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Digital SAT Practice Test 5 — Module 2: Full Walkthrough of All 27 Math Questions with Answers & Explanations

Every question on Digital SAT Practice Test #5, 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.

· 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 5 — Math Module 2? You are in the right place. This is a complete, question-by-question walkthrough of SAT Test #5, 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 unlike Module 1, 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 that typically tops out in the mid-600s. The version reproduced here — from the official College Board Practice Test 5 booklet — is the higher, harder form, which makes it an especially good stress test for students aiming at 700+.

Whether you are a student prepping for the next SAT test date, a parent looking for answer explanations, or a teacher building a review packet, this walkthrough is designed to be the most complete, honest, and student-friendly Digital SAT Math walkthrough on the internet. Written by the same team that teaches Digital SAT prep at SOMATH, a math-focused school on the Upper West Side of New York City run by Northwestern-trained cofounder Marcelo Ambrozio and Harvard-trained cofounder Vivianne Wright.

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Download SAT Practice Test 5 (Full PDF) The complete official College Board digital SAT Practice Test 5 booklet, including Math Module 2. Ready to print.
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How to use this set. Read each question carefully, cover the choices, work it out on paper before you click “Show answer & explanation.” The point is not to read solutions — it is to build the reflex of recognizing which SAT-Math tool the question wants (substitution, factoring, discriminant, similar-figure area ratio, tangent-radius perpendicularity, etc.) before you touch the calculator. If you get stuck for more than 90 seconds, peek at just the correct answer, close the reveal, and try to reconstruct the worked solution on your own.

Sections

  1. Answer key at a glance
  2. Theory refresher · What Math Module 2 tests
  3. Section 1 · Algebra & Linear Systems (Q1–Q8, Q13, Q19–Q20)
  4. Section 2 · Advanced Math & Functions (Q3–Q4, Q7, Q17–Q18, Q23–Q25)
  5. Section 3 · Problem-Solving & Data Analysis (Q1, Q9–Q12, Q15)
  6. Section 4 · Geometry & Trigonometry (Q14, Q16, Q21–Q22, Q26–Q27)
  7. How SOMATH prepares NYC students for the Digital SAT
  8. Digital SAT FAQ

Answer key at a glance

All 27 official College Board answers for Practice Test 5, Math Module 2:

QAnswerQAnswerQAnswer
1B) 8810A) 34 m/s → 5,202 J19D) Table D
2B) 1211C) 62020
3B) 3712D) 982166
4B) 2xy(8x²y + 7)1344/322D) 1,944
5D) c = 25x + 11144,20523D) 42a(k²+1)/k
6615A) Original > New24D) 5x²−14x+49=0
7−30 or 3016A) 11625A) 8
8D) y=4x+617B) 7x³ + 1026C) (10, 2)
9B) Hits water at 1.6 s18B) 48274,176

Theory refresher · What Math Module 2 tests

The Digital SAT Math section has two 35-minute modules of 22 questions each (20 scored + 2 pretest). Module 2 is adaptive — the exact set of questions depends on Module 1 performance. The Bluebook Desmos calculator is available on every single question. Roughly 75% of items are multiple choice (four options each) and 25% are student-produced responses (fill-in-the-blank numeric or fractional answers). The four content domains are:

1 · Algebra

Slope-intercept form. A line with slope m passing through (0, b) has equation y = mx + b. Read m and b directly from the given information.

Percent-of. p% of N means (p/100) × N. So 20% of 440 is 0.20 × 440 = 88.

Linear word problems. Translate rate + fixed amount into y = (rate) · x + (fixed). "25 calories per pound, plus 11 extra" → c = 25x + 11.

Systems of linear equations. Add the equations when adding one pair of opposite coefficients eliminates a variable directly. For "5y = 10x + 11" and "−5y = 5x − 21," adding gives 0 = 15x − 10, so x = 2/3.

Systems of linear inequalities. A point is a solution to the whole system only if it satisfies every inequality. When checking tables, verify each row against the inequality, not just the first row.

Distributing and combining like terms. Slope of y = (1/3)(29x+10) + 5x is found by distributing to get y = (29/3)x + 10/3 + 5x, then combining: slope = 29/3 + 15/3 = 44/3.

2 · Advanced Math

Function evaluation. f(a) means substitute a for every x in the definition of f. f(10) = 4(10) − 3 = 37.

Factoring. Pull out the greatest common factor from every term. 16x³y² + 14xy = 2xy(8x²y + 7).

Difference of squares as an equation. (d−30)(d+30) = 0 means one factor is zero: d = 30 or d = −30. Never forget the sign options.

Discriminant. For ax² + bx + c = 0, the discriminant is b² − 4ac. No real solution ⇒ b² − 4ac < 0.

Radicals as squares. 4√(2x) = 16 → √(2x) = 4 → 2x = 16 → x = 8. Isolate the radical first, then square.

Exponential growth. P(t) = P₀ (1+r)^(kt) grows by 100r% every time kt increases by 1, i.e. every 1/k units of t. For P(t) = 260(1.04)^((6/4)t), the population increases 4% every 4/6 = 2/3 years = 8 months.

Rational expressions. Combine terms over a common denominator: 42a/k + 42ak = 42a/k + 42ak²/k = 42a(1 + k²)/k.

3 · Problem-Solving & Data Analysis

Complement of probabilities. If probabilities of the parts sum to 1, and two are known (0.48 + 0.24 = 0.72), the third is 1 − 0.72 = 0.28. Multiply by the total to get the count: 0.28 × 350 = 98.

Reading a graph. Trace vertically from the given x-value up to the plotted line, then horizontally to the y-axis to read off the corresponding value.

Interpreting function output in context. If K(v) = 5,202 and v represents meters per second, the statement means "an object traveling at v m/s has kinetic energy 5,202 J." Match the units in the answer choices to v and K(v) exactly.

x-intercept interpretation. Where y = 0 on a real-world graph. If y is height above water and x is time, the x-intercept is when the diver hits the water.

Scatterplot: actual vs. predicted. Count only the plotted points that lie strictly above the line of best fit; points on the line don't count.

Effect of adding a data point on the mean. If the new value is less than the original mean, the new mean drops; if greater, the new mean rises; if equal, no change.

4 · Geometry & Trigonometry

Surface area of a box. Full box with edge s: 6s². Box without a lid: 5s². For s = 29: 5(29)² = 5(841) = 4,205.

Triangle angle sum. The three interior angles of any triangle sum to 180°. So if ∠R = 63°, then ∠S + ∠T = 117°, forcing ∠S < 117°.

Rectangle inscribed in a circle. The diagonal of the rectangle is a diameter of the circle. If the diagonal is twice the shortest side, the rectangle is a 30-60-90 arrangement (sides s, s√3, hypotenuse 2s). Set area = s · s√3 = s²√3.

Similar figures & area scaling. If linear dimensions scale by k, areas scale by k². Sides × 6 ⇒ area × 36.

Circles in the xy-plane: tangent-radius perpendicularity. At the point of tangency, the radius is perpendicular to the tangent line. The tangent slope is the negative reciprocal of the radius slope.

Electric flux with a proportional area split. When a uniform electric field passes through a flat surface split into pieces, each piece's flux is proportional to its area. If the larger square has area 9× the smaller one, the larger square carries 9/10 of the total flux.

Every question below is transcribed verbatim from College Board’s Practice Test 5 booklet. Try each one on paper, then click Show answer & explanation to see the correct answer, the full worked solution, and a plain-English explanation for every choice.

Section 1 · The Full 27 Questions

All 27 Module 2 questions, in the order they appear in the College Board booklet.

Question 1 · Percent-of

What is 20% of 440?

  • A) 44
  • B) 88
  • C) 880
  • D) 1,760

Correct answer: B) 88

20% of 440 = (20/100) × 440 = 0.20 × 440 = 88

Choice A is incorrect and would result from computing 10% of 440.

Choice B is correct: 0.20 × 440 = 88.

Choice C is incorrect and would result from multiplying by 2 instead of 0.20.

Choice D is incorrect and would result from multiplying by 4 instead of 0.20.

Question 2 · Reading a Line Graph

See figure in the original College Board PDF above.

Argon is placed inside a container with a constant volume. The graph shows the estimated pressure y, in pounds per square inch (psi), of the argon when its temperature is x kelvins. What is the estimated pressure of the argon, in psi, when the temperature is 600 kelvins?

  • A) 6
  • B) 12
  • C) 300
  • D) 600

Correct answer: B) 12

Trace up from x = 600 on the horizontal (temperature) axis to the line, then across to the vertical (pressure) axis. The line hits about y = 12 psi at x = 600 K.

Choice A is incorrect — 6 psi corresponds to about 300 K, not 600 K.

Choice B is correct: the graph shows pressure ≈ 12 psi when temperature = 600 K.

Choice C confuses temperature and pressure axes.

Choice D confuses temperature and pressure axes.

Question 3 · Function Evaluation

The function f is defined by f(x) = 4x − 3. What is the value of f(10)?

  • A) −30
  • B) 37
  • C) 40
  • D) 43

Correct answer: B) 37

f(x) = 4x − 3 f(10) = 4(10) − 3 = 40 − 3 = 37

Choice A is incorrect — would result from computing −4(10) + 10 = −30 or another sign error.

Choice B is correct: 4(10) − 3 = 37.

Choice C is incorrect — skips subtracting 3.

Choice D is incorrect — adds 3 instead of subtracting.

Question 4 · Factoring by GCF

Which expression is equivalent to 16x³y² + 14xy?

  • A) 2xy(8xy + 7)
  • B) 2xy(8x²y + 7)
  • C) 14xy(2x²y + 1)
  • D) 14xy(8x²y + 1)

Correct answer: B) 2xy(8x²y + 7)

GCF of 16x³y² and 14xy: coefficient GCF = 2, x-power GCF = x, y-power GCF = y ⇒ GCF = 2xy. 16x³y² = (2xy) · (8x²y) 14xy = (2xy) · 7 16x³y² + 14xy = 2xy(8x²y + 7)

Choice A keeps only one x factor inside the parentheses instead of x².

Choice B is correct: the GCF is 2xy, and the leftover factors are 8x²y and 7.

Choice C pulls out too much (14xy is not a common factor of both terms).

Choice D pulls out too much and also mishandles the leftover from the second term.

Question 5 · Linear Word Problem

A veterinarian recommends that each day a certain rabbit should eat 25 calories per pound of the rabbit’s weight, plus an additional 11 calories. Which equation represents this situation, where c is the total number of calories the veterinarian recommends the rabbit should eat each day if the rabbit’s weight is x pounds?

  • A) c = 25x
  • B) c = 36x
  • C) c = 11x + 25
  • D) c = 25x + 11

Correct answer: D) c = 25x + 11

Rate: 25 calories per pound → 25x Fixed additional: 11 calories Total calories c = (rate)(pounds) + fixed = 25x + 11

Choice A omits the additional 11 calories.

Choice B incorrectly combines 25 and 11 into a single rate.

Choice C swaps rate and constant.

Choice D is correct: rate 25 per pound, plus a fixed 11.

Question 6 · Linear Equation (SPR)

If 6n = 12, what is the value of n + 4?

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

Correct answer: 6

6n = 12 n = 12 / 6 = 2 n + 4 = 2 + 4 = 6

Solve the linear equation for n first, then substitute into the expression you were asked to evaluate. A very common mistake is to solve 6n = 12 and stop — you must also add 4.

Question 7 · Difference of Squares (SPR)

(d − 30)(d + 30) − 7 = −7. What is a solution to the given equation?

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

Correct answer: −30 or 30

(d − 30)(d + 30) − 7 = −7 Add 7 to both sides: (d − 30)(d + 30) = 0 Either d − 30 = 0 ⇒ d = 30 or d + 30 = 0 ⇒ d = −30

The Zero-Product Property says if a product of two factors equals 0, at least one factor equals 0. So either d = 30 or d = −30 is a valid answer.

Question 8 · Slope-Intercept Form

Line r in the xy-plane has a slope of 4 and passes through the point (0, 6). Which equation defines line r?

  • A) y = −6x + 4
  • B) y = 6x + 4
  • C) y = 4x − 6
  • D) y = 4x + 6

Correct answer: D) y = 4x + 6

Slope-intercept form: y = mx + b Slope m = 4, y-intercept b = 6 (since line passes through (0, 6)) y = 4x + 6

Choice A has slope −6 (not 4) and y-intercept 4 (not 6).

Choice B has slope 6 (not 4) and wrong y-intercept.

Choice C has correct slope but wrong y-intercept (−6 vs. 6).

Choice D is correct: slope 4, y-intercept 6.

Question 9 · x-Intercept Interpretation

See figure in the original College Board PDF above.

A competitive diver dives from a platform into the water. The graph shown gives the height above the water y, in meters, of the diver x seconds after diving from the platform. What is the best interpretation of the x-intercept of the graph?

  • A) The diver reaches a maximum height above the water at 1.6 seconds.
  • B) The diver hits the water at 1.6 seconds.
  • C) The diver reaches a maximum height above the water at 0.2 seconds.
  • D) The diver hits the water at 0.2 seconds.

Correct answer: B) The diver hits the water at 1.6 seconds.

x-intercept is where y = 0. Here y = height above water; y = 0 ⇒ diver at water level. From the graph, the curve crosses the x-axis at x ≈ 1.6. So the diver hits the water at 1.6 seconds.

Choice A confuses the x-intercept with the vertex (max height).

Choice B is correct: x-intercept = when height = 0 = diver at the water.

Choice C confuses x-intercept with vertex and reads the wrong time.

Choice D misreads the x-intercept as 0.2 seconds — the diver is still climbing at 0.2 s.

Question 10 · Interpreting Function Values

The kinetic energy, in joules, of an object with mass 9 kilograms traveling at a speed of v meters per second is given by the function K, where K(v) = (9/2)v². Which of the following is the best interpretation of K(34) = 5,202 in this context?

  • A) The object traveling at 34 meters per second has a kinetic energy of 5,202 joules.
  • B) The object traveling at 340 meters per second has a kinetic energy of 5,202 joules.
  • C) The object traveling at 5,202 meters per second has a kinetic energy of 34 joules.
  • D) The object traveling at 23,409 meters per second has a kinetic energy of 34 joules.

Correct answer: A) 34 m/s → 5,202 J

In K(v), v is the speed input (m/s) and K(v) is the output (J). K(34) = 5,202 means: input v = 34 m/s output K = 5,202 J Check: K(34) = (9/2)(34²) = 4.5 · 1,156 = 5,202. ✓

Choice A is correct: v = 34 m/s produces K = 5,202 J.

Choice B uses a wrong speed (340 instead of 34).

Choice C swaps the input and output.

Choice D swaps input and output and squares the wrong number.

Question 11 · Scatterplot: Actual vs. Predicted

See figure in the original College Board PDF above.

The scatterplot shows the relationship between two variables x and y. A line of best fit for the data is also shown. For how many of the 10 data points is the actual y-value greater than the y-value predicted by the line of best fit?

  • A) 3
  • B) 4
  • C) 6
  • D) 7

Correct answer: C) 6

Count only the data points that lie STRICTLY ABOVE the line of best fit (points on the line don't count). From the scatterplot, 6 of the 10 points sit above the line.

Choice A undercounts — would result from counting only the clearly obvious "above" points.

Choice B undercounts.

Choice C is correct: 6 of 10 points are above the line of best fit.

Choice D overcounts — would happen if you count points on the line as "above."

Question 12 · Probability & Complement

At a movie theater, there are a total of 350 customers. Each customer is located in either theater A, theater B, or theater C. If one of these customers is selected at random, the probability of selecting a customer who is located in theater A is 0.48, and the probability of selecting a customer who is located in theater B is 0.24. How many customers are located in theater C?

  • A) 28
  • B) 40
  • C) 84
  • D) 98

Correct answer: D) 98

P(A) + P(B) + P(C) = 1 0.48 + 0.24 + P(C) = 1 P(C) = 1 − 0.72 = 0.28 Customers in C = 0.28 × 350 = 98

Choice A is 8% of 350 — wrong probability.

Choice B would be 0.114 × 350 — not derived from a valid complement.

Choice C is 24% of 350 (theater B, not C).

Choice D is correct: P(C) = 0.28, so 0.28 × 350 = 98.

Question 13 · Slope by Combining Like Terms (SPR)

What is the slope of the graph of y = (1/3)(29x + 10) + 5x in the xy-plane?

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

Correct answer: 44/3 (or 14.66 / 14.67)

Distribute the 1/3: y = (29/3)x + 10/3 + 5x Rewrite 5x as (15/3)x to combine: y = (29/3 + 15/3)x + 10/3 y = (44/3)x + 10/3 Slope = 44/3 ≈ 14.67

In y = mx + b, the coefficient of x is the slope. After distributing 1/3 and combining the two x-terms with a common denominator, the coefficient of x is 44/3. College Board accepts 44/3, 14.66, and 14.67 as correct entries.

Question 14 · Surface Area of an Open Box (SPR)

The length of each edge of a box is 29 inches. Each side of the box is in the shape of a square. The box does not have a lid. What is the exterior surface area, in square inches, of this box without a lid?

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

Correct answer: 4,205

A full cube with edge s has 6 square faces of area s². A box without a lid has only 5 faces (bottom + 4 sides). Exterior area = 5 · s² = 5 · 29² = 5 · 841 = 4,205 sq in

The trap is forgetting that "no lid" removes one face. Full box surface area would be 6 · 841 = 5,046, but only 5 faces contribute here: 5 · 841 = 4,205.

Question 15 · Effect of Adding a Data Point on the Mean

See figure in the original College Board PDF above.

Five Eretmochelys imbricata, a type of sea turtle, each have a nest. The table shows an original data set of the number of eggs that each turtle laid in its nest (A: 149, B: 144, C: 148, D: 136, E: 139). A sixth nest with 121 eggs is added to create a new data set. Which of the following correctly compares the means of the two data sets?

  • A) The mean of the original data set is greater than the mean of the new data set.
  • B) The mean of the original data set is less than the mean of the new data set.
  • C) The means of both data sets are equal.
  • D) There is not enough information to compare the means.

Correct answer: A) Original > New

Original mean = (149+144+148+136+139) / 5 = 716 / 5 = 143.2 New mean = (716 + 121) / 6 = 837 / 6 = 139.5 143.2 > 139.5, so original mean > new mean.

Choice A is correct: since the new value (121) is below the original mean (143.2), adding it pulls the mean down.

Choice B is the opposite of what happens.

Choice C would require the new value to equal the original mean exactly.

Choice D is wrong — every needed number is provided.

Question 16 · Triangle Angle Sum

In ▵RST, the measure of ∠R is 63°. Which of the following could be the measure, in degrees, of ∠S?

  • A) 116
  • B) 118
  • C) 126
  • D) 180

Correct answer: A) 116

The three angles of any triangle sum to 180°. ∠R + ∠S + ∠T = 180 63 + ∠S + ∠T = 180 ∠S + ∠T = 117 Since ∠T > 0, we need ∠S < 117°. Only 116° is less than 117°.

Choice A is correct: 116° leaves 1° for ∠T, which is still positive.

Choice B would force ∠T = −1° (impossible).

Choice C would force ∠T = −9° (impossible).

Choice D would force the other two angles to sum to 0 (impossible).

Question 17 · Simplifying Polynomials

Which expression is equivalent to (8x³ + 8) − (x³ − 2)?

  • A) 8x³ + 6
  • B) 7x³ + 10
  • C) 8x³ + 10
  • D) 7x³ + 6

Correct answer: B) 7x³ + 10

(8x³ + 8) − (x³ − 2) = 8x³ + 8 − x³ + 2 (distribute the − sign) = (8x³ − x³) + (8 + 2) = 7x³ + 10

Choice A drops the coefficient combination on the cubic term.

Choice B is correct: 8x³ − x³ = 7x³ and 8 − (−2) = 10.

Choice C forgets to subtract x³ from 8x³.

Choice D fails to distribute the negative onto the constant term.

Question 18 · Radical Equation

If 4√(2x) = 16, what is the value of 6x?

  • A) 24
  • B) 48
  • C) 72
  • D) 96

Correct answer: B) 48

4√(2x) = 16 Divide both sides by 4: √(2x) = 4 Square both sides: 2x = 16 x = 8 6x = 6 · 8 = 48

Choice A is 6 · 4 — forgets the final square step.

Choice B is correct: x = 8, so 6x = 48.

Choice C uses x = 12 (wrong intermediate).

Choice D uses x = 16 (skipped dividing by 4).

Question 19 · Systems of Linear Inequalities (Tables)

See figure in the original College Board PDF above.

2x − y > 883. For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?

  • A) (440, 0), (441, −2), (442, −4)
  • B) (440, 0), (442, −2), (441, −4)
  • C) (442, 0), (440, −2), (441, −4)
  • D) (442, 0), (441, −2), (440, −4)

Correct answer: D) Table D

For each row, compute 2x − y and check if it exceeds 883. Table D: (442, 0): 2(442) − 0 = 884 > 883 ✓ (441, −2): 2(441) − (−2) = 882 + 2 = 884 > 883 ✓ (440, −4): 2(440) − (−4) = 880 + 4 = 884 > 883 ✓ All three rows satisfy the inequality ⇒ Table D.

Choice A fails at (440, 0): 2(440) − 0 = 880, which is NOT greater than 883.

Choice B fails at (440, 0) as well.

Choice C fails at (440, −2): 880 + 2 = 882, not greater than 883.

Choice D is correct: every row gives 2x − y = 884 > 883.

Question 20 · System of Linear Equations (SPR)

5y = 10x + 11; −5y = 5x − 21. The solution to the given system of equations is (x, y). What is the value of 30x?

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

Correct answer: 20

Add the two equations to eliminate y: 5y = 10x + 11 −5y = 5x − 21 ------------------ 0 = 15x − 10 15x = 10 x = 10/15 = 2/3 30x = 30 · (2/3) = 20

Adding the equations eliminates y in one step because the coefficients (5 and −5) are opposites. Then multiply x by 30 to get the requested value.

Question 21 · Rectangle Inscribed in a Circle (SPR)

A rectangle is inscribed in a circle, such that each vertex of the rectangle lies on the circumference of the circle. The diagonal of the rectangle is twice the length of the shortest side of the rectangle. The area of the rectangle is 1,089√3 square units. What is the length, in units, of the diameter of the circle?

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

Correct answer: 66

Let s = shortest side. Diagonal = 2s (given). By Pythagoras: s² + (long side)² = (2s)² = 4s² ⇒ (long side)² = 3s² ⇒ long side = s√3 (This is the classic 30-60-90 triangle: legs s and s√3, hypotenuse 2s.) Area = s · s√3 = s²√3 s²√3 = 1,089√3 s² = 1,089 s = 33 Diagonal = 2s = 66. And in a rectangle inscribed in a circle, the diagonal IS the diameter, so diameter = 66.

The diagonal of any rectangle inscribed in a circle equals the diameter of that circle. Set up the area with the 30-60-90 side ratios, solve for s = 33, then diameter = 2s = 66.

Question 22 · Similar Figures & Area Scaling

Rectangles ABCD and EFGH are similar. The length of each side of EFGH is 6 times the length of the corresponding side of ABCD. The area of ABCD is 54 square units. What is the area, in square units, of EFGH?

  • A) 9
  • B) 36
  • C) 324
  • D) 1,944

Correct answer: D) 1,944

For similar figures, if linear sides scale by k, areas scale by k². k = 6 ⇒ area scale factor = 6² = 36. Area(EFGH) = 36 · area(ABCD) = 36 · 54 = 1,944

Choice A is 54 ÷ 6 — wrong direction and wrong operation.

Choice B uses only the linear factor 6 (not 6²).

Choice C is 54 · 6 — multiplies by the linear factor, not the area factor.

Choice D is correct: 54 · 36 = 1,944.

Question 23 · Rational Expressions

Which expression is equivalent to 42a/k + 42ak, where k > 0?

  • A) 84a/k
  • B) 84ak²/k
  • C) 42a(k+1)/k
  • D) 42a(k²+1)/k

Correct answer: D) 42a(k²+1)/k

42a/k + 42ak Rewrite second term with denominator k: 42a/k + 42ak · (k/k) = 42a/k + 42ak²/k = (42a + 42ak²) / k = 42a(1 + k²) / k = 42a(k² + 1) / k

Choice A just adds numerators as if the two terms had the same denominator.

Choice B equals just 84ak — ignores the first term.

Choice C forgets to square k when rewriting the second term with common denominator.

Choice D is correct: common denominator k gives (42a + 42ak²)/k = 42a(k² + 1)/k.

Question 24 · Discriminant & Real Solutions

Which quadratic equation has no real solutions?

  • A) x² + 14x − 49 = 0
  • B) x² − 14x + 49 = 0
  • C) 5x² − 14x − 49 = 0
  • D) 5x² − 14x + 49 = 0

Correct answer: D) 5x² − 14x + 49 = 0

No real solutions ⇔ discriminant b² − 4ac < 0. A) a=1, b=14, c=−49 → 196 + 196 = 392 > 0 B) a=1, b=−14, c=49 → 196 − 196 = 0 (one real root) C) a=5, b=−14, c=−49 → 196 + 980 = 1,176 > 0 D) a=5, b=−14, c=49 → 196 − 980 = −784 < 0 ✓ Only D has b² − 4ac < 0.

Choice A has discriminant 392 > 0 (two real solutions).

Choice B has discriminant 0 (exactly one real solution — a repeated root, not zero solutions).

Choice C has discriminant 1,176 > 0 (two real solutions).

Choice D is correct: discriminant is −784 < 0, so no real solutions.

Question 25 · Exponential Growth & Time Unit

P(t) = 260(1.04)^((6/4)t). The function P models the population, in thousands, of a certain city t years after 2003. According to the model, the population is predicted to increase by 4% every n months. What is the value of n?

  • A) 8
  • B) 12
  • C) 18
  • D) 72

Correct answer: A) 8

The population is multiplied by 1.04 (a 4% increase) every time the exponent (6/4)t increases by 1. (6/4)t = 1 t = 4/6 = 2/3 year Convert 2/3 year to months: (2/3) · 12 = 8 months.

Choice A is correct: t = 2/3 year = 8 months.

Choice B would be 1 year — ignores the exponent's (6/4) multiplier.

Choice C is 1.5 years — inverts the exponent's multiplier.

Choice D is 6 years — misreads the exponent structure.

Question 26 · Tangent Line to a Circle

A circle in the xy-plane has its center at (−1, 1). Line t is tangent to this circle at the point (5, −4). Which of the following points also lies on line t?

  • A) (0, 6/5)
  • B) (4, 7)
  • C) (10, 2)
  • D) (11, 1)

Correct answer: C) (10, 2)

Slope of radius from (−1, 1) to (5, −4): (−4 − 1) / (5 − (−1)) = −5 / 6 At the point of tangency, the tangent is perpendicular to the radius. Slope of tangent = negative reciprocal of −5/6 = 6/5. Tangent line through (5, −4) with slope 6/5: y − (−4) = (6/5)(x − 5) y + 4 = (6/5)(x − 5) y = (6/5)x − 6 − 4 y = (6/5)x − 10 Test (10, 2): (6/5)(10) − 10 = 12 − 10 = 2 ✓

Choice A is on the y-axis but not on the tangent line ((6/5)(0) − 10 = −10 ≠ 6/5).

Choice B would require y = 24/5 − 10 = −26/5, not 7.

Choice C is correct: y = (6/5)(10) − 10 = 2.

Choice D would require y = 66/5 − 10 = 16/5, not 1.

Question 27 · Proportional Areas & Electric Flux (SPR)

For an electric field passing through a flat surface perpendicular to it, the electric flux of the electric field through the surface is the product of the electric field’s strength and the area of the surface. A certain flat surface consists of two adjacent squares, where the side length, in meters, of the larger square is 3 times the side length, in meters, of the smaller square. An electric field with strength 29.00 volts per meter passes uniformly through this surface, which is perpendicular to the electric field. If the total electric flux of the electric field through this surface is 4,640 volts·meters, what is the electric flux, in volts·meters, of the electric field through the larger square?

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

Correct answer: 4,176

Let smaller side = s. Then larger side = 3s. Areas: smaller = s²; larger = (3s)² = 9s². Total area = s² + 9s² = 10s². Fraction of total area from larger square = 9s² / 10s² = 9/10. Since the field passes uniformly (same strength everywhere), flux is proportional to area: Flux(larger) = (9/10) · total flux = (9/10) · 4,640 = 4,176 volts·meters

Because the field strength is the same over both squares, flux splits in the same ratio as area. The larger square is 9/10 of the total area, so it carries 9/10 of the total flux: (9/10)(4,640) = 4,176. You do not need the explicit 29.00 V/m or the side length — only the area ratio.

How SOMATH prepares NYC students for the Digital SAT

Small-group, in-person Digital SAT prep on the Upper West Side of Manhattan.

SOMATH (School of Math) is a math-focused school on the Upper West Side of Manhattan, at 226 W 79th Street, 1st Floor, right off Broadway between the 1 and 2/3 trains. Our SAT program is built by two educators who have taken the test at the highest levels themselves: Marcelo Ambrozio, a Northwestern-trained engineer and math instructor, and Vivianne Wright, a Harvard alumna and long-time MBA admissions consultant. We use walkthroughs like this one — every question from every official College Board test, worked in full — as the backbone of our small-group SAT Math prep classes (6–8 students max).

Every family that joins SOMATH begins with a free 30-minute in-person diagnostic evaluation. In that session, a SOMATH teacher works one-on-one with your student to identify the exact skills that are keeping their SAT Math score below where it should be — whether that is quadratic manipulation, coordinate-geometry setup, exponential-growth interpretation, or something else — and then explains which of our small-group tracks (SAT, Kid Einsteins, Young Fermats, or a customized private track) will move them fastest. There is no obligation to enroll. If we don’t think we can help, we’ll tell you honestly, and we’ll recommend somewhere else that might be a better fit.

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Book a free 30-minute SAT Math evaluation with SOMATH In-person at 226 W 79th Street on the Upper West Side. No enrollment fee, no obligation, no charge for the evaluation.
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SOMATH course · Grades 9–12

Want your child in an SAT Math class at SOMATH?

Small-group Digital SAT Math prep on the Upper West Side. Full Bluebook + Desmos drilling, timed Modules 1 & 2, weekly score tracking, and a free 30-minute in-person diagnostic before enrollment.

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Digital SAT FAQ

What is on Module 2 of the Digital SAT Math section?

Module 2 of the Digital SAT Math section is a 35-minute, 22-question module (20 scored + 2 pretest). Unlike Module 1, Module 2 is stage-adaptive: the difficulty of the questions you see depends on how you performed on Module 1. It draws from the same four content domains as Module 1 — Algebra, Advanced Math, Problem-Solving & Data Analysis, and Geometry & Trigonometry — but the harder Module 2 form leans more heavily on multi-step algebra, quadratics, exponentials, and coordinate geometry. The Bluebook Desmos calculator is available on every question.

Is Digital SAT Math Module 2 harder than Module 1?

It depends on the student. Module 2 is adaptive: students who score well on Module 1 are routed into a harder version of Module 2 that unlocks access to the full 200–800 scoring range, while students who score lower are routed into an easier version of Module 2 that typically caps in the mid-600s. Practice Test 5’s Module 2, as reproduced in this walkthrough, is the higher (harder) form.

How is the Digital SAT Math section scored?

The Digital SAT Math section is scored from 200 to 800. Module 1 is the same for every student; Module 2 is stage-adaptive — students who perform well on Module 1 are routed to a harder Module 2 with access to the full 800, while students who did not are routed to an easier Module 2 that typically caps in the mid-600s.

Can I use a calculator on Digital SAT Math?

Yes. Every question on the Digital SAT Math section allows a calculator. The Bluebook app includes a built-in Desmos graphing calculator on every question, and you may also bring an approved handheld calculator. Learning to use Desmos efficiently (solving systems, finding zeros, checking answers by graphing) can save 30–60 seconds per question.

What kinds of questions appear on Digital SAT Math Module 2?

About 75% of items are multiple choice (four options each) and 25% are student-produced responses (fill-in-the-blank numeric or fractional answers). Topics include linear equations and systems, quadratic equations (factoring, discriminant, no-real-solutions), exponential growth and doubling-time, function evaluation, similar figures and area scaling, probability from a total, tangent lines and circles in the xy-plane, radicals and rational expressions, and surface area of composite solids.

What is a good Digital SAT Math score?

As a general reference, 600 is around the 75th percentile nationally, 700 is around the 90th percentile, and 750+ is in the top 5%. For MIT, Stanford, Harvard, Ivy League and other highly selective schools, competitive applicants typically score 780+ on Math. For strong state universities such as SUNY Binghamton or UNC, 720+ is competitive. For most CUNY campuses, 600–680 is competitive.

How does SOMATH prepare NYC students for the Digital SAT Math section?

SOMATH is a math-focused school on the Upper West Side of Manhattan led by Northwestern-trained cofounder Marcelo Ambrozio and Harvard-trained cofounder Vivianne Wright. Our SAT Math program includes small-group in-person sessions (6–8 students max) that drill every question type using official College Board practice tests, including this Practice Test 5 walkthrough. We also teach systematic use of the Bluebook Desmos calculator to solve items 2–3 times faster than by hand. Families start with a free 30-minute in-person diagnostic evaluation.

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