35 real, hard-difficulty SAT Math questions from the Algebra and Advanced Math domains, compiled by our tutors. Attempt each one, then reveal the answer and a full explanation.
The Full List
Attempt each problem before revealing the answer and explanation — the struggle is where the learning happens. This is the same challenge set our tutors use to stress-test students who are already scoring well.
Correct Answer: 403
For a linear equation in one variable to have infinitely many solutions, the coefficients of x must be equal on both sides, and the constant terms must also be equal. Rewriting the left side gives 3x + 7 − s13 = rx − 8r. Matching x-coefficients gives r = 3. Matching constants gives 7 − s13 = −8(3) = −24, so s13 = 31, and s = 403.
Correct Answer: 33
The slope between the two given points is −15 − 13(k + 7) − k = −287 = −4. Using the point (k, 13) and the y-intercept (k − 5, b): −4 = 13 − b5. Multiplying both sides by 5 gives −20 = 13 − b, so b = 33.
Correct Answer: A
Rewriting ax + ky = 6 in slope-intercept form gives y = −akx + 6k. The point (0, −3) is the y-intercept, so 6k = −3. Multiplying both sides by k gives 6 = −3k, so k = −2.
Correct Answer: D
In 5 hours the auger removed 24,000 − 19,350 = 4,650 bushels, a rate of 4,6505 = 930 bushels per hour. Solving 24,000 − 930x = 12,840 gives x = 12.
Correct Answer: D
Multiplying the first equation by −2 gives −2ax − 2by = −144. Adding this to the second equation gives (−2a + 6)x = −88. Since x = 4 at the intersection, (−2a + 6)(4) = −88, so −2a + 6 = −22, giving a = 14.
Correct Answer: D
The cost is $270 for the first day plus $135 for each of the remaining (x − 1) days: y = 270 + 135(x − 1), which simplifies to y = 135x + 135.
Correct Answer: D
Taking the bus takes w + 5 minutes in total (wait time plus the ride). Walking is faster exactly when w + 5 > 20.
Correct Answer: 2/7 (also accepted: .2857, 0.285, 0.286)
Multiplying the first equation by 354 gives 72x + 494y = 52, which matches the second equation, so g = 72 and k = 494. Therefore gk = 7/249/4 = 27.
Correct Answer: D
Dividing the second equation by 4 gives 7x + 6y = 5 — identical to the first equation, so any point that satisfies one satisfies both. Substituting r for y and solving for x gives x = −6r7 + 57, so the point is (−6r7 + 57, r).
Correct Answer: D
Distributing and simplifying gives (r − 28)x = 23. For the equation to have exactly one solution, the coefficient (r − 28) cannot equal 0 — so r cannot equal 28.
Correct Answer: D
At $4 per gallon and 25 miles per gallon, the cost per mile is 425 dollars. Reducing weekly spending by $5 means 425m = 5.
Correct Answer: A
An increase of 9.10 kelvins increases F(x) by 95(9.10) = 16.38 degrees Fahrenheit.
Correct Answer: B
The first session is free ($0), the second is half off ($12.70), and the remaining (x − 2) sessions are full price: f(x) = 0 + 12.70 + 25.40(x − 2), or f(x) = 25.40(x − 2) + 12.70.
Correct Answer: D
Setting F(x) = 2.74 gives 0 = −0.19(x − 3), so x = 3 — which is December 1, 2014 (3 months after September 1). So 2.74 is the average price per gallon on that date.
Correct Answer: D
Staining twice covers 2w square feet. Dividing by 170 square feet per gallon gives 2w170, which simplifies to w85.
Correct Answer: C
The triangle inequality gives three conditions: 6 + x > 12, 6 + 12 > x, and 12 + x > 6. Together, these simplify to 6 < x < 18.
Correct Answer: 2/43 (also accepted: .0465, 0.046, 0.047)
Since f(cx) = x − 8 for all x, setting cx = 2 gives x = 2c. Substituting into f(2) = 35: 2c − 8 = 35, so 2c = 43, and c = 243.
Correct Answer: 20
The condition is 6n500 ≤ 0.25, so 6n ≤ 125, which gives n ≤ 20.833. Since n must be a whole number, the greatest possible value is 20.
Correct Answer: 609
Since a < 0, |a|a = −1. Then g(15a) = 15(−1) − 14 = −29, and g(7a) = 7(−1) − 14 = −21. The product of g(15a) and g(7a) is (−29)(−21) = 609.
Correct Answer: 20
Substituting x = 0, y = −757 into y = −ax + b − 12 gives −757 = −1 + b − 12, so b = 167. Since ab = 3207, a = 320/716/7 = 20.
Correct Answer: 27
Combining the right-hand side over the common denominator x(x + 51) gives (p + w)x + 51px(x + 51). Matching numerators with 29x + 102 gives p + w = 29 and 51p = 102 (so p = 2). Therefore w = 29 − 2 = 27.
Correct Answer: D
Since f(−9) = f(3), the vertex's x-coordinate is halfway between −9 and 3, so h = −3. Writing f in vertex form and matching to ax² + 4x + c shows a = 23, so a ≥ 1 is false (II is not true). It also shows k = c − 6; since k < 0 just means c < 6, c could still be zero or positive, so c < 0 is not necessarily true (I is not true either). Neither statement must be true.
Correct Answer: A
The population increases by 4% each time the exponent 64t increases by 1 — that is, each time t increases by 23 of a year. Since 23 of a year is 8 months, n = 8.
Correct Answer: D
Substituting x = −5 into x² + y² = 36 gives y² = 11, and since y < 0, y = −√11. Substituting x = −5 and y = −√11 into the second equation: −√11 = −5m + b4. Adding 5m to both sides and multiplying by 4 gives b = 20m − 4√11.
Correct Answer: A
Since the graph passes through (0, 22), g(0) = −a−b = 22, so a = 22b. Since g(−22) = 0, the numerator must be 0: (−22)² − (−22) − 22b = 0, or 484 + 22 = 22b, so 22 + 1 = b (after dividing by 22), giving b = 23.
Correct Answer: A
For x² − cx − b to equal x + b, x² − c must factor as (x + b)(x − b) = x² − b² — a difference of squares. So c must be a perfect square; only choice A (4) is one.
Correct Answer: A
The left side factors as (x + b)(57x + a) = 0, giving solutions x = −b and x = −a57. Their product is ab57. Since this equals kab, k = 157.
Correct Answer: 15.5 (also accepted: 31/2)
Expanding (3x² + a)(2x² + b) and matching coefficients gives 3b + 2a = 31 and ab = 35, whose positive-integer solution is a = 5, b = 7. Repeating the same relationships while allowing noninteger constants gives c = 10.5, d = 35/10.5. So a + c = 5 + 10.5 = 15.5.
Correct Answer: 31
Completing the square on each equation gives r = −3 − √2, s = −3 + √2, t = −4 − 2√2, u = −4 + 2√2. Then r + t = −7 − 3√2 and s + u = −7 + 3√2. Since c equals the product of the new roots, c = (−7)² − (3√2)² = 49 − 18 = 31.
Correct Answer: B
A population that is 150% more each year is multiplied by 1 + 1.5 = 2.5 annually, so n = a(2.5)ᵗ for some initial value a. Since n = 180 when t = 1, 180 = 2.5a, so a = 72. The model is n = 72(2.5)ᵗ.
Correct Answer: D
Each choice can be written as (3x)(x) + (7)(2b) terms, so factoring as (3x + 7)(x + 2b) gives 3x² + (7 + 6b)x + 14b. Testing each choice's x-coefficient (7, 28, 42, 49) for a positive-integer b shows only 49 works, where 7 + 6b = 49 gives b = 7.
Correct Answer: B
Multiplying every term by the common denominator 12xyz and simplifying gives y² = 4x − 3z12x²z². Since y is positive, y = √4x − 3z12x²z².
Correct Answer: D
Expanding (hx + k)(x + j) and matching the constant term gives kj = −45. Since h, k, j are integers, j = −45k must be an integer — so 45k must be an integer too.
Correct Answer: 41/81 (also accepted: .5061, .5062, 0.506)
The left side is equivalent to p2/3, so p2/3 = t9/7. Substituting t = p3n−1 gives p2/3 = p(9/7)(3n−1), so 23 = 97(3n − 1). Multiplying both sides by 21 gives 14 = 27(3n − 1), or 14 = 81n − 27. Adding 27 and dividing by 81 gives n = 4181.
Correct Answer: D
Writing the parabola as y = a(x − 9)² − 14 and expanding shows a + b + c = 64a − 14. Since the vertex is below the x-axis and the parabola crosses it twice, the parabola must open upward, so a > 0. Only choice D (−12) gives a positive a (a = 132).
Book a free consultation and we will match your student with the perfect tutor.
