SHSAT 2025 Official Test: Math Tutoring Walkthrough (Q63–Q114)

Direct answer: Below is the full math tutoring walkthrough of the SHSAT 2025 official test, covering questions 63 through 114. Every answer has been independently verified. Each problem gets a one-line method so students can see the pattern, not just the answer — because on the SHSAT, recognizing the pattern is what saves you 30 seconds per question.

How to use this SHSAT math walkthrough

The 2025 SHSAT math section is not harder than previous years in raw difficulty — but the wording is tighter, and the answer choices are engineered so that a student who stops one step too early sees their intermediate result in the options. That is the entire trap.

At SOMATH, the Upper West Side math tutoring program at 226 W 79th St, we tell every SHSAT student the same thing: the arithmetic is easy; the reading is the test. So when you go through this walkthrough, do not just read the answers. Read the method. Each method is the routine we teach in class — a quick, repeatable check that closes the gap between "got the math right" and "picked the right answer."

SHSAT 2025 — math walkthrough, Q63 to Q114

Q63 — Reverse percent. Answer: A. 50.

Question. Mr. Jones has 550 goats, which is 10% more than Mr. King has. How many more goats does Mr. Jones have than Mr. King?

Jones has 550 goats, which is 10% more than King. So King has 550 ÷ 1.10 = 500. Difference: 50. The trap is computing 10% of 550 (= 55) — that is forward percent, not reverse percent.

Q64 — Fractions with a common denominator. Answer: G. 3y.

Question. If 2y/x − y/(2x) = ☐/(2x), what expression is represented by ☐?

2y/x − y/(2x) = 4y/(2x) − y/(2x) = 3y/(2x). The box equals 3y.

Q65 — Square area to quarter circle. Answer: B.

Question. The area of square PQRS is 4 cm². S is the center of the circle. Find the area of the shaded quarter-circle.

Square area = 4, so side = 2, which is also the radius. Quarter-circle area = π · r² / 4 = π · 4 / 4 = π.

Q66 — Consecutive integers count. Answer: H. 67.

Question. A list contains consecutive integers from m to n. If n − m = 66, how many integers are in the list?

For a list of consecutive integers from m to n, the count is (n − m) + 1 = 66 + 1 = 67. The "+1" is the only thing students forget here.

Q67 — Algebraic simplification. Answer: A. x.

Question. Simplify ( 39(x − 3)/3 + 39 ) ÷ 13.

(39(x − 3)/3 + 39) / 13 = (13(x − 3) + 39) / 13 = (13x − 39 + 39) / 13 = 13x / 13 = x. Cancel before you distribute and this is a five-second problem.

Q68 — Pouring fractions. Answer: G. 5/8 cup.

Question. Jar 1 and Jar 2 each contain ½ cup of liquid. One-fourth of the liquid in Jar 1 is poured into Jar 2. How much is now in Jar 2?

¼ of ½ cup = ⅛ cup is poured from Jar 1 into Jar 2. Jar 2 now holds ½ + ⅛ = 5/8 cup.

Q69 — Reciprocal inequality. Answer: D. n = 99.

Question. 0.01 lies between 1/n and 1/(n + 2). What is the value of n?

0.01 = 1/100. For 1/100 to sit between 1/n and 1/(n+2), pick n near 100: 1/99 > 1/100 > 1/101 works. n = 99.

Q70 — Backwards from a percent. Answer: H. 200.

Question. In a poll, 72% of those surveyed said yes and 56 people did not say yes. How many people were polled in total?

72% said yes means 28% said no. 28% of total = 56, so total = 56 ÷ 0.28 = 200.

Q71 — Scale drawing area. Answer: A. 1 ⅕ in.

Question. The actual floor area is 960 sq ft. A scale drawing uses 1 inch = 20 feet. If the length in the drawing is 2 inches, what is the width in the drawing?

Actual length = 2 in × 20 ft/in = 40 ft. Actual width = 960 ÷ 40 = 24 ft. Drawing width = 24 ÷ 20 = 1.2 in = 1 ⅕ in.

Q72 — Probability prediction. Answer: H. 167.

Question. A sequence of 300 numbers is generated from the integers 1 through 9. Predict approximately how many of those 300 numbers are odd.

Odd numbers between 1 and 9: {1, 3, 5, 7, 9} — 5 out of 9. So 5/9 × 300 ≈ 166.67, which rounds to 167.

Q73 — Discount, base fee, and miles. Answer: D. 35 miles.

Question. A rental costs a $40 base fee plus $1 per mile. With a 20% off coupon, the customer spent $60. How many miles did she drive?

She paid 80% of the original (20% off coupon), so original = 60 ÷ 0.80 = 75. Subtract the $40 base fee: $35 paid for mileage, which equals 35 miles at $1/mile.

Q74 — Probability after removal. Answer: H. 1/4.

Question. A jar contains 25 candies. The probability of drawing a red candy is 2/5. If 5 red candies are removed, what is the new probability of drawing a red candy?

Start with 2/5 × 25 = 10 red out of 25. Remove 5 red: 5 red out of 20 total. 5/20 = 1/4.

Q75 — Sequence rule, two directions. Answer: C. 112.

Question. Each term in a sequence is given by 2 × (previous term) + 1. The 9th term is 63. Find (10th term) − (7th term).

Rule: term = 2 × previous + 1. 9th term = 63. Going forward: 10th = 2·63 + 1 = 127. Going backward (invert: previous = (term − 1)/2): 8th = 31, 7th = 15. So 10th − 7th = 127 − 15 = 112. The skill is being comfortable running the rule in both directions.

Q76 — Drop high and low, then mean. Answer: G. 8.64.

Question. Seven scores are recorded: 8.9, 8.2, 8.5, 9.0, 8.4, 8.6, 8.8. Drop the highest and lowest, then find the mean of the remaining scores.

Drop 9.0 (high) and 8.2 (low). Mean of the remaining five (8.9, 8.5, 8.4, 8.6, 8.8) = 43.2 / 5 = 8.64.

Q77 — Range for the shorter piece. Answer: A. 0 < x < 2 ¼.

Question. A 4½-foot piece of wood is cut into two pieces of different lengths. If x is the length of the shorter piece, which inequality describes x?

Wood is 4 ½ ft, cut into two unequal pieces. If x is the shorter, x must be positive but strictly less than half the total length (otherwise it would be the longer or equal). So 0 < x < 2 ¼.

Q78 — Trapezoid on a grid. Answer: G. 150.

Question. Find the area of the shaded region: a trapezoid with bases 10 (top, from x = 5 to x = 15) and 20 (bottom, from x = 0 to x = 20), with height 10 (from y = 5 to y = 15).

Bases 10 (top) and 20 (bottom), height 10. Area = ½ (10 + 20)(10) = 150.

Q79 — Fahrenheit to Celsius difference. Answer: A. 10°.

Question. Two temperatures are 86°F and 68°F. What is the difference between them in degrees Celsius?

86°F = 30°C; 68°F = 20°C; difference = 10°C. Shortcut: a Fahrenheit difference of 18° equals a Celsius difference of 10° because the slope is 9/5.

Q80 — Sum of even neighbors. Answer: F. 2x.

Question. If x is an odd integer, what is the sum of the two even integers closest to x?

If x is odd, the two closest even numbers are x − 1 and x + 1, which sum to 2x. Don't reach for algebraic substitution — read the structure.

Q81 — Per capita debt. Answer: B. $14,400.

Question. A nation's debt is $3.6 trillion and its population is 250 million. What is the debt per person?

3.6 × 10¹² ÷ 2.5 × 10⁸ = 1.44 × 10⁴ = $14,400. Big-number division becomes easy when you switch to scientific notation.

Q82 — Midpoint segment combination. Answer: H. MP.

Question. On segment MQ, N is the midpoint and P lies between N and Q. MN = x and PQ = y. Which segment has length 2x − y?

MN = x and N is the midpoint of MQ, so NQ = x and MQ = 2x. PQ = y, so NP = x − y. Then MP = MN + NP = x + (x − y) = 2x − y.

Q83 — Parallelogram area. Answer: C. 500 sq ft.

Question. Find the area of a parallelogram with base 25 ft and vertical height 20 ft.

Area = base × vertical height (not the slanted side). 25 × 20 = 500.

Q84 — Three-day total with relations. Answer: H. 160.

Question. Wednesday's count is 100 more than Tuesday's. Thursday's count is 50 less than Tuesday's. The three days total 230. What is Wednesday's count?

Let Tuesday = T. Wed = T + 100, Thu = T − 50. Sum: 3T + 50 = 230, so T = 60 and Wed = 160. Always solve for the variable you defined, then substitute back — students lose points by reporting T instead of Wed.

Q85 — Weighted mean of quiz scores. Answer: C. 75.

Question. On a quiz: 9 students scored 60, 7 scored 70, 4 scored 80, 5 scored 90, and 3 scored 100. What is the mean score?

Total points: 60·9 + 70·7 + 80·4 + 90·5 + 100·3 = 2,100. Total students: 28. Mean = 2,100 / 28 = 75. Treat the frequency table like an x-bar problem and you cannot get it wrong.

Q86 — Inequality graph. Answer: G.

Question. Which number-line graph represents the solution to x + 4 ≥ 3?

x + 4 ≥ 3 simplifies to x ≥ −1: closed circle at −1, shaded to the right.

Q87 — Reciprocal of a sum of reciprocals. Answer: A. 3/13.

Question. Add the reciprocal of ¼ to the reciprocal of 3. Find the reciprocal of that sum.

Reciprocals of ¼ and 3 are 4 and ⅓. Sum: 4 + ⅓ = 13/3. Reciprocal: 3/13. Three nested operations — students who slow down on "reciprocal of" twice rarely miss this.

Q88 — Difference of squares. Answer: E. 56.

Question. Two square posters have side lengths 13 in and 15 in. What is the difference in their areas?

15² − 13² = (15 − 13)(15 + 13) = 2 × 28 = 56. The difference of squares factoring is faster than the direct subtraction.

Q89 — Unit rate from mixed fractions. Answer: B. 8/39.

Question. A recipe uses 2/3 cup of oats for every 3¼ cups of water. What is the unit rate (cups of oats per cup of water)?

Oats per water = (2/3) ÷ (13/4) = (2/3) × (4/13) = 8/39. The skill is converting "3 ¼" to 13/4 before dividing.

Q90 — Solving a fractional equation. Answer: H. 55/6.

Question. Solve for x: (3/5 − 1/2) x = 1/4 + 2/3.

(3/5 − 1/2)x = 1/4 + 2/3 becomes (1/10)x = 11/12, so x = 110/12 = 55/6. Common-denominator both sides first.

Q91 — Tax rate change. Answer: A. $1.

Question. The sales tax rate increases from 7.0% to 7.5%. On a $200 purchase, by how much does the tax increase?

The rate change is 0.5 percentage points (7.5% − 7.0%). 0.005 × $200 = $1. Notice the trap word: it is the change in tax, not the new total tax.

Q92 — Absolute value chain. Answer: F. −18.

Question. Evaluate |(−8) − 12 + (−17) − (−31)| − |24|.

Inside the first | · |: −8 − 12 − 17 + 31 = −6, so | · | = 6. Then 6 − |24| = 6 − 24 = −18. Yes, the final answer is negative — absolute value bars do not make the outer expression positive.

Q93 — Percent pie with algebra. Answer: B. 20%.

Question. A pie chart shows four colors with sizes 8k%, 30%, (30 − 2k)% and (k + 5)%. They sum to 100%. What percent is blue (the (30 − 2k)% slice)?

8k + 30 + (30 − 2k) + (k + 5) = 100 → 7k + 65 = 100 → k = 5. Blue = 30 − 2k = 30 − 10 = 20%. The setup is "everything must add to 100."

Q94 — Number line midpoint. Answer: E. 1/2.

Question. On a number line, the distance from X to Y is 1 and the distance from Y to Z is 2 (with X, Y, Z in that order). What is the distance from Y to the midpoint of XZ?

Place Y at 0. Then X = −1 (XY = 1) and Z = 2 (YZ = 2). Midpoint of XZ = (−1 + 2)/2 = 0.5. Distance from Y to that midpoint = 0.5.

Q95 — Percent increase. Answer: B. 8%.

Question. A price rises from $1.25 to $1.35. What is the percent increase?

Increase = 1.35 − 1.25 = 0.10. Percent increase = 0.10 / 1.25 = 0.08 = 8%. Divide by the original, never the new value.

Q96 — Surface area of a pyramid. Answer: G. 340.

Question. Find the total surface area of a square pyramid with base side 10 and triangular face (slant) height 12.

Square base: 10 × 10 = 100. Four triangle faces: 4 × (½ × 10 × 12) = 240. Total = 340.

Q98 — Fifth-game score from means. Answer: G. 12.

Question. The mean score over 5 games is 8. The mean over the first 4 games is 7. What was the score in the fifth game?

Total over 5 games = 5 × 8 = 40. Total over first 4 = 4 × 7 = 28. Fifth game = 40 − 28 = 12. The "total = mean × count" identity is one of the most reused tricks on the SHSAT.

Q100 — Decimal division. Answer: H. 200.

Question. Compute 3.6 ÷ 0.018.

3.6 ÷ 0.018 = 3,600 ÷ 18 = 200. Multiply both numerator and denominator by 1,000 to clear decimals.

Q101 — Tank filling, percent full. Answer: C. 60%.

Question. A 500-gallon tank starts with 75 gallons. Water is added at 5 gallons per minute for 45 minutes. What percent full is the tank now?

Starting 75 gallons + (5 gal/min × 45 min) = 300 gallons. 300 / 500 = 60%.

Q114 — Constant of proportionality. Answer: F. 3.

Question. 16 cups cost $48 and 4 cups cost $12. What is the constant of proportionality (cost per cup)?

$48 / 16 cups = $3/cup; $12 / 4 cups = $3/cup. k = 3.

Patterns in the SHSAT 2025 math section

If you read the answers above as a set, the patterns jump out — and recognizing patterns is exactly what good SHSAT math tutoring builds. A handful of question families produced most of the 2025 problems:

• Reverse percent (Q63, Q70, Q73, Q91, Q95). Students who only know forward percent ("X% of Y") lose 3–5 points here every test.
• Sequence rules run in both directions (Q75). The forward direction is easy; the backward direction is what separates the 95th-percentile scorer.
• Weighted means and total = mean × count (Q76, Q85, Q98). One identity, three different disguises.
• Geometry with the right base or radius (Q65, Q78, Q83, Q96). Students drop points by using the slanted side as the height or confusing radius with diameter.
• "What is the question really asking?" (Q70, Q73, Q84, Q91). Students compute correctly and then choose the intermediate value. This is the most expensive habit on the test.

None of these are obscure topics. They are all things a strong 7th-grader has technically seen. What turns "seen" into "scored" is structured practice with feedback — which is what a math tutoring program is actually for.

How SOMATH teaches the 2025 SHSAT

At SOMATH, our SHSAT math tutoring is built around three things: diagnostic accuracy, pattern practice, and timed mixed sets. Every new student starts with a free 30-minute evaluation; within 48 hours, the family receives a written diagnostic that names the specific question families their child is missing (e.g., reverse percent, sequence backward, area-vs-perimeter confusion). Generic "practice more" advice is what we replace.

From there, sessions in our 226 W 79th St classroom on the Upper West Side mix two modes: short concept blocks (one pattern, ten focused problems) and timed mixed sets that force the student to switch between question families — because on test day, the SHSAT does not warn you when the topic changes. That ability to switch quickly is the real difference between a student who knows the math and a student who scores well.

FAQ

Is the SHSAT 2025 math section harder than previous years?

Not in absolute difficulty. The 2025 SHSAT math section continues to emphasize the same recurring patterns — reverse percent, sequence rules, simple geometry with hidden traps, and weighted means — but the wording is tighter, so reading misses cost more points than arithmetic misses.

What math topics show up most on the SHSAT?

Percent (both forward and reverse), ratios and unit rates, basic algebra (one-variable equations and inequalities), geometry of squares, circles, trapezoids and parallelograms, weighted means, sequences defined by a rule, and absolute value. The 2025 test hits all of these across Q63–Q114.

How should my child use this walkthrough?

Have them attempt each question cold first, then check the one-line method here. The goal is not to memorize answers — it is to recognize which family each question belongs to so they can match the right method on test day.

When should my child start SHSAT math tutoring?

Most students start structured SHSAT math tutoring in late 6th grade or early 7th grade. If your child is currently behind in fractions, ratios, or percent fluency, starting earlier reduces pressure later. We see plenty of strong students who start in 8th grade and still hit their target score — what matters is consistency, not just timing.

Where is SOMATH located?

SOMATH (School of Math) is at 226 W 79th St on the Upper West Side in Manhattan. We offer in-person and online math tutoring for SHSAT, AP Calculus, SAT, and K–12 enrichment.

Related reading

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