Bluebook SAT Digital Assessment — Questions and Answers

Question 1: In right triangle ABC with the right angle at C, AB = 17 and BC = 8. What is the value of tan(A)?

15/17
15/8
8/15 (Correct answer)
8/17

Correct answer: 8/15

AC = √(17² − 8²) = √(289 − 64) = √225 = 15; tan(A) = opposite/adjacent = BC/AC = 8/15.

Question 2: A fruit punch recipe mixes orange juice and pineapple juice in a ratio of 3:2. How many liters of pineapple juice are needed to make 25 liters of punch?

12 liters
10 liters (Correct answer)
8 liters
15 liters

Correct answer: 10 liters

Pineapple juice fraction = 2/(3+2) = 2/5; 2/5 × 25 = 10 liters.

Question 3: The following sentence contains a punctuation error. Identify it. "The novelist's three most celebrated works — a sparse debut set in rural Appalachia, a sprawling postcolonial epic, and a novella about grief — has each been adapted for film." A) The em dashes around the appositive list should be replaced with commas. B) The verb 'has' should be 'have' to agree with the plural subject modified by 'each.' C) 'each' should be removed because it contradicts the plural subject. D) A colon should replace the first em dash to introduce the list.

The verb 'has' should be 'have' to agree with the plural subject modified by 'each.' (Correct answer)
A colon should replace the first em dash to introduce the list.
'each' should be removed because it contradicts the plural subject.
The em dashes around the appositive list should be replaced with commas.

Correct answer: The verb 'has' should be 'have' to agree with the plural subject modified by 'each.'

The subject 'works' is plural; the verb 'has' is singular. While 'each' can sometimes take a singular verb when it follows the noun ('they each has' is wrong), the construction here — 'works…have each been' — requires a plural verb because 'each' is a post-positioned modifier, not the head noun. The em dashes correctly set off the long appositive list (commas would create ambiguity with the commas inside the list). Removing 'each' doesn't fix the subject-verb disagreement. A colon is incorrect here because em dashes properly interrupt mid-sentence.

Question 4: A student reads the following passage excerpt: "The committee's decision, though ostensibly driven by fiscal prudence, was met with skepticism by those who recalled similar austerity measures from a decade prior—measures that had, paradoxically, increased long-term expenditures." Which of the following best describes the rhetorical function of the word 'ostensibly' in this sentence?

It confirms that fiscal prudence was the primary and undisputed driver of the decision.
It indicates that the author personally endorses the committee's reasoning.
It suggests that the committee was unaware of the historical precedent mentioned later.
It signals that the stated reason for the decision may not reflect the actual motivation. (Correct answer)

Correct answer: It signals that the stated reason for the decision may not reflect the actual motivation.

'Ostensibly' means 'apparently' or 'on the surface,' and is used to cast doubt on the stated reason. The sentence structure—combined with the reference to paradoxical outcomes—implies the committee's real motivation may differ from the fiscal prudence they cited. The word is a subtle qualifier that invites the reader to question the official explanation.

Question 5: The scientist's hypothesis, which challenged decades of established research, _______ initially dismissed by the academic community.

were
are
was (Correct answer)
is

Correct answer: was

The subject 'hypothesis' is singular; the relative clause is a modifier, so the singular past tense 'was' is correct.

Question 6: A store marks up items by 40%. If the cost price of an item is $50, what is the selling price?

$80
$90
$60
$70 (Correct answer)

Correct answer: $70

Markup = 40% × $50 = $20; selling price = $50 + $20 = $70.

Question 7: A chord connects two points on a circle of radius 10, and its perpendicular bisector passes through the center. The midpoint of the chord is 6 units from the center. What is the chord's length?

8
20
12
16 (Correct answer)

Correct answer: 16

Half-chord = √(r²−d²) = √(100−36) = √64 = 8; full chord = 16.

Question 8: The board of trustees _______ meeting next Thursday to review the organization's annual financial statement.

is (Correct answer)
have been
are
were

Correct answer: is

'Board' is a singular collective noun, so 'is' is the correct verb form.

Question 9: The documentary, as well as the three follow-up episodes, _______ nominated for several industry awards.

were
was (Correct answer)
have been
are

Correct answer: was

The subject is 'documentary' (singular); 'as well as the three follow-up episodes' is a parenthetical phrase, so the singular verb 'was' is correct.

Question 10: A company's annual revenue R(t), in thousands of dollars, is modeled by R(t) = −2t² + 20t + 50, where t is the number of years since founding. Between which pair of consecutive years does the revenue increase by the greatest amount?

Years 2 and 3
Years 4 and 5
Years 5 and 6
Years 0 and 1 (Correct answer)

Correct answer: Years 0 and 1

The year-over-year increase is R(t + 1) − R(t). Expanding: R(t+1) − R(t) = −2(t+1)² + 20(t+1) + 50 − (−2t² + 20t + 50) = −4t + 18. This expression is a decreasing linear function of t, meaning the annual increase is greatest when t is smallest — at t = 0 (years 0 to 1), where the gain is 18 thousand dollars. Many students intuitively pick years 4–5 (the year before the revenue peak at t = 5), but that interval has only a 2-thousand-dollar increase. For a concave-down parabola, the steepest gains occur at the beginning of the increasing phase, not near the peak.

Question 11: If ax + b = cx + d, where a ≠ c, which of the following expresses x in terms of a, b, c, and d?

x = (d − b) / (a − c) (Correct answer)
x = (b − d) / (a − c)
x = (a − c) / (d − b)
x = (d + b) / (a + c)

Correct answer: x = (d − b) / (a − c)

Collecting x terms on the left: ax − cx = d − b, so x(a − c) = d − b. Dividing both sides by (a − c) gives x = (d − b)/(a − c). Choice B is the negative of the correct answer. Choice C incorrectly adds rather than subtracts. Choice D inverts the fraction.

Question 12: For the equation x² + 5x = 14, what are both solutions?

x = 2 and x = 7
x = 7 and x = −2
x = −7 and x = 2 (Correct answer)
x = −7 and x = −2

Correct answer: x = −7 and x = 2

x² + 5x − 14 = 0 → (x + 7)(x − 2) = 0; x = −7 or x = 2.

Question 13: The values in a dataset increase uniformly. A dot plot shows values at 10, 20, 30, 40, 50. If 30 is added a second time to the dataset, which measure of center changes?

Both mean and median
Mean only
Neither mean nor median (Correct answer)
Median only

Correct answer: Neither mean nor median

With the original dataset, mean = 30 and median = 30. Adding a second 30 gives {10,20,30,30,40,50}: new mean = 180/6 = 30, new median = (30+30)/2 = 30. Both remain 30, so neither changes.

Question 14: A company models its monthly profit, in thousands of dollars, with the function P(t) = -2t² + 16t - 24, where t is the number of months after launch. An analyst claims the company first becomes profitable at t = 2 months. Which of the following best evaluates this claim?

The claim is correct; P(2) = 0 indicates the company transitions from loss to profit at t = 2.
The claim is correct; P(2) = 8 > 0 confirms profitability begins at t = 2.
The claim is incorrect; the company first becomes profitable at t = 3 months, since P(3) = 6 > 0 and P(2) = 0 means the company breaks even at t = 2. (Correct answer)
The claim is incorrect; the company is never profitable because the parabola opens downward.

Correct answer: The claim is incorrect; the company first becomes profitable at t = 3 months, since P(3) = 6 > 0 and P(2) = 0 means the company breaks even at t = 2.

Evaluating P(2) = -2(4) + 16(2) - 24 = -8 + 32 - 24 = 0, which means the company breaks even — it does not yet have positive profit. P(3) = -2(9) + 48 - 24 = -18 + 48 - 24 = 6 > 0, so the company first earns positive profit at t = 3. The zero of P at t = 2 marks break-even, not profitability.

Question 15: If f(x) = −5x + 2 and g(x) = 3x − 6, for what value of x does f(x) = g(x)?

2
0
−1
1 (Correct answer)

Correct answer: 1

−5x + 2 = 3x − 6 → 8 = 8x → x = 1.

Question 16: A parking lot charges $3 for the first hour and $1.50 for each additional hour. Daria paid $9. How many total hours did she park?

4 hours
5 hours (Correct answer)
7 hours
6 hours

Correct answer: 5 hours

3 + 1.50(h − 1) = 9 → 1.50(h−1) = 6 → h−1 = 4 → h = 5.

Question 17: A circle has a circumference of 20π. What is its radius?

5
20
10 (Correct answer)
40

Correct answer: 10

C = 2πr → 20π = 2πr → r = 10.

Question 18: In the xy-plane, the graph of the function p(x) = -(x - 1)(x + 5) intersects the x-axis at two points. What is the x-coordinate of the midpoint of these two intersection points, and what does this value represent on the graph?

x = 2; it is where the parabola reaches its minimum value.
x = 2; it is the x-coordinate of the vertex, where the parabola reaches its maximum value.
x = -2; it is where the parabola crosses the y-axis.
x = -2; it is the x-coordinate of the vertex, where the parabola reaches its maximum value. (Correct answer)

Correct answer: x = -2; it is the x-coordinate of the vertex, where the parabola reaches its maximum value.

The zeros occur at x = 1 and x = -5. The midpoint of the zeros is (1 + (-5))/2 = -4/2 = -2. The axis of symmetry of a parabola passes through the midpoint of the zeros, so x = -2 is the axis of symmetry and the x-coordinate of the vertex. Since the leading coefficient is negative (the parabola opens downward), the vertex is a maximum. The y-intercept is found by setting x = 0, which gives a different point entirely.

Question 19: A salesperson earns a base salary of $2,000 per month plus a 5% commission on all sales. If the salesperson earns $2,750 in a month, what were their total sales that month?

$15,000 (Correct answer)
$13,750
$15,500
$17,500

Correct answer: $15,000

2000 + 0.05s = 2750 → 0.05s = 750 → s = $15,000.

Question 20: A model y = kx² fits a dataset. When x = 3, y = 36. What is the value of k?

4 (Correct answer)
36
12
9

Correct answer: 4

36 = k(9) → k = 36/9 = 4.

Question 21: The salaries of 10 employees at a small company are listed. Nine employees earn between $40,000 and $60,000, while the CEO earns $350,000. Which measure of center would be most affected by the CEO's salary?

The mode
The median
The range
The mean (Correct answer)

Correct answer: The mean

The mean is calculated by summing all values and dividing by the count, so it is highly sensitive to extreme values (outliers). The CEO's salary is a significant outlier and will pull the mean much higher. The median is the middle value when the data is ordered, so it is resistant to outliers. Therefore, the mean is the measure that is most affected.

Question 22: A two-way table records 500 households by internet speed and streaming usage. Streams Daily Does Not Stream High-speed internet 180 70 Low-speed internet 60 190 What is the probability a household streams daily?

12/25 (Correct answer)
9/25
3/5
18/50

Correct answer: 12/25

Total streaming = 180 + 60 = 240 out of 500; P = 240/500 = 12/25.

Question 23: Two right triangles are similar. Their areas are 50 and 200 square units. What is the ratio of their corresponding sides?

1:√3
1:4
1:√2
1:2 (Correct answer)

Correct answer: 1:2

Area ratio = 50:200 = 1:4; side ratio = √(1/4) = 1:2.

Question 24: A passage about ocean acidification states: 'The ocean absorbs approximately 30% of human CO2 emissions. When CO2 dissolves in seawater, it forms carbonic acid, lowering the ocean's pH. Since industrialization began, ocean pH has fallen from 8.2 to 8.1 — a 26% increase in acidity. This threatens shellfish and coral by dissolving the calcium carbonate structures of their shells and skeletons.' A student claims: 'Ocean acidification poses a specific, measurable threat to marine organisms.' Which evidence best supports this?

'This threatens shellfish and coral by dissolving the calcium carbonate structures of their shells and skeletons.' (Correct answer)
'ocean pH has fallen from 8.2 to 8.1 — a 26% increase in acidity'
'When CO2 dissolves in seawater, it forms carbonic acid, lowering the ocean's pH.'
'The ocean absorbs approximately 30% of human CO2 emissions.'

Correct answer: 'This threatens shellfish and coral by dissolving the calcium carbonate structures of their shells and skeletons.'

The sentence identifying the specific organisms threatened (shellfish, coral) and the specific mechanism of harm (dissolving calcium carbonate structures) directly supports a 'specific, measurable threat to marine organisms.'

Question 25: The angles of a triangle are in the ratio 2:3:7. What is the measure of the largest angle?

90°
30°
105° (Correct answer)
60°

Correct answer: 105°

2k+3k+7k=180° → k=15°; largest = 7×15° = 105°.

Question 26: A function f defined for all real numbers satisfies f(x + y) = f(x) · f(y) for all real x and y, and f(1) = 3. What is f(0) + f(−1)?

4/3 (Correct answer)
2
7/3
4

Correct answer: 4/3

For f(0): set x = y = 0 → f(0) = [f(0)]², so f(0) = 0 or 1. Since f(1) = 3 ≠ 0, f(0) = 0 would force f(everything) = 0 — contradiction. So f(0) = 1. For f(−1): set x = 1, y = −1 → f(0) = f(1)·f(−1) → 1 = 3·f(−1) → f(−1) = 1/3. Therefore f(0) + f(−1) = 1 + 1/3 = 4/3.

Question 27: The physicist's lecture — which drew scholars from three continents — focused on the paradox that time, ______ its role as the universe's most fundamental dimension, remains poorly understood at the quantum level. (A) despite (B) regardless of (C) notwithstanding (D) even with

despite
even with
regardless of
notwithstanding (Correct answer)

Correct answer: notwithstanding

Choice C, 'notwithstanding,' is correct. When used as a preposition immediately following the noun it modifies ('time, notwithstanding its role'), 'notwithstanding' functions in a postpositive construction that the SAT tests. 'Despite' and 'regardless of' require the prepositional phrase to precede or directly follow 'time' but sound natural only in different sentence constructions. 'Even with' is informal and imprecise. 'Notwithstanding its role' correctly indicates that despite time's role, it remains poorly understood — the postpositive placement is the advanced usage being tested here.

Question 28: In triangle ABC, m∠A = 60°. The exterior angle at vertex B is exactly twice the exterior angle at vertex C. What is m∠B?

60°
20° (Correct answer)
40°
80°

Correct answer: 20°

Let the exterior angle at C = e°, so the exterior angle at B = 2e°. The corresponding interior angles are (180 − 2e)° at B and (180 − e)° at C. Applying the triangle angle sum: 60 + (180 − 2e) + (180 − e) = 180 → 420 − 3e = 180 → e = 80°. Therefore m∠B = 180° − 2(80°) = 20°. Check: m∠C = 180° − 80° = 100°, and 20° + 100° + 60° = 180° ✓.

Question 29: If f(x) = 3x - 2 and g(x) = x + 4, what is f(g(2))?

16 (Correct answer)
4
14
10

Correct answer: 16

g(2) = 2+4 = 6. f(6) = 3(6)-2 = 16.

Question 30: What is the product (2 + 3i)(1 − i)?

5 − i
2 + 3i
5 + i (Correct answer)
−1 + 5i

Correct answer: 5 + i

Expanding: 2 − 2i + 3i − 3i² = 2 + i − 3(−1) = 2 + i + 3 = 5 + i.

Question 31: A circle has center (−3, 2). Which equation could represent this circle?

(x − 3)² + (y − 2)² = 10
(x − 3)² + (y + 2)² = 10
(x + 3)² + (y + 2)² = 10
(x + 3)² + (y − 2)² = 10 (Correct answer)

Correct answer: (x + 3)² + (y − 2)² = 10

Center (−3, 2) gives (x−(−3))²+(y−2)²=r² → (x+3)²+(y−2)²=r².

Question 32: For x > 0, which expression is equivalent to x^(2/3) ÷ x^(1/6)?

x^(1/3)
x^(1/2) (Correct answer)
x^(5/6)
x^(1/6)

Correct answer: x^(1/2)

When dividing expressions with the same base, subtract exponents: x^(2/3) ÷ x^(1/6) = x^(2/3 − 1/6). Convert to a common denominator: 2/3 = 4/6, so 4/6 − 1/6 = 3/6 = 1/2. The result is x^(1/2), or equivalently √x. A common error is adding exponents (giving 5/6) or subtracting in the wrong direction (giving 1/6).

Question 33: If (2x − 1)² = 49, what are the two values of x?

x = 3 and x = −4
x = 4 and x = −4
x = 3 and x = −3
x = 4 and x = −3 (Correct answer)

Correct answer: x = 4 and x = −3

2x − 1 = ±7; case 1: 2x = 8, x = 4; case 2: 2x = −6, x = −3.

Question 34: A student reads: 'Sleep deprivation impairs cognitive function in ways that mirror alcohol intoxication. After 17–19 hours without sleep, performance on tests resembles that of someone with a blood alcohol level of 0.05%. After 24 hours awake, impairment equals a blood alcohol level of 0.10%, which is above the legal driving limit in most countries.' What supporting detail reinforces the central claim that sleep deprivation seriously impairs cognition?

After 24 hours awake, impairment equals illegal intoxication levels (Correct answer)
Sleep deprivation is a common problem in modern society
Cognitive tests measure a variety of mental skills
Many countries set blood alcohol limits for drivers

Correct answer: After 24 hours awake, impairment equals illegal intoxication levels

The comparison to exceeding legal blood alcohol limits is a concrete detail that reinforces how severely sleep loss impairs cognitive function.

Question 35: Which expression is equivalent to (x^(1/3) + y^(1/3))(x^(2/3) − x^(1/3)·y^(1/3) + y^(2/3))?

x^(2/3) + y^(2/3)
x + y (Correct answer)
(x + y)^(2/3)
x^(1/3) + y^(1/3)

Correct answer: x + y

Let a = x^(1/3) and b = y^(1/3). The expression becomes (a + b)(a² − ab + b²), which is the sum-of-cubes factoring pattern in reverse: a³ + b³. Substituting back: (x^(1/3))³ + (y^(1/3))³ = x + y. The other choices confuse the pattern with difference of squares or partial factorizations.

Question 36: Solve: 2^(3x−1) = 16

x = 1
x = 5/3 (Correct answer)
x = 4/3
x = 2

Correct answer: x = 5/3

16 = 2⁴, so 3x − 1 = 4 → x = 5/3.

Question 37: Passage: "Every summer for twelve years, Renata sent her niece a handwritten letter. This summer, the niece received a short email instead." Which of the following inferences requires the MOST additional evidence to be considered valid?

The niece's contact information or address may have changed.
Renata no longer values her relationship with her niece. (Correct answer)
Something about Renata's circumstances may have changed this summer.
Renata may have had difficulty writing by hand this summer.

Correct answer: Renata no longer values her relationship with her niece.

Inferring that Renata 'no longer values the relationship' is a sweeping conclusion about internal emotional states that requires far more evidence than a single medium change. The other inferences each propose plausible, concrete explanations (changed circumstances, physical difficulty, updated contact info) that require only minimal additional context to validate. Judging someone's feelings about a relationship from one behavioral data point is a logical leap unsupported by the passage.

Question 38: The following text is from a 2019 article about deep-sea exploration. "The remotely operated vehicle's cameras, though ______ by the crushing pressure at 4,000 meters, captured footage of a previously unknown species of siphonophore drifting past the lights." Which choice completes the text with the most logical and precise word or phrase?

unaffected
improved
constrained (Correct answer)
liberated

Correct answer: constrained

"Though" signals the cameras worked despite a limitation imposed by pressure.

Question 39: A diagonal of a rectangle has length 20 and makes a 30° angle with the longer side. What is the area of the rectangle?

100√3 (Correct answer)
100
200
150√3

Correct answer: 100√3

The longer side = 20·cos(30°) = 20·(√3/2) = 10√3. The shorter side = 20·sin(30°) = 20·(1/2) = 10. Area = 10√3 × 10 = 100√3.

Question 40: The line of best fit for a dataset is y = −3x + 90. Which of the following best describes the relationship between x and y?

As x increases, y decreases (Correct answer)
x and y have no relationship
As x increases, y stays constant
As x increases, y increases

Correct answer: As x increases, y decreases

A negative slope means y decreases as x increases.

Question 41: If 3x − 7 = 14, what is the value of x?

7 (Correct answer)
21
−7
3

Correct answer: 7

Add 7 to both sides: 3x = 21; divide by 3: x = 7.

Question 42: Let f(x) = |2x − 3| − |x + 1|. How many integers in the closed interval [−5, 5] satisfy f(x) > 0?

5
6
8
7 (Correct answer)

Correct answer: 7

The critical points are x = −1 and x = 3/2, creating three regions. For x < −1: f(x) = −x + 4 > 0 for all x < 4, so integers −5, −4, −3, −2 all qualify (4 values). For −1 ≤ x < 3/2: f(x) = −3x + 2 > 0 when x < 2/3, so integers −1 and 0 qualify (2 values). For x ≥ 3/2: f(x) = x − 4 > 0 only when x > 4, giving just x = 5 (1 value). Total: 4 + 2 + 1 = 7.

Question 43: A ship sails 60 miles east and then 80 miles north. What is the straight-line distance from its starting point, and what angle (to the nearest degree) does this path make with the east-west axis?

100 miles, 53° (Correct answer)
100 miles, 37°
100 miles, 45°
140 miles, 45°

Correct answer: 100 miles, 53°

Distance = √(60² + 80²) = √(3600+6400) = √10000 = 100 miles. tan(θ) = 80/60 = 4/3, so θ ≈ 53°.

Question 44: Which of the following is equivalent to 4x² − 20x + 19?

2(x − 5/2)² − 6
(2x − 5)² + 6
(2x − 5)² − 6 (Correct answer)
4(x − 5)² − 6

Correct answer: (2x − 5)² − 6

Notice that (2x − 5)² = 4x² − 20x + 25. Since the original expression is 4x² − 20x + 19, we subtract 6 from (2x − 5)²: (2x − 5)² − 6 = 4x² − 20x + 25 − 6 = 4x² − 20x + 19. Choice C incorrectly factors out 4 with a wrong inner term; choice D has an incorrect coefficient.

Question 45: Passage 1: "Literary critic Saoirse Tully claims that unreliable narrators in 20th-century fiction function primarily as a formal device for thematic irony — the gap between what the narrator reports and what the reader infers creates a productive space for ideological critique." Passage 2: "Narratologist Hideo Mura argues that framing unreliable narrators solely as vehicles for irony flattens their cognitive complexity. His research suggests readers engage with unreliable narrators through 'epistemic simulation' — modeling the narrator's flawed belief system as a coherent internal world — which is a fundamentally empathetic rather than critical process." How would Mura most likely respond to a reader who, having finished a novel with an unreliable narrator, says: 'I never trusted the narrator, so I was always looking for the ironic gap'?

Mura would affirm this response as evidence that readers naturally detect narrative unreliability through critical distance.
Mura would suggest this reader may have missed a deeper layer of engagement: simulating the narrator's worldview from the inside rather than critiquing it from the outside. (Correct answer)
Mura would argue that this response confirms Tully's thesis and therefore contradicts his own theory.
Mura would contend that the reader's distrust is an authorial failure, since successful unreliable narrators should never be detected.

Correct answer: Mura would suggest this reader may have missed a deeper layer of engagement: simulating the narrator's worldview from the inside rather than critiquing it from the outside.

Mura's theory holds that readers engage with unreliable narrators through empathetic epistemic simulation — inhabiting the narrator's flawed logic — not through detached ironic critique. A reader who is 'always looking for the ironic gap' is doing exactly what Tully describes and exactly what Mura says flattens the cognitive complexity of the experience. Mura would suggest something richer is being bypassed.

Question 46: A scatterplot of study hours (x) vs. exam score (y) shows the line y = 8x + 45. A student studies 0 hours. What does the y-intercept tell us about this student?

The model predicts a score of 45 even with no study time. (Correct answer)
The student cannot score above 45.
The student will fail with a score of 0.
Study hours do not affect the score.

Correct answer: The model predicts a score of 45 even with no study time.

The y-intercept (45) is the model's predicted value of y when x = 0, representing the expected baseline score.

Question 47: Which of the following is equivalent to 2x³ − 2x?

2x(x² − 1)
2x(x − 1)(x + 1)
All of the above (Correct answer)
2x(x − 1)(x + 1)

Correct answer: All of the above

Factor: 2x(x²−1) = 2x(x−1)(x+1); the first two listed forms represent the same expression.

Question 48: Novelist Toni Morrison, who received the Nobel Prize in Literature in ______ was the first African American woman to win the award. Which choice completes the text so that it conforms to the conventions of Standard English?

1993;
1993, (Correct answer)
1993:
1993

Correct answer: 1993,

The nonessential clause beginning with "who" must be closed with a comma.

Question 49: Two parallel lines are cut by a transversal. Co-interior angles are (4n + 10)° and (2n + 26)°. What is n?

30
12
24 (Correct answer)
18

Correct answer: 24

(4n+10)+(2n+26)=180 → 6n+36=180 → 6n=144 → n=24.

Question 50: If x² + 12x + c = (x + 6)², what is the value of c?

36 (Correct answer)
6
72
12

Correct answer: 36

(x + 6)² = x² + 12x + 36, so c = 36.

Question 51: A student is revising a passage about the history of cartography. The current sentence reads: 'Early mapmakers, lacking access to aerial perspectives, relied on traveler accounts and dead reckoning, techniques that introduced systematic error into their representations of coastlines.' The student wants to add a phrase that acknowledges a counterintuitive finding without undermining the sentence's critical tone. Which addition best accomplishes this? A) '—yet produced works of surprising navigational utility—' B) ', which were nonetheless considered artistic masterpieces,' C) ', even though travelers were often unreliable sources,' D) '—a limitation that was largely unrecognized at the time—'

, which were nonetheless considered artistic masterpieces,
, even though travelers were often unreliable sources,
—yet produced works of surprising navigational utility— (Correct answer)
—a limitation that was largely unrecognized at the time—

Correct answer: —yet produced works of surprising navigational utility—

Choice A introduces a genuine counterintuitive finding (these error-prone methods still yielded useful navigational tools) while preserving the critical analytical tone about systematic error. Choice B shifts focus to aesthetics, which is neither counterintuitive nor critical. Choice C amplifies the criticism rather than countering it. Choice D adds detail about recognition but offers no counterintuitive finding—it deepens the negative claim rather than complicating it.

Question 52: A line segment has endpoints A(−2, 5) and B(4, −3). What are the coordinates of the point that divides AB in the ratio 2:1 from A to B?

(2, −1)
(2, −1/3) (Correct answer)
(1, 1)
(0, 1)

Correct answer: (2, −1/3)

Section formula with ratio 2:1: x = (−2+8)/3 = 2; y = (5−6)/3 = −1/3; point = (2, −1/3).

Question 53: If 2^x = 32, what is the value of 2^(x−3)?

4 (Correct answer)
8
2
16

Correct answer: 4

2^x = 2^5 so x = 5. Then 2^(5−3) = 2² = 4.

Question 54: In the xy-plane, line ℓ passes through the origin and is perpendicular to the line 3x − 5y = 15. Which of the following points lies on line ℓ?

(−3, −5)
(−5, 3)
(3, −5) (Correct answer)
(5, −3)

Correct answer: (3, −5)

Rewrite 3x − 5y = 15 as y = (3/5)x − 3, giving a slope of 3/5. The perpendicular slope is the negative reciprocal: −5/3. Since line ℓ passes through the origin, its equation is y = (−5/3)x. Testing (3, −5): y = (−5/3)(3) = −5 ✓. The other choices fail: (5, −3) gives −25/3; (−3, −5) gives 5; (−5, 3) gives 25/3.

Question 55: A gym charges a $40 registration fee plus $25 per month. A competing gym charges no registration fee but $35 per month. After how many months will the total cost at both gyms be equal?

5
4 (Correct answer)
2
3

Correct answer: 4

Set 40 + 25m = 35m → 40 = 10m → m = 4 months.

Question 56: A well-designed, double-blind randomized experiment with 500 participants finds that a new medication reduces headache frequency by a statistically significant margin (p = 0.03). A critic argues the result is not practically meaningful. Which of the following, if true, would most strongly support the critic's position?

The p-value threshold should have been set at 0.01, not 0.05.
The experiment was not replicated in a second study.
The participants were not representative of the general population.
The medication reduced headaches by an average of 0.2 episodes per month compared to placebo. (Correct answer)

Correct answer: The medication reduced headaches by an average of 0.2 episodes per month compared to placebo.

Statistical significance (p < 0.05) indicates that a result is unlikely to be due to chance alone, but it does not measure practical or clinical significance. A reduction of only 0.2 headaches per month — even if real — may be too small to meaningfully improve patients' lives. Effect size, not just p-value, determines practical relevance.

Question 57: An education researcher randomly assigns 40 classrooms to use a new math curriculum and 40 to use the standard curriculum. At year-end, the new curriculum classrooms score significantly higher. What is the most valid conclusion?

The new curriculum is superior to all other existing math curricula.
Every student who uses the new curriculum will score higher on math tests.
The new curriculum causes higher math scores in classrooms like those in the study. (Correct answer)
There is only a correlation between curriculum type and scores, not causation.

Correct answer: The new curriculum causes higher math scores in classrooms like those in the study.

Random assignment of classrooms supports a causal conclusion for classrooms similar to those in the study, though it may not generalize to all possible classroom environments.

Question 58: If 𝑓(𝑥)=2𝑥+3, what is 𝑓(4)?

8
11 (Correct answer)
7
9

Correct answer: 11

To find 𝑓(4) for the function 𝑓(𝑥)=2𝑥+3, substitute the value 4 for 𝑥 in the expression. This means calculating 2 multiplied by 4, and then adding 3. Following the order of operations, 2(4) equals 8, and 8+3 equals 11, so 𝑓(4)=11.

Question 59: In right triangle XYZ with hypotenuse XZ = 2, angle X = θ, and angle Z = 90° − θ. Which expression is equivalent to sin²(θ) + sin²(90° − θ)?

sin(2θ)
1 (Correct answer)
cos(2θ) + 1
2sin(θ)cos(θ)

Correct answer: 1

Since sin(90° − θ) = cos(θ), the expression becomes sin²(θ) + cos²(θ), which equals 1 by the Pythagorean identity. This is true for all values of θ, making it a fundamental trigonometric identity.

Question 60: Two cars start at the same point and drive in opposite directions. Car A travels at 55 mph and Car B at 45 mph. After how many hours are they 300 miles apart?

4 hours
3 hours (Correct answer)
3.5 hours
2.5 hours

Correct answer: 3 hours

Combined separation rate = 55 + 45 = 100 mph; time = 300 ÷ 100 = 3 hours.

Question 61: When using the built-in Desmos graphing calculator within a Bluebook full practice simulation, which of the following represents a genuine limitation that students often discover too late?

Desmos is available on all Math questions in both modules, but it cannot solve equations symbolically — it only graphs and computes numerically, so students relying on it to algebraically isolate variables will be misled. (Correct answer)
The graphing calculator is disabled for any question classified as a 'no-calculator' question, which comprises approximately half of all Math module questions.
Students must activate Desmos at the start of each module separately; failing to do so locks them out for the entire module.
The Desmos calculator is only available during Math Module 2, not Module 1, regardless of question type.

Correct answer: Desmos is available on all Math questions in both modules, but it cannot solve equations symbolically — it only graphs and computes numerically, so students relying on it to algebraically isolate variables will be misled.

In the digital SAT, the Desmos graphing calculator is available on every Math question in both Module 1 and Module 2 — the old paper SAT's 'no-calculator section' does not exist in this format. However, Desmos is a graphing and numerical computation tool, not a computer algebra system (CAS). It cannot symbolically rearrange or solve equations for a variable the way tools like Wolfram Alpha can. Students who attempt to 'solve for x' by entering an equation and expecting an algebraic solution may get a graph or numerical approximation rather than the exact simplified form the question requires, leading to errors.

Question 62: [READING] The author of a passage argues that urban parks improve mental health. Which of the following, if true, would most strengthen this argument?

A study found that city residents with access to parks report 30% lower stress levels (Correct answer)
Some parks are located in areas with high crime rates
Urban parks require significant maintenance costs
Parks were more common in cities 50 years ago

Correct answer: A study found that city residents with access to parks report 30% lower stress levels

Direct evidence that parks reduce stress specifically supports the claim about mental health benefits.

Question 63: If a fair six-sided die is rolled twice, what is the probability of getting a sum of 8?

1/4
1/6
5/36 (Correct answer)
7/36

Correct answer: 5/36

Pairs summing to 8: (2,6),(3,5),(4,4),(5,3),(6,2) = 5 pairs. Total = 36. P = 5/36.

Question 64: A student is writing an essay and includes the following two sentences: 'Early clinical trials showed the compound suppressed tumor growth in mice. Oncologists remain cautious about extrapolating these results to human patients.' Which transition word or phrase, inserted at the beginning of the second sentence, most precisely captures the logical relationship between the sentences? A) Nevertheless, B) Consequently, C) That said, D) By the same token,

Consequently,
Nevertheless, (Correct answer)
That said,
By the same token,

Correct answer: Nevertheless,

'Nevertheless' signals a concessive contrast — the promising mouse-trial results exist, yet caution persists despite them. 'Consequently' implies the caution is a direct result of the trials, which reverses the logic. 'That said' is colloquial and vague. 'By the same token' introduces a parallel point, not a contrast. Only 'Nevertheless' captures the tension between positive data and clinical restraint.

Question 65: A researcher randomly surveys 100 employees at a company and finds that 40% use public transportation, with a margin of error of ±8 percentage points at 95% confidence. To cut the margin of error to ±4 percentage points, the researcher must survey how many employees?

50
400 (Correct answer)
800
200

Correct answer: 400

Halving the margin of error requires quadrupling the sample size: 100 × 4 = 400.

Question 66: What is the modulus (absolute value) of the complex number 3 + 4i?

5 (Correct answer)
1
7
25

Correct answer: 5

|3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5.

Question 67: If 5y − 3 = 2y + 9, what is the value of y?

4 (Correct answer)
2
3
6

Correct answer: 4

Combining like terms: 3y = 12, so y = 4.

Question 68: Parallel lines a and b are cut by transversal t. The angle at the intersection with a is 148°. What is the size of the alternate exterior angle at b?

32°
58°
148° (Correct answer)
180°

Correct answer: 148°

Alternate exterior angles are equal when lines are parallel, so the angle is 148°.

Question 69: How many solutions does the system y = x² + 3x + 2 and y = x + 2 have?

2 (Correct answer)
1
Infinitely many
0

Correct answer: 2

x² + 3x + 2 = x + 2 → x² + 2x = 0 → x(x+2) = 0 → x = 0 or x = −2; two solutions.

Question 70: Which of the following is equivalent to (4x + 2)(x − 3)?

4x² − 10x + 6
4x² − 6x − 6
4x² − 10x − 6 (Correct answer)
4x² + 10x − 6

Correct answer: 4x² − 10x − 6

FOIL: 4x·x=4x², 4x·(−3)=−12x, 2·x=2x, 2·(−3)=−6; combine: 4x²+(−12x+2x)−6 = 4x²−10x−6.

Question 71: A function g satisfies g(g(x)) = x for all real numbers x. If g(3) = 7, what is the value of g(7) + g(3)?

10 (Correct answer)
7
14
21

Correct answer: 10

Since g(g(x)) = x, g is its own inverse. Applying this with x = 3: g(g(3)) = 3, so g(7) = 3. Therefore g(7) + g(3) = 3 + 7 = 10.

Question 72: A coffee shop creates a 30-pound blend using Type X beans at $12/lb and Type Y beans at $18/lb. The blend is priced at $15.60/lb. How many pounds of Type Y beans are in the blend?

8 pounds
10 pounds
18 pounds (Correct answer)
12 pounds

Correct answer: 18 pounds

Let y = pounds of Type Y. Then (30 − y) = pounds of Type X. Equation: 12(30 − y) + 18y = 15.60 × 30. Expanding: 360 − 12y + 18y = 468 → 6y = 108 → y = 18 pounds.

Question 73: The two-way table shows 480 shoppers by age and purchase type. Online In-Store Under 40 180 60 40 and over 80 160 Given a shopper buys in-store, what is the probability they are 40 or over?

8/11 (Correct answer)
4/9
5/11
4/11

Correct answer: 8/11

In-store total = 60 + 160 = 220; P(40+ | in-store) = 160/220 = 8/11.

Question 74: The circle (x + 4)² + (y − 1)² = 36 has been graphed. Which point lies on the circle?

(2, 1)
(4, 1)
(0, 1)
(−4, 7) (Correct answer)

Correct answer: (−4, 7)

For (−4, 7): (−4+4)² + (7−1)² = 0 + 36 = 36 ✓.

Question 75: Evaluate: lim_{x→2} (x³ − 8)/(x − 2).

0
4
6
12 (Correct answer)

Correct answer: 12

Factor: x³ − 8 = (x−2)(x² + 2x + 4); cancel (x−2) to get x² + 2x + 4; at x = 2: 4 + 4 + 4 = 12.

Question 76: Which of the following is equivalent to 2x(3x − 1) − 3x(2x − 1)?

−x
6x² − 5x
x (Correct answer)
12x² − 5x

Correct answer: x

Expand: 6x²−2x − (6x²−3x) = 6x²−2x−6x²+3x = x.

Question 77: Two data sets, A and B, are created. Data set A consists of 10 consecutive even integers. Data set B is created by multiplying each element of data set A by -2 and then adding 5. Which of the following statements correctly compares the standard deviations of the two data sets?

The standard deviations are equal.
The standard deviation of set B is twice the standard deviation of set A. (Correct answer)
The standard deviation of set A is half the standard deviation of set B.
The standard deviation of set B is -2 times the standard deviation of set A.

Correct answer: The standard deviation of set B is twice the standard deviation of set A.

Standard deviation measures the spread or dispersion of a data set. Adding a constant to every value in a data set does not change the spread, so the '+5' operation has no effect on the standard deviation. However, multiplying each value by a constant 'c' multiplies the standard deviation by the absolute value of that constant, |c|. In this case, each element of set A is multiplied by -2. Therefore, the standard deviation of set B will be |-2| times the standard deviation of set A. This means the standard deviation of B is 2 times the standard deviation of A.

Question 78: Which expression is equivalent to (1/x − 1/y) / (1/x + 1/y), where x ≠ 0, y ≠ 0, and x ≠ −y?

(xy − 1) / (xy + 1)
(x − y) / (x + y)
(x + y) / (x − y)
(y − x) / (y + x) (Correct answer)

Correct answer: (y − x) / (y + x)

Combine the numerator: (1/x − 1/y) = (y − x)/(xy). Combine the denominator: (1/x + 1/y) = (y + x)/(xy). Dividing gives [(y − x)/(xy)] ÷ [(y + x)/(xy)] = (y − x)/(y + x). Choice A, (x − y)/(x + y), has the opposite sign in the numerator — a common trap.

Question 79: Simplify ∛54 in simplest radical form.

6∛2
3∛6
3∛2 (Correct answer)
2∛7

Correct answer: 3∛2

∛54 = ∛(27 × 2) = 3∛2.

Question 80: Which of the following is equivalent to x^(2/3)?

√(x³)
x^(3/2)
x² ÷ x³
∛(x²) (Correct answer)

Correct answer: ∛(x²)

The rational exponent 2/3 means the cube root of x squared: x^(2/3) = ∛(x²).

Question 81: The equation x² + y² + 6x − 4y + k = 0 represents a circle with area 12π. What is the value of k?

1 (Correct answer)
−2
−1
2

Correct answer: 1

Complete the square: (x² + 6x + 9) + (y² − 4y + 4) = −k + 9 + 4, giving (x + 3)² + (y − 2)² = 13 − k. For area = 12π, we need πr² = 12π, so r² = 12. Setting 13 − k = 12 gives k = 1.

Question 82: Solve for x: 5(x + 2) = 35

x = 33
x = 9
x = 5 (Correct answer)
x = 7

Correct answer: x = 5

Divide both sides by 5: x + 2 = 7; subtract 2: x = 5.

Question 83: A company reports employee salaries. The CEO earns $500,000 while most employees earn around $50,000. Which measure of center should HR use to most accurately represent the typical employee salary?

Range
Mode
Mean
Median (Correct answer)

Correct answer: Median

The CEO's high salary would skew the mean upward, so the median better represents the typical employee salary.

Question 84: A rectangular room is 12 ft long and 10 ft wide. What area of carpet covers the floor?

44 ft²
22 ft²
240 ft²
120 ft² (Correct answer)

Correct answer: 120 ft²

Area = 12 × 10 = 120 ft².

Question 85: Which word could replace 'novel' in the following sentence without changing the meaning? 'The researcher's novel approach to treating the disease involved combining traditional methods with cutting-edge technology.'

fictional
published
innovative (Correct answer)
lengthy

Correct answer: innovative

In this context, 'novel' means new, original, or innovative. The phrase 'combining traditional methods with cutting-edge technology' supports this meaning, indicating a fresh, innovative approach.

Question 86: A researcher collects a random sample of 400 adults and constructs a 95% confidence interval for the mean commute time: (22 min, 30 min). Which of the following statements about this interval is correct?

There is a 95% probability that the true mean is between 22 and 30 minutes.
The true mean commute time is guaranteed to be between 22 and 30 minutes.
95% of adults in the sample have a commute time between 22 and 30 minutes.
If the sampling process is repeated many times, approximately 95% of resulting intervals will contain the true mean. (Correct answer)

Correct answer: If the sampling process is repeated many times, approximately 95% of resulting intervals will contain the true mean.

The correct interpretation of a 95% confidence interval is that the procedure produces intervals that capture the true parameter about 95% of the time across repeated samples.

Question 87: A researcher studies the relationship between hours of sleep and test scores. The scatter plot shows a correlation coefficient of r = 0.73. Which statement best describes this relationship?

There is a weak negative correlation between sleep and test scores
There is a strong positive correlation between sleep and test scores (Correct answer)
The correlation is moderate but the direction cannot be determined
There is no significant correlation between sleep and test scores

Correct answer: There is a strong positive correlation between sleep and test scores

A correlation coefficient of r = 0.73 indicates a strong positive correlation (closer to 1.0), meaning as hours of sleep increase, test scores tend to increase significantly.

Question 88: A table shows the percentage of adults in four age groups who reported exercising regularly: ages 18–25: 62%; ages 26–35: 55%; ages 36–50: 47%; ages 51+: 38%. Which conclusion is directly supported?

Older adults exercise less because they have more health problems.
People under 26 exercise more than any other age group in the general population.
The difference between age groups is caused by differing access to gyms.
Regular exercise participation decreases with age in this sample. (Correct answer)

Correct answer: Regular exercise participation decreases with age in this sample.

The data shows a consistent decline in the percentage from youngest to oldest group, directly supporting the trend.

Question 89: A student's test scores are 78, 82, 91, 85, and 79. What score must the student earn on the 6th test to achieve a mean of exactly 85?

90
92
95 (Correct answer)
99

Correct answer: 95

Required total = 85 × 6 = 510; current sum = 78+82+91+85+79 = 415; needed score = 510 − 415 = 95.

Question 90: A scatterplot shows a strong negative linear relationship between hours of TV watched (x) and GPA (y). The regression line is y = −0.3x + 4.2. Predict the GPA for a student who watches 8 hours of TV per day.

2.1
1.8 (Correct answer)
2.4
4.2

Correct answer: 1.8

y = −0.3(8) + 4.2 = −2.4 + 4.2 = 1.8.

Question 91: Text 1 argues that social media amplifies misinformation. Text 2 reports that platforms have reduced false content by 40% through content moderation policies. How does Text 2 relate to Text 1?

Text 2 suggests that the problem Text 1 describes is being actively addressed. (Correct answer)
Text 2 contradicts Text 1 by showing that social media prevents misinformation.
Text 2 confirms that 60% of content on social media is false.
Text 2 proves that social media no longer amplifies misinformation.

Correct answer: Text 2 suggests that the problem Text 1 describes is being actively addressed.

A 40% reduction shows efforts to address the problem, neither fully refuting nor ignoring Text 1's concern.

Question 92: A circle has radius 6. A chord subtends a central angle of 120°. What is the area of the circular segment (the region between the chord and the arc)?

6π − 9√3
12π − 9√3 (Correct answer)
12π − 18√3
12π − 9

Correct answer: 12π − 9√3

Sector area = (120/360)·π(6²) = (1/3)·36π = 12π. The triangle formed by the two radii and the chord has area = (1/2)(6)(6)sin(120°) = 18·(√3/2) = 9√3. Segment area = sector − triangle = 12π − 9√3.

Question 93: In similar triangles, the ratio of similarity is 3:4. The smaller triangle has altitude 12. What is the larger triangle's altitude?

16 (Correct answer)
18
12
9

Correct answer: 16

3/4 = 12/h → h = 48/3 = 16.

Question 94: A shop sells two types of gift bags. Type X sells for $6 and Type Y for $9. A customer buys 8 bags and pays $57. How many Type X bags were purchased?

6
3
5 (Correct answer)
4

Correct answer: 5

Let x be Type X: 6x + 9(8 − x) = 57 → 6x + 72 − 9x = 57 → −3x = −15 → x = 5.

Question 95: A study finds that a random sample of 225 high school students has a mean screen time of 4.5 hours per day with a margin of error of ±0.3 hours at 95% confidence. How would the margin of error change if the sample size increased to 900 students, assuming the same standard deviation?

It would decrease to ±0.15 hours. (Correct answer)
It would remain ±0.3 hours.
It would decrease to ±0.075 hours.
It would increase to ±0.6 hours.

Correct answer: It would decrease to ±0.15 hours.

Quadrupling the sample size from 225 to 900 halves the margin of error: 0.3 ÷ 2 = 0.15 hours.

Question 96: Two similar cones have radii in the ratio 2:3. The smaller cone has volume 16π cm³. What is the volume of the larger cone?

54π cm³ (Correct answer)
81π cm³
36π cm³
24π cm³

Correct answer: 54π cm³

Volume ratio = (2/3)³ = 8/27; larger volume = 16π × (27/8) = 54π cm³.

Question 97: A taxi charges $2.50 plus $1.80 per mile. The total fare was $11.50. How many miles did the passenger travel?

6 miles
4 miles
5 miles (Correct answer)
7 miles

Correct answer: 5 miles

2.50 + 1.80m = 11.50 → 1.80m = 9 → m = 5.

Question 98: A rectangle has perimeter 40 cm and width 8 cm. What is its area?

96 cm² (Correct answer)
192 cm²
48 cm²
128 cm²

Correct answer: 96 cm²

2(l + 8) = 40 → l = 12 cm; Area = 12 × 8 = 96 cm².

Bluebook SAT Digital Assessment

The SAT is a standardized college admissions test covering evidence-based reading, writing, and mathematics.

Questions

195 min

Time Limit

65%

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