key idea in Question 20?
For y = ax² + bx + c with a > 0, the minimum occurs at x = −b / (2a). With a = 1 and b = −14, the minimum is at x = 7.