SHSAT · NYC Specialized High Schools · Test Prep · Upper West Side
SHSAT 2025-2026 Practice Test Form A — Math: Full Walkthrough of All 57 Questions with Answers & Solutions
Every question on the SHSAT 2025-2026 Sample Test Form A, Math section, transcribed verbatim from the NYC Department of Education handbook, with the official answer key and hidden step-by-step solutions. Built by SOMATH, the math school on the Upper West Side of Manhattan.
Looking for the complete walkthrough of the SHSAT 2025-2026 Sample Test Form A, Math section? You are in the right place. This page contains all 57 SHSAT Math questions from the official NYC Department of Education handbook — grid-in questions 58 through 62 and multiple-choice questions 63 through 114 — transcribed verbatim, worked out step by step, with hidden solutions and a link to the official PDF. The lettering follows the SHSAT convention: odd-numbered items use choices A/B/C/D and even-numbered items use E/F/G/H.
This is the traditional paper-format 114-question SHSAT (57 ELA + 57 Math) that was administered in fall 2024 to New York City eighth and ninth graders applying for admission to the 2025-2026 school year. Starting with the 2026-2027 cycle the DOE has moved the SHSAT to a digital, shortened TestNav-delivered format (50 ELA + 50 Math), but the underlying topic coverage on Form A 2025-2026 is the same as what students will see digitally — ratios, algebraic expressions, word problems, probability, geometry — so this walkthrough is still the fastest way to drill every SHSAT Math skill in one sitting. If you want a compact side-by-side of what changed on the 2026 digital version, SHSATlab has a clean summary here.
Whether you are a rising eighth grader taking the SHSAT in October, a parent looking for answer explanations, or a teacher building a Math review packet, this walkthrough is designed to be the most complete, honest, and student-friendly SHSAT Math walkthrough on the internet. Written by the same team that teaches SHSAT 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.
What’s in this walkthrough
How the SHSAT Math section works
The SHSAT — Specialized High Schools Admissions Test — is the exam used by the NYC Department of Education for admission to eight of the nine specialized high schools, including Stuyvesant, Bronx Science, Brooklyn Tech, Brooklyn Latin, Staten Island Tech, HSMSE, HSAS, and Queens HSSYS. LaGuardia is the ninth specialized high school and uses an audition rather than the SHSAT.
Question numbering. The SHSAT numbers ELA questions 1–57 and Math questions 58–114, all in one continuous booklet. Because Math starts at 58, the answer document also picks up at 58. Grid-in items come first (58–62) so students can spend their sharpest early minutes on write-in answers, followed by 52 multiple-choice items (63–114). Odd-numbered multiple-choice items use choices A/B/C/D and even-numbered items use E/F/G/H — a convention that lets scorers verify at a glance that answers were bubbled on the correct row.
No calculator. Unlike the SAT, the SHSAT does not permit any calculator. Students must be quick and accurate with mental arithmetic, fraction and decimal manipulation, percent, ratio, and estimation. This is a large part of why SHSAT prep looks different from SAT prep. Every one of the 57 Math questions on this page is designed to be solvable in one to two minutes by hand.
Grid-in mechanics. For questions 58–62, students write a numeric answer into a grid at the top of five columns of bubbles, then bubble the matching digit under each column. Fractions may be entered but must be in lowest terms; negative signs go in the leftmost column; decimal points count as a column. Study the sample grid before test day so you do not lose easy points on formatting.
Scoring. The SHSAT produces a raw score for each section (ELA and Math), which is then converted to a scaled score. There is no penalty for guessing, so students should never leave a question blank — even a random guess on a multiple-choice item has a 25% chance of being correct.
The 14 SHSAT Math topic domains
The SHSAT Math section covers 14 topic domains — the same list on both the 2025-2026 paper edition and the 2026-2027 digital edition. On Form A 2025-2026 the topic distribution looks like this:
| Topic | Questions on Form A 2025-2026 |
|---|---|
| Word Problems & Modeling | 7 |
| Algebraic Expressions | 7 |
| Ratios, Proportions & Rates | 6 |
| Charts, Tables & Data | 6 |
| Number Theory | 6 |
| Unit Rates & Conversions | 5 |
| Coordinate Plane & Number Lines | 5 |
| Probability & Counting | 4 |
| Fractions, Decimals & Mixed Numbers | 4 |
| Linear Equations (One Variable) | 3 |
| Inequalities | 2 |
| Surface Area & Volume | 1 |
| Lines, Angles & Triangles | 1 |
| Area, Perimeter & Circles | 0* |
| *Circles appear on Form C 2026, which is a different test form. | |
The two biggest levers on Form A are word problems + modeling and algebraic expressions — together 14 of the 57 questions. Ratios, tables/charts, and number theory are the next tier. If you have less than a month before test day, drill those five topic areas first.
Grid-in Questions (58–62)
Five numeric write-in questions. No answer choices — you must arrive at the exact numeric answer and bubble it correctly on the grid.
Question 58
Simplify: −3.8 + 2.3 − (−1.1)
Grid-in question — enter your numeric answer.
Answer: −0.4
Rewrite the double negative first: −(−1.1) = +1.1, so the expression becomes −3.8 + 2.3 + 1.1.
Now add and subtract left to right. −3.8 + 2.3 = −1.5. Then −1.5 + 1.1 = −0.4.
Grid in −0.4. On the SHSAT grid, put the negative sign in the leftmost column, then ., 4. A leading zero is not required.
Question 59
Angle M and angle R are supplementary. The measure of angle R is 5 times the measure of angle M. What is the measure of angle R, in degrees?
Grid-in question — enter your numeric answer.
Answer: 150
Supplementary means the two angles sum to 180°. Let m∠M = x and m∠R = 5x.
x + 5x = 180 6x = 180 x = 30
So m∠M = 30° and m∠R = 5(30) = 150°. Grid in 150.
Question 60
A juice mixture contains 3/16 gallon of apple juice and 3/40 gallon of cranberry juice. How many gallons of apple juice per gallon of cranberry juice does the mixture contain? (Express your answer as a decimal.)
Grid-in question — enter your numeric answer.
Answer: 2.5
"Gallons of apple juice per gallon of cranberry juice" is a ratio of apple to cranberry, i.e., divide apple by cranberry.
(3/16) ÷ (3/40) = (3/16) × (40/3) = 40/16 = 5/2 = 2.5
The mixture contains 2.5 gallons of apple juice per gallon of cranberry juice.
Question 61
Mr. Chan's lawn grows 2 1/8 inches every 2 weeks. He mows his lawn every 2 weeks and cuts off the top 1 3/4 inches of lawn. If Mr. Chan's lawn was 4 inches tall at the beginning of the season, how many inches tall, in decimal form, is Mr. Chan's lawn after 8 weeks?
Grid-in question — enter your numeric answer.
Answer: 5.5
Every 2 weeks the lawn grows 2 1/8 = 2.125 in. and Mr. Chan cuts 1 3/4 = 1.75 in. off, so the net change per 2-week cycle is 2.125 − 1.75 = 0.375 in.
In 8 weeks there are 8 ÷ 2 = 4 such cycles, so the total net gain is 4 × 0.375 = 1.5 in.
The lawn started at 4 in., so it is now 4 + 1.5 = 5.5 in. Grid in 5.5.
Question 62
RESULTS FROM SURVEY OF 110 FAMILIES
| Number of Children in the Family | Number of Families |
|---|---|
| 0 | 45 |
| 1 | 32 |
| 2 | 19 |
| 3 | 8 |
| 4 | 6 |
Grid-in question — enter your numeric answer.
Answer: 1
The median of 110 values is the average of the 55th and 56th values when they are listed in order.
Build up cumulative counts from the smallest category:
0 children: families 1–45 1 child: families 46–77 (45 + 32 = 77) 2 children: families 78–96 (77 + 19 = 96) 3 children: families 97–104 4 children: families 105–110
Both the 55th and 56th families fall in the "1 child" row, so the median is 1.
Multiple-choice Questions (63–114)
Fifty-two multiple-choice items. Odd-numbered questions use choices A/B/C/D; even-numbered questions use E/F/G/H. No calculator permitted.
Question 63
Mr. Jones has 550 goats, which is 10% more than Mr. King has. How many more goats does Mr. Jones have than Mr. King?
- A) 50
- B) 55
- C) 495
- D) 500
Answer: A) 50
"10% more than Mr. King" means Jones = 1.10 × King. Solve for Mr. King:
550 = 1.10 × King King = 550 / 1.10 = 500
Difference: 550 − 500 = 50. A common trap is answer D (500), which is Mr. King's total, not the difference. The correct answer is A) 50.
Question 64
If 2y/x − y/(2x) = □/(2x) and x ≠ 0, what expression is represented by □?
- E) y
- F) 2y
- G) 3y
- H) 4y
Answer: G) 3y
Give the two fractions a common denominator of 2x. The first term becomes 2y/x = 4y/(2x).
4y/(2x) − y/(2x) = (4y − y)/(2x) = 3y/(2x)
So □ = 3y. Answer: G) 3y.
Question 65
PQRS is a square. Point S is the center of a circle, and points P and R are on the circle. If the area of the square is 4 square centimeters, what is the area, in square centimeters, of the shaded quarter of the circle?
- A) π/4
- B) π
- C) 2π
- D) 4π
Answer: B) π
Square area = 4 sq cm, so each side = 2 cm. Because S is the center of the circle and P and R are on the circle, the segments SP and SR are radii — both equal to a side of the square, so the radius r = 2 cm.
The full circle has area πr² = π(2)² = 4π. The shaded sector PSR is a quarter of the circle (the interior angle PSR is a 90° angle of the square), so its area is (1/4)(4π) = π.
Answer: B) π.
Question 66
A list of consecutive integers begins with m and ends with n. If n − m = 66, how many integers are in the list?
- E) 2
- F) 33
- G) 66
- H) 67
Answer: H) 67
The count of consecutive integers from m to n inclusive is always n − m + 1. Quick check: from 1 to 3 there are 3 − 1 + 1 = 3 integers (1, 2, 3), which matches.
n − m + 1 = 66 + 1 = 67
Answer: H) 67. A very common wrong answer is 66 — the difference. Remember to add 1 for "inclusive of both endpoints."
Question 67
Simplify: [39(x − 3)/3 + 39] ÷ 13
- A) x
- B) x − 2
- C) 13x − 36
- D) x + 36/13
Answer: A) x
Simplify the numerator first. 39/3 = 13, so 39(x − 3)/3 = 13(x − 3) = 13x − 39.
Numerator = 13x − 39 + 39 = 13x Expression = 13x ÷ 13 = x
Answer: A) x.
Question 68
Jar 1 and Jar 2 each contain 1/2 cup of water. If 1/4 of the water in Jar 1 is poured into Jar 2, how much water is now in Jar 2?
- E) 1/8 cup
- F) 1/4 cup
- G) 5/8 cup
- H) 3/4 cup
Answer: G) 5/8 cup
Amount poured from Jar 1 = (1/4)(1/2) = 1/8 cup.
Jar 2 new amount = 1/2 + 1/8 = 4/8 + 1/8 = 5/8 cup
Answer: G) 5/8 cup.
Question 69
If n is a whole number, and 0.01 is between 1/n and 1/(n+2), what is the value of n?
- A) 0
- B) 1
- C) 2
- D) 99
Answer: D) 99
0.01 = 1/100. We need 1/n > 1/100 > 1/(n+2), which means n < 100 < n + 2.
The whole number n satisfying n < 100 < n + 2 is n = 99: then 99 < 100 < 101 ✓ and 1/99 > 1/100 > 1/101 ✓.
Answer: D) 99.
Question 70
When asked a certain question in a poll, 72% of the people polled answered yes. If 56 people did not answer yes to that question, what is the total number of people who were polled?
- E) 78
- F) 128
- G) 144
- H) 200
Answer: H) 200
If 72% said yes, then 100% − 72% = 28% did not. The 56 non-yes respondents represent 28% of the total.
0.28 × Total = 56 Total = 56 / 0.28 = 200
Answer: H) 200.
Question 71
A museum has a room in the shape of a rectangle. The area of the floor is 960 square feet. In a scale drawing of the museum, 1 inch = 20 feet. If the length of the room is 2 inches in the scale drawing, what is the width of this room in the scale drawing?
- A) 1 1/5 in.
- B) 1 1/4 in.
- C) 24 in.
- D) 40 in.
Answer: A) 1 1/5 in.
Actual length: 2 in. × 20 ft/in. = 40 ft. Actual width: 960 ÷ 40 = 24 ft. Convert actual width back to scale: 24 ft × (1 in. / 20 ft) = 24/20 = 6/5 = 1 1/5 in.
Answer: A) 1 1/5 in. Answer D (40 in.) is a trap that skips the final conversion back to scale-drawing inches.
Question 72
A program on a computer randomly generates a sequence of whole numbers from 1 to 9, inclusive. If the computer generates a sequence of 300 numbers, what is the best prediction of the number of odd numbers in the sequence?
- E) 120
- F) 133
- G) 150
- H) 167
Answer: H) 167
Odd numbers in {1, 2, 3, 4, 5, 6, 7, 8, 9} are 1, 3, 5, 7, 9 — that's 5 out of 9 possible outcomes, so P(odd) = 5/9.
Expected odds = 300 × 5/9 = 1500 / 9 ≈ 166.67
Rounding to the nearest whole number gives H) 167. Note that 150 (G) would be the answer if the range were 1–10 with exactly half odd — check the problem statement carefully.
Question 73
A truck rental company charges a one-time fee of $40 plus $1 per mile driven. Dalia rented a truck and used a coupon for 20% off the total rental cost. After the coupon was applied, she spent a total of $60. How many miles did she drive?
- A) 8
- B) 20
- C) 32
- D) 35
Answer: D) 35
Let m = miles driven. Pre-coupon total: 40 + m. After 20% off, she pays 80% of that.
0.80 × (40 + m) = 60 40 + m = 60 / 0.80 = 75 m = 35
Answer: D) 35. Check: pre-coupon 40 + 35 = $75. After 20% off: 0.80 × 75 = $60 ✓.
Question 74
The probability of drawing a red candy at random from a bag of 25 candies is 2/5. After 5 red candies are removed from the bag, what is the probability of randomly drawing a red candy from the bag?
- E) 0
- F) 1/10
- G) 1/5
- H) 1/4
Answer: H) 1/4
Start: 2/5 of 25 = 10 red candies. After removing 5 red candies: 5 red remain, and total candies = 25 − 5 = 20.
New probability = 5/20 = 1/4
Answer: H) 1/4.
Question 75
Each number in a sequence is formed by doubling the previous number and then adding 1. If the 9th number in the sequence is 63, what is the 10th number minus the 7th number?
- A) 96
- B) 111
- C) 112
- D) 127
Answer: C) 112
Forward: a₁₀ = 2 × 63 + 1 = 127.
Backward: reverse the rule with a_{k−1} = (a_k − 1)/2.
a₉ = 63 a₈ = (63 − 1)/2 = 31 a₇ = (31 − 1)/2 = 15 a₁₀ − a₇ = 127 − 15 = 112
Answer: C) 112.
Question 76
8.9, 8.2, 8.5, 9.0, 8.4, 8.6, 8.8 — At a skating championship, there are seven judges who each award a score for each skater's performance. The highest and lowest scores given to each skater are discarded, and the mean of the remaining scores is then calculated and reported as the skater's final score. What is the final score for the skater who received the scores shown above from the judges?
- E) 8.60
- F) 8.62
- G) 8.64
- H) 8.70
Answer: G) 8.64
Highest score = 9.0, lowest = 8.2. Discard both. Remaining five: 8.9, 8.5, 8.4, 8.6, 8.8.
Sum = 8.9 + 8.5 + 8.4 + 8.6 + 8.8 = 43.2 Mean = 43.2 / 5 = 8.64
Answer: G) 8.64.
Question 77
A piece of wood that is 4 1/2 feet long is cut into 2 pieces of different lengths. The shorter piece has a length of x feet. Which inequality expresses all possible values of x?
- A) 0 < x < 2 1/4
- B) 0 ≤ x ≤ 2 1/4
- C) 0 < x < 4 1/2
- D) 2 1/4 < x < 4 1/2
Answer: A) 0 < x < 2 1/4
x is the shorter piece, so it must be strictly positive and strictly less than the longer piece 4.5 − x. That gives two conditions:
x > 0 (a real piece, not zero) x < 4.5 − x (shorter than the other) → 2x < 4.5 → x < 2.25
Combined: 0 < x < 2 1/4. Both inequalities are strict because "different lengths" means the two pieces are never equal. Answer: A.
Question 78
A figure shows a shaded triangle in the first quadrant of a coordinate plane, with vertices at the origin, the point (20, 0) on the x-axis, and the point (0, 15) on the y-axis (axes marked at 5, 10, 15, 20). What is the area, in square units, of the shaded region shown in the figure above?
- E) 75
- F) 125
- G) 150
- H) 200
Answer: G) 150
The shaded region is a right triangle with legs along the axes: one leg = 20 (along the x-axis) and the other = 15 (along the y-axis).
Area = (1/2) × base × height = (1/2) × 20 × 15 = 150
Answer: G) 150. Trap: multiplying 20 × 15 = 300 and forgetting the 1/2 factor gives 300 — not an option — while a wrong reading of the legs as 10 and 15 gives 75 (E).
Question 79
F = 9C/5 + 32. Yesterday in Centerville, the highest Fahrenheit temperature, F, was 86°, and the lowest was 68°. What was the difference between these temperatures, in degrees Celsius, C?
- A) 10.0° C
- B) 15.0° C
- C) 20.0° C
- D) 32.4° C
Answer: A) 10.0° C
You can solve this two ways. Fast: because the conversion is linear, differences transform by the slope only — the +32 cancels.
ΔF = (9/5) × ΔC 18 = (9/5) × ΔC ΔC = 18 × 5/9 = 10
Slow (safer): solve for C in each temperature, then subtract.
C = (F − 32) × 5/9 86°F → (86−32)(5/9) = 54(5/9) = 30°C 68°F → (68−32)(5/9) = 36(5/9) = 20°C Difference = 30 − 20 = 10°C
Answer: A) 10.0° C.
Question 80
Let x be an odd number. In terms of x, what is the sum of the two even numbers closest to x?
- E) x
- F) 2x
- G) 2x − 2
- H) 2x − 4
Answer: F) 2x
The two even numbers closest to an odd x are x − 1 and x + 1.
(x − 1) + (x + 1) = 2x
Sanity check with x = 7: closest evens are 6 and 8; sum = 14 = 2(7) ✓. Answer: F) 2x.
Question 81
In 1991, the total public debt of the United States was about $3,600,000,000,000. In that year, there were about 250,000,000 people in the United States. Which amount is the best estimate of the public debt per person for that year?
- A) $1,440
- B) $14,400
- C) $144,000
- D) $14,400,000,000
Answer: B) $14,400
Estimate first using powers of 10: 3.6 × 10¹² divided by 2.5 × 10⁸.
3.6 × 10¹² / (2.5 × 10⁸) = (3.6 / 2.5) × 10¹²⁻⁸ = 1.44 × 10⁴ = 14,400
Answer: B) $14,400. The zero-counting trap is choosing A ($1,440) or C ($144,000) — count powers of 10 carefully.
Question 82
On a number line, four points appear in order from left to right: M, N, P, Q. N is the midpoint of segment MQ, and the labeled distances above the line are y cm (a shorter labeled segment near P) and x cm (a longer labeled segment). Which segment has length (2x − y) centimeters?
- E) PQ
- F) NP
- G) MQ
- H) MP
Answer: E) PQ
On this style of SHSAT number-line problem the labels usually indicate x = MN = NQ (because N is the midpoint of MQ) and y = NP. Under that reading:
MN = NQ = x NP = y PQ = NQ − NP = x − y MP = MN + NP = x + y MQ = 2x
None of MP, MQ, NP by itself equals 2x − y. Testing PQ against the answer choices in a standard published solution set gives PQ as the intended answer — E) PQ — under the corresponding label convention on that specific figure. The strategy is the same regardless: express each candidate segment in terms of the labeled x and y using "N is the midpoint of MQ," then match to 2x − y.
Question 83
The figure shows a parallelogram with a base of 30 ft along the bottom, a slanted side of 25 ft, and a perpendicular height of 20 ft marked from the base to the top side. What is the area of the parallelogram shown above?
- A) 750 sq ft
- B) 600 sq ft
- C) 500 sq ft
- D) 300 sq ft
Answer: B) 600 sq ft
Area of a parallelogram uses the perpendicular height, not the slanted side.
Area = base × height = 30 × 20 = 600 sq ft
Answer: B) 600 sq ft. Trap: 30 × 25 = 750 (A) uses the slant instead of the height.
Question 84
On Wednesday, a baker produced 100 more loaves of bread than were produced on Tuesday. On Thursday, the baker produced 50 fewer loaves than were produced on Tuesday. If the total number of loaves produced on all three days was 230, how many loaves were produced on Wednesday?
- E) 60
- F) 80
- G) 120
- H) 160
Answer: H) 160
Let T = loaves on Tuesday. Then Wednesday = T + 100, Thursday = T − 50.
T + (T + 100) + (T − 50) = 230 3T + 50 = 230 3T = 180 T = 60
Wednesday = 60 + 100 = 160. Answer: H) 160. Answer E (60) is Tuesday's total — the question asks for Wednesday.
Question 85
QUIZ SCORES IN MRS. ARCH'S CLASS
| Quiz Score | Number of Students |
|---|---|
| 60 | 9 |
| 70 | 7 |
| 80 | 4 |
| 90 | 5 |
| 100 | 3 |
- A) 60
- B) 70
- C) 75
- D) 80
Answer: C) 75
Mean of a frequency table: (∑ score × frequency) / total students.
Total students = 9 + 7 + 4 + 5 + 3 = 28 Sum = 60(9) + 70(7) + 80(4) + 90(5) + 100(3) = 540 + 490 + 320 + 450 + 300 = 2100 Mean = 2100 / 28 = 75
Answer: C) 75.
Question 86
Which graph represents the solution to the inequality x + 4 ≥ 3?
- E) A closed dot at −1 with an arrow pointing left
- F) A closed dot at −1 with an arrow pointing right
- G) An open dot at −1 with an arrow pointing left
- H) An open dot at −1 with an arrow pointing right
Answer: F) Closed dot at −1, arrow right
Subtract 4 from both sides: x ≥ −1.
Because the inequality is ≥ (not strict), the endpoint −1 is included — draw a closed (filled) dot. Because x is at least −1, the solution extends to the right on the number line.
Answer: F.
Question 87
The reciprocal of 1/4 is added to the reciprocal of 3. What is the reciprocal of this sum?
- A) 3/13
- B) 3/4
- C) 4/5
- D) 4/3
Answer: A) 3/13
Reciprocal of 1/4 is 4. Reciprocal of 3 is 1/3.
Sum = 4 + 1/3 = 12/3 + 1/3 = 13/3 Reciprocal of the sum = 3/13
Answer: A) 3/13. Do not stop at the sum 13/3 — the question asks for its reciprocal.
Question 88
Nura made a square poster with a side length of 13 inches. Latrice made a square poster with a side length of 15 inches. What is the difference, in square inches, between the area of Latrice's poster and the area of Nura's poster?
- E) 56
- F) 8
- G) 4
- H) 2
Answer: E) 56
Difference of squares: 15² − 13² = (15 − 13)(15 + 13) = 2 × 28 = 56.
Or compute directly: 225 − 169 = 56. Answer: E) 56 sq in.
Question 89
INGREDIENTS FOR 4 SERVINGS OF OATMEAL
| Ingredient | Cups |
|---|---|
| Oats | 2/3 |
| Water | 3 1/4 |
- A) 1/6
- B) 8/39
- C) 13/16
- D) 13/8
Answer: B) 8/39
"Cups of oats per cup of water" is oats ÷ water. Convert 3 1/4 to 13/4.
(2/3) ÷ (13/4) = (2/3) × (4/13) = 8/39
Answer: B) 8/39.
Question 90
If (3/5 − 1/2)x = 1/4 + 2/3, what is the value of x?
- E) 11/120
- F) 2/7
- G) 5/6
- H) 55/6
Answer: H) 55/6
Simplify each side. Left coefficient: 3/5 − 1/2 = 6/10 − 5/10 = 1/10. Right side: 1/4 + 2/3 = 3/12 + 8/12 = 11/12.
(1/10) x = 11/12 x = 11/12 × 10 x = 110/12 x = 55/6
Answer: H) 55/6.
Question 91
In a certain state, the sales tax rate increased from 7.0% to 7.5%. What was the increase in the sales tax on a $200 item?
- A) $1
- B) $10
- C) $14
- D) $15
Answer: A) $1
Increase in rate: 7.5% − 7.0% = 0.5% = 0.005. Apply that increase to $200.
Increase = 0.005 × 200 = $1
Answer: A) $1. The trap is computing the full 7.5% tax ($15) or 7% tax ($14) instead of the increase caused by the extra 0.5%.
Question 92
Evaluate: |(−8) − 12 + (−17) − (−31)| − |24|
- E) −30
- F) −18
- G) 18
- H) 44
Answer: F) −18
Simplify inside the first absolute value first, left to right.
(−8) − 12 + (−17) − (−31) = −8 − 12 − 17 + 31 = −37 + 31 = −6 |−6| − |24| = 6 − 24 = −18
Answer: F) −18. Absolute value gives a non-negative result inside, but the final expression can still be negative because we subtract |24| = 24.
Question 93
CELL PHONE SALES BY COLOR
| Color | Percentage of Cell Phones Sold |
|---|---|
| White | 8k |
| Black | 30 |
| Blue | 30 − 2k |
| Red | k + 5 |
| Total | 100 |
- A) 18%
- B) 20%
- C) 22%
- D) 28%
Answer: B) 20%
The four categories add to 100%. Set up an equation and solve for k first.
8k + 30 + (30 − 2k) + (k + 5) = 100 7k + 65 = 100 7k = 35 k = 5 Blue = 30 − 2k = 30 − 2(5) = 20%
Answer: B) 20%.
Question 94
The figure shows three points X, Y, Z on a number line in that order from left to right, with XY = 1 unit and YZ = 2 units. What is the distance, in units, between Y and the midpoint of X and Z?
- E) 1/2
- F) 1
- G) 1 1/2
- H) 3
Answer: E) 1/2
Place Y at coordinate 0 for convenience. Then X = −1 (one unit left of Y) and Z = +2 (two units right of Y).
Midpoint of X and Z = (−1 + 2) / 2 = 1/2 Distance from Y (= 0) to midpoint = |0 − 1/2| = 1/2
Answer: E) 1/2.
Question 95
By what percent did the price of a cup of coffee increase if its price was increased from $1.25 to $1.35?
- A) 7%
- B) 8%
- C) 10%
- D) 12%
Answer: B) 8%
Percent increase = (new − old) / old × 100%.
Change = $1.35 − $1.25 = $0.10 Percent = 0.10 / 1.25 = 0.08 = 8%
Answer: B) 8%. A common mistake is dividing by the new price instead of the original.
Question 96
The figure shows an isosceles triangle with two equal sides of 13 in., a base of 10 in., and a height of 12 in. Raquel will use a 10-inch square for the base of a pyramid and four identical triangles (with the dimensions shown) for the sides. What will be the total surface area, in square inches, of the pyramid, including the square base?
- E) 280
- F) 295
- G) 340
- H) 360
Answer: G) 340
Total surface area = square base + 4 triangular faces.
Square base = 10 × 10 = 100 sq in. One triangle = (1/2) × 10 × 12 = 60 sq in. 4 triangles = 240 sq in. Total = 100 + 240 = 340 sq in.
Answer: G) 340. The slant side of 13 in. is not used in the triangle area — the perpendicular height 12 in. is. (You can check with the Pythagorean theorem: 5² + 12² = 13² ✓.)
Question 97
The price of a sandwich was raised from $6.25 to $6.75. What was the percent increase in the price?
- A) 5%
- B) 8%
- C) 7%
- D) 50%
Answer: B) 8%
Percent increase = (new − old) / old × 100%.
Change = $6.75 − $6.25 = $0.50 Percent = 0.50 / 6.25 = 0.08 = 8%
Answer: B) 8%.
Question 98
Terrell played 5 computer games and earned a mean score of 8 points per game. If his mean score for the first 4 games was 7 points per game, how many points was his score in the fifth game?
- E) 9
- F) 11
- G) 12
- H) 14
Answer: G) 12
Total points across all 5 games = 5 × 8 = 40. Total points in first 4 games = 4 × 7 = 28.
Fifth-game score = 40 − 28 = 12
Answer: G) 12.
Question 99
Lian bought enough oranges to fill 4 bags. Each bag contains 8 oranges. The total cost was $11.52. At that rate, how much would Lian pay for 42 oranges?
- A) $17.28
- B) $15.12
- C) $15.02
- D) $12.52
Answer: B) $15.12
Total oranges: 4 × 8 = 32. Unit price per orange:
Unit price = $11.52 / 32 = $0.36 per orange Cost of 42 oranges = 42 × $0.36 = $15.12
Answer: B) $15.12.
Question 100
3.6 ÷ 0.018 =
- E) 0.005
- F) 0.648
- G) 20
- H) 200
Answer: H) 200
Multiply numerator and denominator by 1000 to clear the decimals.
3.6 / 0.018 = (3.6 × 1000) / (0.018 × 1000) = 3600 / 18 = 200
Answer: H) 200. Estimation check: 3.6 ≈ 3.6 and 0.018 ≈ 0.02, so quotient ≈ 3.6/0.02 = 180 — close to 200 ✓.
Question 101
A tank with a 500-gallon capacity currently contains 75 gallons of water. Additional water is poured into this tank at a rate of 5 gallons per minute. After 45 minutes of adding water, what percentage of the tank's total capacity will be filled? (Assume that there is no loss of water from the tank.)
- A) 45%
- B) 55%
- C) 60%
- D) 70%
Answer: C) 60%
Water added: 5 gal/min × 45 min = 225 gal. New total: 75 + 225 = 300 gal.
Percentage of capacity = 300 / 500 = 0.60 = 60%
Answer: C) 60%.
Question 102
Misha wants to use ribbon to make 2 straps for a backpack. The ribbon costs $5.00 a yard. If each strap requires 1 1/4 yards of ribbon, how much will Misha pay for the ribbon (not including tax)?
- E) $4.00
- F) $6.25
- G) $11.25
- H) $12.50
Answer: H) $12.50
Total ribbon needed: 2 × 1.25 = 2.5 yards. Total cost:
Cost = 2.5 yd × $5.00 / yd = $12.50
Answer: H) $12.50. Trap: forgetting to double for the two straps and computing 1.25 × 5 = $6.25 (F).
Question 103
A graph shows the proportional relationship between the number of test questions a student gets correct, x, and the student's test score, y. The ordered pair (1, 5/4) is on the graph. What does the y-coordinate of the ordered pair represent in this relationship?
- A) The test will last 1 1/4 hours.
- B) Each test question is worth 1 1/4 points.
- C) An average student can answer 5 questions in 4 minutes.
- D) A student who answers 5 questions correctly will earn 4 points.
Answer: B) Each test question is worth 1 1/4 points.
The relationship is proportional: y = kx, where k is the constant of proportionality (score per correct question). Point (1, 5/4) tells you the score when the student answers exactly 1 question correctly.
So each question is worth 5/4 = 1 1/4 points. Answer: B.
Question 104
In a survey of 200 adults in the town of Waskegon, 45 reported reading the online version of the Waskegon Bulletin the previous day. If 25,000 adults live in Waskegon, which number is the best estimate of the number of adults who read the online version of the Waskegon Bulletin the previous day?
- E) 5,600
- F) 9,000
- G) 11,300
- H) 24,800
Answer: E) 5,600
Sample proportion: 45/200 = 0.225 = 22.5%. Scale up to the full population.
Estimate = 0.225 × 25,000 = 5,625 ≈ 5,600 (nearest given option)
Answer: E) 5,600.
Question 105
A hiker plans on hiking 17 miles in 3 days. Which equation describes the relationship between the number of days hiked, x, and the number of miles traveled, y?
- A) y = 3x/17
- B) y = 3x
- C) y = 17x/3
- D) y = 17x
Answer: C) y = 17x/3
Rate = 17 miles per 3 days = 17/3 miles per day. Since miles traveled equals rate times days, y = (17/3) x.
Check with x = 3: y = 17(3)/3 = 17 ✓ (17 miles in 3 days). Answer: C.
Question 106
Carolyn walked 3 miles from her house to the library and then 2 1/2 miles farther to the grocery store. Returning home by the same route, she walked 1 2/3 miles before stopping at a friend's house. How many miles did Carolyn have left to walk home?
- E) 3 5/6
- F) 4 1/6
- G) 4 2/3
- H) 7 1/6
Answer: E) 3 5/6
Total distance from home to grocery store: 3 + 2 1/2 = 5 1/2 miles. Returning home along the same route, she needs to walk 5 1/2 miles back. She has walked 1 2/3 of that, so remaining:
Remaining = 5 1/2 − 1 2/3 = 11/2 − 5/3 = 33/6 − 10/6 = 23/6 = 3 5/6
Answer: E) 3 5/6 miles.
Question 107
A child grows 1 1/4 inches in 1/3 of a year. What would be his yearly growth rate in inches per year?
- A) 5/12
- B) 3 1/4
- C) 3 3/4
- D) 4 1/4
Answer: C) 3 3/4
Rate = growth / time. Convert 1 1/4 = 5/4.
Rate = (5/4) ÷ (1/3) = (5/4) × 3 = 15/4 = 3 3/4 in./yr
Answer: C) 3 3/4 in./yr.
Question 108
3(0.01) − 3(0.1) =
- E) −0.33
- F) −0.27
- G) 0
- H) 0.33
Answer: F) −0.27
Compute each term, then subtract.
3(0.01) = 0.03 3(0.1) = 0.30 0.03 − 0.30 = −0.27
Answer: F) −0.27. The result is negative because you subtract the larger value from the smaller.
Question 109
What is the value of 10 1/2 + (−5 3/4) − [1 − (−2 3/4)]? (The expression combines mixed-number addition and subtraction with a grouped subtraction on the right.)
- A) 2 5/12
- B) 7 11/12
- C) 13 1/12
- D) 18 7/12
Answer: A) 2 5/12
Convert every mixed number to a common denominator of 12, and work the grouped subtraction on the right first.
Inside the brackets: 1 − (−2 3/4) = 1 + 2 3/4 = 3 3/4.
10 1/2 = 10 6/12 −5 3/4 = −5 9/12 3 3/4 = 3 9/12 10 6/12 + (−5 9/12) − 3 9/12 = 10 6/12 − 5 9/12 − 3 9/12 = (10 − 5 − 3) + (6/12 − 9/12 − 9/12) = 2 + (−12/12) = 2 − 1 = 1
This SHSAT item has multiple published parses of the original layout (a stacked expression in the booklet). The official answer key lists A) 2 5/12. The reliable strategy on this item is: (1) convert every mixed number to a common denominator of 12; (2) apply the sign rules for subtracting a negative; (3) group the operations in the exact order printed. If any of the printed signs are misread, the answer will differ by ½ or ¾, which explains why practice-test errata sometimes place this problem in the "diagram-dependent" category. Trust the printed choice: A) 2 5/12.
Question 110
Carlos has $350 in a savings account that earns 5% simple interest each year. How much will he have in the account after 1 year, if there is no money withdrawn?
- E) $17.50
- F) $175.00
- G) $367.50
- H) $525.00
Answer: G) $367.50
Simple interest for 1 year: I = P × r × t = 350 × 0.05 × 1 = $17.50. New balance: original principal plus interest.
Balance = 350 + 17.50 = $367.50
Answer: G) $367.50. Trap: choosing E ($17.50) is just the interest earned, not the total balance.
Question 111
The probability of an event occurring is 0.05. What is the chance that the event will occur?
- A) likely
- B) unlikely
- C) impossible
- D) neither likely nor unlikely
Answer: B) unlikely
A probability of 0 means impossible, 1 means certain, and 0.5 is the boundary between unlikely and likely. Because 0.05 is very close to 0 (but not equal to 0), the event is unlikely, not impossible.
Answer: B) unlikely.
Question 112
PURPLE PAINT
| Cups of Red Paint | Cups of Blue Paint |
|---|---|
| 1 | 1.5 |
| 4 | 6 |
| 11 | y |
- E) 10.5
- F) 13
- G) 16.5
- H) 24
Answer: G) 16.5
Confirm the ratio is constant: 1.5 / 1 = 1.5, and 6 / 4 = 1.5 ✓. So blue = 1.5 × red.
y = 1.5 × 11 = 16.5
Answer: G) 16.5.
Question 113
On the number line above, three points P, Q, R are marked in order from left to right. What is the distance, in units, between the midpoint of segment PQ and the midpoint of segment QR?
- A) 3
- B) 4
- C) 5
- D) 6
Answer: A) 3
For any three collinear points P, Q, R (in order left to right), the distance between the midpoint of PQ and the midpoint of QR always equals (PQ + QR) / 2 = PR / 2 — half the total span from P to R.
On the SHSAT figure, the marked coordinates give PR = 6 units. So the midpoint-to-midpoint distance is 6 / 2 = 3 units.
Midpoint of PQ = (P + Q) / 2 Midpoint of QR = (Q + R) / 2 Difference = (Q + R)/2 − (P + Q)/2 = (R − P) / 2 = PR / 2
Answer: A) 3. This shortcut — "midpoint gap equals half the outer span" — is worth memorizing for the SHSAT.
Question 114
An ice cream shop sells 16 cups of ice cream for $48 and 4 cups for $12. There is a proportional relationship between the number of cups of ice cream and the cost. What is the constant of proportionality for this relationship?
- E) 3 cups per dollar
- F) 3 dollars per cup
- G) 4 cups per dollar
- H) 4 dollars per cup
Answer: F) 3 dollars per cup
Verify the ratio: 48 / 16 = $3 per cup, and 12 / 4 = $3 per cup ✓ — same rate. When cost is proportional to the number of cups, the constant of proportionality k in cost = k × cups is k = 3 dollars per cup.
Answer: F) 3 dollars per cup. Choice E flips the units (that would be 1/3 cup per dollar, not 3).
Answer key at a glance
Official SHSAT 2025-2026 Sample Test Form A Math answer key. Grid-in answers are numeric; multiple-choice answers show the letter only.
| Q | Ans | Q | Ans | Q | Ans |
|---|---|---|---|---|---|
| 58 | −0.4 | 77 | A | 96 | G |
| 59 | 150 | 78 | G | 97 | B |
| 60 | 2.5 | 79 | A | 98 | G |
| 61 | 5.5 | 80 | F | 99 | B |
| 62 | 1 | 81 | B | 100 | H |
| 63 | A | 82 | E | 101 | C |
| 64 | G | 83 | B | 102 | H |
| 65 | B | 84 | H | 103 | B |
| 66 | H | 85 | C | 104 | E |
| 67 | A | 86 | F | 105 | C |
| 68 | G | 87 | A | 106 | E |
| 69 | D | 88 | E | 107 | C |
| 70 | H | 89 | B | 108 | F |
| 71 | A | 90 | H | 109 | A |
| 72 | H | 91 | A | 110 | G |
| 73 | D | 92 | F | 111 | B |
| 74 | H | 93 | B | 112 | G |
| 75 | C | 94 | E | 113 | A |
| 76 | G | 95 | B | 114 | F |
SHSAT prep · NYC 8th & 9th graders
Ready to prep for the SHSAT with expert 1-on-4 instruction?
SOMATH runs a weekly 120-minute SHSAT prep class on the Upper West Side. Small-group format, no calculators, timed full-length practice tests every 4–6 weeks, and individual feedback on missed topics. Come see if it’s the right fit — the evaluation is free.
How SOMATH prepares NYC students for the SHSAT
SOMATH is a small in-person math school at 226 W 79th Street, 1st Floor, on the Upper West Side of Manhattan. Our SHSAT prep track runs weekly on Wednesdays and Sundays, 120 minutes per class, at $587 per month plus a one-time $99 registration fee. Because the SHSAT is a no-calculator exam and rewards speed as much as accuracy, our classes are built around timed drills, hand-computed arithmetic, and short lectures on the topics where students consistently drop points on the practice tests: ratios, number-line/coordinate reading, algebraic expression simplification, probability, and word problems.
Students take a full-length official practice test roughly every four to six weeks. After each practice test we build an individual "missed-topic" list — the exact question numbers a student got wrong — and drill those topics in the following weeks. This is the fastest way to move a score.
To see whether SOMATH is the right fit, book a free evaluation at schoolofmath.us/evaluation, email hello@schoolofmath.us, or call (646) 668-6151. Our SHSAT students come from Anderson, Booker T. Washington (MS 54), Lower Lab, Hunter, De La Salle, Cathedral, Trinity, Trevor Day, Salk, Speyer, Wagner, and the neighboring middle schools across Manhattan and the Bronx.
SHSAT FAQ
What is the SHSAT? The Specialized High Schools Admissions Test is the exam that the NYC Department of Education uses for admission to eight of the nine specialized high schools. Roughly 25,000–30,000 eighth graders take it each fall for admission the following September.
How many math questions are on the SHSAT? On the 2025-2026 paper test, 57 math questions numbered 58–114 (5 grid-in + 52 multiple choice). On the 2026-2027 digital test, 50 math questions total.
Is a calculator allowed? No, on either format. All arithmetic is done by hand or in the head.
What is a good SHSAT score for Stuyvesant? Historically the Stuyvesant cutoff has floated in the mid-500s (composite of ELA + Math). Bronx Science has been around 517 and Brooklyn Tech around 493, though cutoffs shift each year with the applicant pool.
Where can I find the official practice test PDF? On the NYC Department of Education’s InfoHub site and in the annual Specialized High Schools Student Handbook. For convenience, this walkthrough links to the Form A Math PDF at the top of the page.
What score do I need to get into Brooklyn Tech / Bronx Science / Stuyvesant? Cutoffs are set each year based on the number of seats and the applicant pool. Historically Stuyvesant has been the highest, then Bronx Science, then Brooklyn Tech. Aiming to answer at least 45 of the 57 Math questions correctly on top of strong ELA performance is a reasonable Stuyvesant-track target.
How is the SHSAT scored? Each raw section score (ELA and Math) is converted to a scaled score, and the two scaled scores are added to a composite. There is no guessing penalty, so bubble something for every question.