Digital SAT · Math · Exam Prep · NYC Test Prep

Digital SAT Practice Test 5 — Module 1: Full Walkthrough of All 27 Math Questions with Answers & Explanations

Every question on Digital SAT Practice Test #5, Math Module 1, 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 1? You are in the right place. This is a complete, question-by-question walkthrough of SAT Test #5, Math Module 1 — 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 practice test for offline studying.

Module 1 is the first Math module on the Digital SAT, and unlike Module 2, it is not adaptive: every student sees the same questions. Your performance on Module 1 is what routes you into the harder or the easier version of Module 2. That means Module 1 is arguably the single most important 35 minutes of the Math section — and drilling it question by question is the fastest way to raise your composite score.

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 Digital PDF) The complete official College Board practice test, including Math Module 1. Ready to print and use alongside this walkthrough.
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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, exponent rules, Pythagorean theorem, discriminant, 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 1 tests
  3. Section 1 · Algebra & Linear Systems
  4. Section 2 · Advanced Math & Functions
  5. Section 3 · Geometry, Trig & Circles
  6. How SOMATH prepares NYC students for the Digital SAT
  7. Digital SAT FAQ

Answer key at a glance

All 27 official College Board answers for Practice Test 5, Math Module 1, in one table for quick review:

QAnswerQAnswerQAnswer
1D) (5, 0)10B) 32019A) −√3
2B) 13011C) −62011/28 (or .3928, .3929)
3D) (0, 8)12D) 0.6 ≤ s ≤ 1.821336
4B) (3, 6)13622B) −14
5B) Increasing linear144.5123D) V(x) = 9x(x − 7)
6415C) 726π24D) Neither I nor II
72916D)25A) 12
8A) The value of x is less than 145.17B) 26x26A) −6
9B) 77018D) Kaylani purchased 6 yards more fabric than she used to make the suits.2725

Theory refresher · What Math Module 1 tests

The Digital SAT Math section has two 35-minute modules of 22 questions each (20 scored + 2 pretest). Module 1 is not adaptive — every student sees the same items. 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

Linear equations in one variable. Isolate the variable using inverse operations. Watch for two-step traps like “what is the value of x − 7” — once you have x, do not stop; finish the extra step the prompt asks for.

Systems of linear equations. Substitution is fastest when one variable is already isolated (e.g. r = 3). Otherwise use elimination: line up matching coefficients and add or subtract to cancel a variable.

Interpreting linear models & graphs. In y = mx + b, the slope m is the rate of change and b is the starting value. To read a y-intercept off a graph, look at where the line crosses the y-axis (x = 0).

Ratios & proportions. A ratio like “coaches to athletes is 1 to 26” means athletes = 26 · coaches. If there are x coaches, there are 26x athletes.

2 · Advanced Math

Function notation. If the graph of y = g(x) passes through (24, 0), then g(24) = 0 — plug 24 into the definition and solve for the constant.

Vertical translations. y = f(x) + 5 shifts every point on the graph up by 5 units. Every y-value increases by 5; x-values are unchanged.

Exponential functions. f(x) = a · bx: a is the value at x = 0. Exponential functions with positive base b take every value between their horizontal asymptote and their vertical extreme — they have no minimum value in the interior.

Quadratics & factoring. If f(x) = ax² + bx + c has roots 7 and −3, factor as f(x) = a(x − 7)(x + 3). Expanding gives b = −4a and c = −21a.

Repeated roots & substitution. If g(x) has roots r1, r2, r3 and g(7 − w) = 0, then 7 − w equals one of those roots — giving one w per distinct root.

3 · Problem-Solving & Data Analysis

Percent increase. Increasing x by 400% means adding 4x, giving 5x. So “5x = 60” ⇒ x = 12.

Sampling. If a random sample of 20 gives 16 successes (80%), scale to the full population of 400: 0.8 · 400 = 320.

Range. Range of a data set = maximum − minimum. Order the numbers first if they are not sorted.

Rates. Total cost = fixed fee + rate · quantity. Solve for one unknown given the others.

4 · Geometry & Trigonometry

Parallel lines & transversals. When line k crosses parallel lines m and n, corresponding and alternate interior angles are equal, and angles on a straight line sum to 180°.

Right-triangle trigonometry. For angle θ in a right triangle: cos θ = adjacent leg ÷ hypotenuse. Identify the hypotenuse first (opposite the right angle), then the adjacent leg (the one touching θ that is not the hypotenuse).

Unit-circle values. tan(θ) is periodic with period π. Reduce θ mod π before evaluating: tan(92π/3) ≡ tan(2π/3) = −√3.

Circles. The circle (x − h)² + (y − k)² = r² has center (h, k) and radius r. Any point (a, b) on it has h − r ≤ a ≤ h + r.

Volumes. Cylinder: V = πr²h. Right rectangular prism: V = ℓ · w · h. Diameter is twice the radius.

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. Questions with figures — graphs, tables, triangles, and diagrams — are marked with a note pointing you to the exact figure in the official Practice Test 5 PDF above.

Section 1 · Algebra & Linear Systems

Systems, linear functions, ratios, sampling, and interpreting equations in context.

Question 1 · Systems (Linear + Nonlinear)

See figure in the original College Board PDF above (linear and exponential curve).

The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution (x, y) to this system?

  • A) (0, 0)
  • B) (0, 4)
  • C) (4, 5)
  • D) (5, 0)

Answer: D) (5, 0)

The solution of a system on the coordinate plane is the point where the two graphs intersect. The line and the nonlinear curve meet on the x-axis at x = 5, y = 0. So the solution is (5, 0).

✗ A) (0, 0)
The origin is not on either graph — neither the line nor the curve passes through (0, 0).

✗ B) (0, 4)
Only the curve appears to pass near this point, but the line does not — not a shared point.

✗ C) (4, 5)
The curve is near (4, 5), but the line is not — the two graphs do not meet here.

✓ D) (5, 0)
Both graphs cross the x-axis at (5, 0). This is where they intersect, so it is the solution to the system.

Question 2 · Arithmetic Sequences / Linear Growth

On the first day of a semester, a film club has 90 members. Each day after the first day of the semester, 10 new members join the film club. If no members leave the film club, how many total members will the film club have 4 days after the first day of the semester?

  • A) 400
  • B) 130
  • C) 94
  • D) 90

Answer: B) 130

Starting total = 90. Each of the 4 days adds 10 new members. Total after 4 days = 90 + 4(10) = 90 + 40 = 130.

✗ A) 400
Would require ~31 days of growth, not 4 — way too many additions.

✓ B) 130
90 starting + 10 per day × 4 days = 130.

✗ C) 94
Adds only 4 total members — treats 10 members/day as 1 member/day.

✗ D) 90
Ignores the 4 days of new members entirely.

Question 3 · Linear Functions · y-intercept

See figure in the original College Board PDF above (line with negative slope).

The graph of the linear function f is shown, where y = f(x). What is the y-intercept of the graph of f?

  • A) (0, 0)
  • B) (0, −16/11)
  • C) (0, −8)
  • D) (0, 8)

Answer: D) (0, 8)

The y-intercept is the point where the graph crosses the y-axis (x = 0). The line in the figure crosses the y-axis at y = 8, so the y-intercept is (0, 8).

✗ A) (0, 0)
The line does not pass through the origin.

✗ B) (0, −16/11)
A negative y-intercept, but the graph clearly crosses the y-axis above zero.

✗ C) (0, −8)
Correct magnitude, wrong sign — the line does not cross the negative y-axis.

✓ D) (0, 8)
The graph crosses the y-axis at 8. That is the y-intercept.

Question 4 · Systems by Substitution

s + 7r = 27, r = 3. What is the solution (r, s) to the given system of equations?

  • A) (6, 3)
  • B) (3, 6)
  • C) (3, 27)
  • D) (27, 3)

Answer: B) (3, 6)

Substitute r = 3 into the first equation: s + 7(3) = 27 s + 21 = 27 s = 6. So the solution is (r, s) = (3, 6).

✗ A) (6, 3)
Swaps r and s in the answer pair.

✓ B) (3, 6)
r = 3 is given; substituting yields s = 6. The pair (r, s) is (3, 6).

✗ C) (3, 27)
Uses 27 (the RHS of the first equation) instead of solving for s.

✗ D) (27, 3)
Confuses r with 27 and puts s = 3.

Question 5 · Linear vs. Exponential

See the table of x-values (−1, 0, 1, 2) and f(x)-values (16, 17, 18, 19) in the original College Board PDF above.

The table shows selected values from function f. Which of the following is the best description of function f?

  • A) Decreasing linear
  • B) Increasing linear
  • C) Decreasing exponential
  • D) Increasing exponential

Answer: B) Increasing linear

Check the differences between consecutive f(x) values: 17 − 16 = 1 18 − 17 = 1 19 − 18 = 1. The differences are constant → linear. The values are going up → increasing. So f is increasing linear.

✗ A) Decreasing linear
The values are going up, not down.

✓ B) Increasing linear
Constant common difference (+1) each step — the hallmark of a linear function — and increasing.

✗ C) Decreasing exponential
Exponential functions have a constant common ratio, not a constant difference. And these values are increasing, not decreasing.

✗ D) Increasing exponential
Would require a common ratio (e.g., ×2 each step). Here 17/16 ≠ 18/17 ≠ 19/18 — so it is not exponential.

Question 6 · Systems from a Graph

See figure in the original College Board PDF above (two lines crossing).

The graph of a system of linear equations is shown. The solution to the system is (x, y). What is the value of x?

Student-produced response — enter as a fraction or decimal.

Answer: 4

The solution to a system of two linear equations shown as a graph is the intersection point of the two lines. Reading the graph, the two lines intersect at the point (4, −1). The x-coordinate of that intersection is x = 4.

Why this works.
Every point on a line satisfies that line’s equation. The intersection satisfies both equations at once, which is exactly what “solution to the system” means. Read the x-coordinate of the crossing point directly off the graph.

Question 7 · Statistics · Range

23, 27, 27, 32, 35, 36, 52. What is the range of the 7 scores shown?

Student-produced response — enter as a fraction or decimal.

Answer: 29

Range = maximum − minimum. Maximum value = 52. Minimum value = 23. Range = 52 − 23 = 29.

Why this works.
Range measures the spread of a data set as the difference between its largest and smallest values. Sort or scan the list, pick the max (52) and min (23), and subtract.

Question 8 · Parallel Lines & Angles

See figure in the original College Board PDF above (line k crossing parallel lines m and n; angle marked 145°). Note: figure not drawn to scale.

In the figure, line m is parallel to line n, and line k intersects both lines. Which of the following statements is true?

  • A) The value of x is less than 145.
  • B) The value of x is greater than 145.
  • C) The value of x is equal to 145.
  • D) The value of x cannot be determined.

Answer: A) The value of x is less than 145.

Lines m and n are parallel and both cut by transversal k. The 145° angle and the angle x° sit on the same side of the transversal, at line n, and together form a linear pair on line n. Angles on a straight line sum to 180°: x + 145 = 180 x = 35. 35 < 145, so x is less than 145.

✓ A) The value of x is less than 145.
x = 35, which is less than 145.

✗ B) The value of x is greater than 145.
x = 35 is less than 145, not greater.

✗ C) The value of x is equal to 145.
x = 145 would require x and 145° to be equal angles (corresponding or alternate). They are supplementary, not equal.

✗ D) The value of x cannot be determined.
It can — the parallel-line angle rules pin x exactly at 35°.

Question 9 · Linear Equations in 2 Vars

The equation x + y = 1,440 represents the number of minutes of daylight (between sunrise and sunset), x, and the number of minutes of non-daylight, y, on a particular day in Oak Park, Illinois. If this day has 670 minutes of daylight, how many minutes of non-daylight does it have?

  • A) 670
  • B) 770
  • C) 1,373
  • D) 1,440

Answer: B) 770

Substitute x = 670 into x + y = 1,440: 670 + y = 1,440 y = 1,440 − 670 y = 770.

✗ A) 670
That is the daylight amount, not the non-daylight amount.

✓ B) 770
1,440 − 670 = 770 minutes of non-daylight.

✗ C) 1,373
Does not come from any correct arithmetic on the given values.

✗ D) 1,440
That is the total minutes in a day (24 × 60), not the non-daylight portion.

Question 10 · Sampling & Proportion

Scott selected 20 employees at random from all 400 employees at a company. He found that 16 of the employees in this sample are enrolled in exactly three professional development courses this year. Based on Scott’s findings, which of the following is the best estimate of the number of employees at the company who are enrolled in exactly three professional development courses this year?

  • A) 4
  • B) 320
  • C) 380
  • D) 384

Answer: B) 320

Sample proportion enrolled in exactly three courses = 16 / 20 = 0.8. Apply this proportion to the whole company of 400: 0.8 × 400 = 320.

✗ A) 4
The number NOT enrolled in exactly three courses in the sample, not scaled to the population.

✓ B) 320
Scaling the 80% success rate to 400 employees gives 320.

✗ C) 380
Does not come from any correct scaling.

✗ D) 384
Would require a 96% rate (384 = 0.96 × 400), inconsistent with the sample 16/20 = 80%.

Question 11 · Linear Equations in 1 Var

If 4x − 28 = −24, what is the value of x − 7?

  • A) −24
  • B) −22
  • C) −6
  • D) −1

Answer: C) −6

Divide the whole equation 4x − 28 = −24 by 4: x − 7 = −6. So x − 7 = −6 directly. (Alternate approach: solve for x first — 4x = 4, x = 1, then x − 7 = 1 − 7 = −6.)

✗ A) −24
That is the right-hand side of the equation, not x − 7.

✗ B) −22
Comes from a subtraction slip — not a valid step.

✓ C) −6
Dividing 4x − 28 = −24 by 4 gives x − 7 = −6 in one step.

✗ D) −1
The value of x is 1 — that is x, not x − 7.

Question 12 · Inequalities in Context

For a snowstorm in a certain town, the minimum rate of snowfall recorded was 0.6 inches per hour, and the maximum rate of snowfall recorded was 1.8 inches per hour. Which inequality is true for all values of s, where s represents a rate of snowfall, in inches per hour, recorded for this snowstorm?

  • A) s ≥ 2.4
  • B) s ≥ 1.8
  • C) 0 ≤ s ≤ 0.6
  • D) 0.6 ≤ s ≤ 1.8

Answer: D) 0.6 ≤ s ≤ 1.8

Every recorded rate is between the minimum and the maximum: minimum = 0.6, maximum = 1.8. Every s satisfies 0.6 ≤ s ≤ 1.8.

✗ A) s ≥ 2.4
The maximum recorded rate was 1.8, so no value ever reached 2.4.

✗ B) s ≥ 1.8
Would exclude all rates below the max — but rates as low as 0.6 were recorded.

✗ C) 0 ≤ s ≤ 0.6
Rates went up to 1.8, so many recorded values exceed 0.6.

✓ D) 0.6 ≤ s ≤ 1.8
Every recorded rate falls between the minimum (0.6) and the maximum (1.8), inclusive.

Question 13 · Nonlinear Systems

y = 4x, y = x² − 12. A solution to the given system of equations is (x, y), where x > 0. What is the value of x?

Student-produced response — enter as a fraction or decimal.

Answer: 6

Set the two expressions for y equal: 4x = x² − 12. Bring everything to one side: x² − 4x − 12 = 0. Factor: (x − 6)(x + 2) = 0. So x = 6 or x = −2. The problem requires x > 0, so x = 6.

Why this works.
At any solution to the system, the y-values match, so 4x = x² − 12. Solving that quadratic gives two roots, but x > 0 selects x = 6.

Question 14 · Interpreting Coefficients

A store sells two different-sized containers of blueberries. The store’s sales of these blueberries totaled 896.86 dollars last month. The equation 4.51x + 6.07y = 896.86 represents this situation, where x is the number of smaller containers sold and y is the number of larger containers sold. According to the equation, what is the price, in dollars, of each smaller container?

Student-produced response — enter as a fraction or decimal.

Answer: 4.51

In 4.51x + 6.07y = 896.86: x = number of smaller containers, so the coefficient of x is the price of one smaller container. Coefficient of x = 4.51. Price of one smaller container = $4.51.

Why this works.
In a total-sales equation of the form (price) · (quantity) + (price) · (quantity) = total, each coefficient is the unit price. Smaller container sits with x, so its price is the 4.51 on x.

Question 15 · Cylinder Volume

A right circular cylinder has a base diameter of 22 centimeters and a height of 6 centimeters. What is the volume, in cubic centimeters, of the cylinder?

  • A) 132π
  • B) 264π
  • C) 726π
  • D) 2,904π

Answer: C) 726π

Diameter = 22, so radius r = 22 / 2 = 11. Height h = 6. V = πr²h = π(11)²(6) = π(121)(6) = 726π.

✗ A) 132π
132 = 22 · 6 = diameter · height — treats diameter as radius and drops the square.

✗ B) 264π
264 = 22² / (something) — not the correct combination for πr²h.

✓ C) 726π
π(11)²(6) = 726π.

✗ D) 2,904π
2,904 = π(22)²(6) — uses the diameter as the radius, which quadruples the correct value.

Section 2 · Advanced Math & Functions

Function translations, ratios, exponents, quadratics, and repeated roots.

Question 16 · Vertical Translation of a Graph

See the four candidate graphs in the original College Board PDF above.

The graph of the rational function f is shown, where y = f(x) and x ≥ 0. Which of the following is the graph of y = f(x) + 5, where x ≥ 0?

  • A) (see figure)
  • B) (see figure)
  • C) (see figure)
  • D) (see figure)

Answer: D)

y = f(x) + 5 shifts every point on the graph of y = f(x) UP by 5 units. The original curve starts high (around y ≈ 12) and decreases toward y ≈ 0 (a horizontal asymptote near the x-axis) as x grows. After shifting up by 5, the curve should start ~5 higher and decrease toward a horizontal asymptote at y ≈ 5. Only choice D shows a curve with the same shape starting high and decreasing to a horizontal asymptote at y = 5 (not at y = 0 or a shifted-down asymptote).

✗ A)
Shifts the curve DOWN or leaves it unchanged — opposite of “+ 5”.

✗ B)
Same as the original graph — no shift.

✗ C)
Small shift, but the asymptote is not at y = 5.

✓ D)
Curve identical in shape, shifted up by 5 units, with horizontal asymptote at y = 5.

Question 17 · Ratios in Symbolic Form

At a particular track meet, the ratio of coaches to athletes is 1 to 26. If there are x coaches at the track meet, which of the following expressions represents the number of athletes at the track meet?

  • A) x/26
  • B) 26x
  • C) x + 26
  • D) 26/x

Answer: B) 26x

Ratio of coaches to athletes = 1 : 26 means for every 1 coach there are 26 athletes. So if there are x coaches, athletes = 26 · x = 26x. (Sanity check: x = 1 gives 26 athletes; x = 2 gives 52 athletes. Correct ratio 1 : 26.)

✗ A) x/26
That would be athletes-per-26-coaches divided out — a fraction of x, not 26 × x.

✓ B) 26x
26 athletes per coach × x coaches = 26x athletes.

✗ C) x + 26
Adds a fixed 26 rather than multiplying by 26 — not a ratio.

✗ D) 26/x
Inverts the relationship — would decrease as x grows.

Question 18 · Interpreting Constants

Kaylani used fabric measuring 5 yards in length to make each suit for a men’s choir. The relationship between the number of suits that Kaylani made, x, and the total length of fabric that she purchased y, in yards, is represented by the equation y − 5x = 6. What is the best interpretation of 6 in this context?

  • A) Kaylani made 6 suits.
  • B) Kaylani purchased a total of 6 yards of fabric.
  • C) Kaylani used a total of 6 yards of fabric to make the suits.
  • D) Kaylani purchased 6 yards more fabric than she used to make the suits.

Answer: D) Kaylani purchased 6 yards more fabric than she used to make the suits.

y − 5x = 6 means (total fabric purchased) − (fabric used for suits) = 6. 5x = yards used to make x suits (5 yards each). y = total fabric purchased. So 6 = purchased − used = extra fabric beyond what was used.

✗ A) Kaylani made 6 suits.
x, not 6, represents the number of suits.

✗ B) Kaylani purchased a total of 6 yards of fabric.
Total purchased is y, not 6.

✗ C) Kaylani used a total of 6 yards of fabric to make the suits.
Used is 5x, not 6.

✓ D) Kaylani purchased 6 yards more fabric than she used to make the suits.
y − 5x is exactly “purchased − used,” and that difference equals 6.

Question 19 · Trigonometry · Unit Circle

What is the value of tan(92π/3)?

  • A) −√3
  • B) −√3/3
  • C) √3/3
  • D) √3

Answer: A) −√3

tan has period π. Reduce 92π/3 modulo π: 92π/3 = 92π/3 − k · π for integer k that makes the result lie in (0, π). Subtract 30π = 90π/3: 92π/3 − 90π/3 = 2π/3. So tan(92π/3) = tan(2π/3). 2π/3 = 120°. tan(120°) = −tan(60°) = −√3. Therefore tan(92π/3) = −√3.

✓ A) −√3
Reducing mod π to 2π/3 gives tan(120°) = −√3.

✗ B) −√3/3
That is tan(150°), not tan(120°) — different reference angle.

✗ C) √3/3
That is tan(30°), not tan(120°).

✗ D) √3
tan(60°) = √3, but tan(120°) = −√3 — sign flip because 2π/3 is in quadrant II.

Question 20 · Right-Triangle Trigonometry

See figure in the original College Board PDF above (right triangle with legs 11 and hypotenuse 28, angle x° at the top vertex). Note: figure not drawn to scale.

In the triangle shown, what is the value of cos x°?

Student-produced response — enter as a fraction or decimal.

Answer: 11/28 (or .3928, .3929)

The right angle sits at the bottom corner. The hypotenuse (28) is opposite the right angle. The angle x° is at the top vertex. From x°, the ADJACENT leg (the one touching x° that is NOT the hypotenuse) is 11. cos(x°) = adjacent / hypotenuse = 11 / 28. As a decimal (rounded per SPR rules to 4 digits): 11 / 28 ≈ 0.3928 or 0.3929.

Why this works.
Cosine of an acute angle in a right triangle is the leg adjacent to that angle divided by the hypotenuse. Identify the hypotenuse (28, opposite the right angle), then the adjacent leg for x° (11, the other leg touching x°), and divide.

Question 21 · Function Values from Roots

The function g is defined by g(x) = (x + 14)(t − x), where t is a constant. In the xy-plane, the graph of y = g(x) passes through the point (24, 0). What is the value of g(0)?

Student-produced response — enter as a fraction or decimal.

Answer: 336

"(24, 0) is on the graph" means g(24) = 0: g(24) = (24 + 14)(t − 24) = 38 · (t − 24) = 0. Since 38 ≠ 0, we need t − 24 = 0 → t = 24. Now evaluate g(0): g(0) = (0 + 14)(24 − 0) = 14 · 24 = 336.

Why this works.
A point on the graph gives an equation g(input) = output. Solve for the unknown constant t, then plug 0 in to find g(0). 14 · 24 = 336.

Question 22 · Circles in the Plane

(x + 4)² + (y − 19)² = 121. The graph of the given equation is a circle in the xy-plane. The point (a, b) lies on the circle. Which of the following is a possible value for a?

  • A) −16
  • B) −14
  • C) 11
  • D) 19

Answer: B) −14

The circle (x + 4)² + (y − 19)² = 121 has center (−4, 19) and radius √121 = 11. Any x-coordinate of a point on the circle must satisfy: center_x − r ≤ x ≤ center_x + r −4 − 11 ≤ x ≤ −4 + 11 −15 ≤ x ≤ 5. Check each choice: A) −16 → less than −15, outside range. B) −14 → between −15 and 5, allowed. C) 11 → greater than 5, outside range. D) 19 → greater than 5, outside range.

✗ A) −16
x = −16 is 1 unit past the leftmost point of the circle (x = −15).

✓ B) −14
−15 ≤ −14 ≤ 5, so x = −14 is a valid x-coordinate for some point on the circle.

✗ C) 11
x = 11 is past the rightmost point (x = 5).

✗ D) 19
19 is the y-coordinate of the center, not a valid x on this circle.

Question 23 · Volume of a Rectangular Prism

A right rectangular prism has a height of 9 inches. The length of the prism’s base is x inches, which is 7 inches more than the width of the prism’s base. Which function V gives the volume of the prism, in cubic inches, in terms of the length of the prism’s base?

  • A) V(x) = x(x + 9)(x + 7)
  • B) V(x) = x(x + 9)(x − 7)
  • C) V(x) = 9x(x + 7)
  • D) V(x) = 9x(x − 7)

Answer: D) V(x) = 9x(x − 7)

Height h = 9. Length ℓ = x. Width w: "length is 7 more than the width" → x = w + 7 → w = x − 7. Volume V = length · width · height = x · (x − 7) · 9 = 9x(x − 7).

✗ A) V(x) = x(x + 9)(x + 7)
Puts height as x + 9 (wrong — height is 9) and width as x + 7 (wrong sign).

✗ B) V(x) = x(x + 9)(x − 7)
Uses (x + 9) for height — but height is the constant 9.

✗ C) V(x) = 9x(x + 7)
Uses width = x + 7 — but the length is 7 more than the width, so width = x − 7.

✓ D) V(x) = 9x(x − 7)
Height 9, length x, width x − 7 — multiplied gives 9x(x − 7).

Question 24 · Extrema of Functions

Which of the following functions has(have) a minimum value at −3?
I. f(x) = −6(3)x − 3
II. g(x) = −3(6)x

  • A) I only
  • B) II only
  • C) I and II
  • D) Neither I nor II

Answer: D) Neither I nor II

Function I: f(x) = −6(3)^x − 3. As x → +∞, 3^x → +∞, so −6·3^x → −∞, so f(x) → −∞. As x → −∞, 3^x → 0, so f(x) → 0 − 3 = −3 (approaches but never reaches). So f(x) < −3 for all real x, and f has NO minimum value (values go arbitrarily far below −3). It does approach −3 as an asymptotic MAXIMUM (supremum), not a minimum. So f does NOT have minimum value −3. Function II: g(x) = −3(6)^x. As x → +∞, 6^x → +∞, so g(x) → −∞. As x → −∞, 6^x → 0, so g(x) → 0. g takes every value in (−∞, 0), never a minimum value. Certainly not a minimum at −3. Neither function has a minimum value at −3.

✗ A) I only
f approaches −3 as a supremum, not a minimum, and it decreases without bound below −3.

✗ B) II only
g = −3·6^x approaches 0 from below and goes to −∞; no minimum exists.

✗ C) I and II
Neither function has a minimum value.

✓ D) Neither I nor II
Both are strictly decreasing exponentials of the form −A·b^x (± constant), which have a horizontal asymptote as a supremum and no minimum.

Question 25 · Percent Increase

The result of increasing the quantity x by 400% is 60. What is the value of x?

  • A) 12
  • B) 15
  • C) 240
  • D) 340

Answer: A) 12

"Increasing x by 400%" means adding 400% of x to x itself: x + 4x = 5x. Set equal to 60: 5x = 60 x = 12.

✓ A) 12
5 · 12 = 60. Confirmed.

✗ B) 15
15 · 4 = 60, but the correct equation is 5x = 60, not 4x = 60.

✗ C) 240
60 / 0.25 = 240 — treats a 400% increase as a 4× decrease.

✗ D) 340
60 − something? Not a valid derivation for this problem.

Question 26 · Quadratic from Its Roots

The function f is defined by f(x) = ax² + bx + c, where a, b, and c are constants. The graph of y = f(x) in the xy-plane passes through the points (7, 0) and (−3, 0). If a is an integer greater than 1, which of the following could be the value of a + b?

  • A) −6
  • B) −3
  • C) 4
  • D) 5

Answer: A) −6

Roots of f are 7 and −3, so f factors as f(x) = a(x − 7)(x + 3). Expand: (x − 7)(x + 3) = x² + 3x − 7x − 21 = x² − 4x − 21. Multiply by a: f(x) = ax² − 4ax − 21a. Compare to f(x) = ax² + bx + c: b = −4a c = −21a. Then a + b = a + (−4a) = −3a. a is an integer greater than 1, so a ∈ {2, 3, 4, ...} and a + b = −3a ∈ {−6, −9, −12, ...}. Only −6 is among the choices (corresponding to a = 2).

✓ A) −6
a = 2 gives a + b = 2 − 8 = −6. Valid.

✗ B) −3
Would require a = 1, but a must be an integer greater than 1.

✗ C) 4
a + b must be negative for a > 1 — 4 is positive, impossible.

✗ D) 5
Same issue — positive, so impossible.

Question 27 · Repeated-Root Substitution

The function g is defined by g(x) = x(x − 2)(x + 6)². The value of g(7 − w) is 0, where w is a constant. What is the sum of all possible values of w?

Student-produced response — enter as a fraction or decimal.

Answer: 25

g(x) = 0 when x = 0, or x = 2, or x = −6 (double root — counted once for "distinct values that produce zero"). g(7 − w) = 0 means 7 − w is one of {0, 2, −6}. Case 1: 7 − w = 0 → w = 7. Case 2: 7 − w = 2 → w = 5. Case 3: 7 − w = −6 → w = 13. Sum of all possible values of w = 7 + 5 + 13 = 25.

Why this works.
A product is zero exactly when one of its factors is zero. Solving 7 − w = each root of g gives three distinct values of w. Their sum is 25. The repeated factor (x + 6)² does not add a new value of w — it just means x = −6 is a double zero.

How SOMATH prepares NYC students for the Digital SAT

SOMATH (School of Math) is a math-focused school on the Upper West Side of Manhattan, cofounded by Marcelo Ambrozio (Northwestern-trained lead math teacher) and Vivianne Wright (Harvard-trained, also runs MBA House, one of the largest international GMAT/EA prep firms). Vivianne’s track record with standardized tests — hundreds of admits to Harvard Business School, Stanford GSB, Wharton, MIT Sloan, and INSEAD — is the same rigor we bring to SAT prep for high schoolers.

Our Digital SAT track:

Manhattan families: If your student is targeting a 1500+ Digital SAT score for Ivy League, MIT, Stanford, or top-15 admissions, book a free diagnostic evaluation or call (646) 668-6151. We are two blocks from the 79th Street 1 train and three blocks from the B/C at the American Museum of Natural History.

Ready to raise your SAT Math score?

Start with a free 30-minute in-person diagnostic at our Upper West Side classroom. Includes a written diagnostic report within 48 hours — yours to keep whether you enroll or not. A one-time $99 enrollment fee applies if you join a course.

Book Free Evaluation →   or call (646) 668-6151

SOMATH course · Grades 9–12

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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 1 of the Digital SAT Math section?

Module 1 is a 35-minute, 20 scored + 2 pretest question module that every student sees in the same fixed form. It draws from four content domains: Algebra, Advanced Math, Problem-Solving & Data Analysis, and Geometry & Trigonometry. The Bluebook Desmos calculator is available on every question.

How is the Digital SAT Math section scored?

The 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, and students who did not are routed to an easier Module 2 that typically caps in the mid-600s.

How long is the Digital SAT Math section?

70 minutes total, split into two 35-minute modules of 22 questions each (20 scored + 2 pretest). Together with the Reading & Writing section (64 minutes), the full Digital SAT is 2 hours 14 minutes.

Can I use a calculator on Digital SAT Math?

Yes, on every single question. The Bluebook app includes a built-in Desmos graphing calculator, 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 is a good Digital SAT Math score?

As a general reference: 600 ≈ 75th percentile, 700 ≈ 90th percentile, 750+ ≈ top 5%. For MIT, Stanford, Harvard, and the Ivy League, competitive applicants score 780+ on Math. For SUNY Binghamton or UNC, 720+. For most CUNY campuses, 600–680.

What test dates does SOMATH prepare for?

All U.S. Digital SAT dates — August, October, November, December, March, May, and June each year. Our small-group SAT classes run on rolling 12-week cohorts, so a student can start any month.

Where is SOMATH located?

226 West 79th Street, 1st Floor, New York, NY 10024. Upper West Side of Manhattan, between Amsterdam and Broadway, right by the 1 train at 79th and the B/C at 81st.

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