Digital SAT Prep · Retired Paper SAT · No-Calculator Math · Grades 10–12
Paper SAT May 2023 — No-Calculator Math (Section 3) — 20 Original SOMATH Problems & Digital SAT Bridge
The May 2023 US paper SAT was the last May sitting before College Board retired the paper test for the Digital SAT. Its No-Calculator math section (Section 3, 20 questions in 25 minutes) covered a skill list that is still ~80% of what shows up on today's Digital SAT math. This post walks that skill list with 20 original SOMATH problems — not reprints of the real exam — each with a full step-by-step SOMATH solution and a note on how the same concept appears on the current Digital SAT.
This post is part of SOMATH's Digital SAT tutoring in NYC series. If you're prepping now, the concepts below are what you should actually be drilling — the real, current Digital SAT lives in College Board's Bluebook app. Book a free 30-minute SAT evaluation at 226 W 79th St, Upper West Side, or call (646) 668-6151.
Concept-walkthrough slides — free download
14 slides covering all 20 skills the retired May 2023 No-Calculator section tested. Each slide names the concept, the fastest solving move, and an original SOMATH problem with the answer. Bridge to Digital SAT included.
20-question print worksheet — free download
All 20 original SOMATH problems in a clean print-ready 5-page PDF (25 minutes, no calculator). Answer key with skill labels on the last page. Perfect for a timed pre-Digital-SAT diagnostic.
What's in this post
1. What the May 2023 paper SAT was — and why it still matters
The last May of paper
The May 2023 US SAT was a paper-and-pencil exam administered at high schools nationwide. It had four scored sections: Reading (52 questions, 65 minutes), Writing & Language (44 questions, 35 minutes), Math No-Calculator (Section 3, 20 questions, 25 minutes), and Math Calculator (Section 4, 38 questions, 55 minutes). College Board retired the paper SAT in March 2024, replacing it with the shorter, adaptive, Bluebook-native Digital SAT.
The paper SAT is gone as a live test — you cannot register for it anymore. But its released practice exams and QAS (Question-and-Answer Service) booklets are still the largest bank of College-Board-authored SAT math questions in existence. That makes them useful for concept drill even if the format is out of date.
2. Section 3 concept spread (what No-Calculator tested)
Across the 20 questions of a typical No-Calculator section, the College Board consistently tested a fixed skill list. Here is the spread from the May 2023 US administration (paraphrased from the concept level — not the question text):
| Skill domain | Typical count | Sample skill |
|---|---|---|
| Heart of Algebra (linear) | 7–8 | Solve a two-equation linear system; interpret slope and intercept |
| Passport to Advanced Math | 6–7 | Rearrange a formula with a squared term; count solutions of a linear equation; interpret x-intercepts |
| Problem Solving & Data Analysis | 1–2 | Read a scatterplot; percent-change per step |
| Additional Topics (geometry, trig, complex) | 3–4 | Isosceles right triangle; tangent-to-circle slope; cylinder-vs-sphere volume |
3. Bridge to the Digital SAT — what carried over
| Paper SAT No-Calc concept | On Digital SAT? | Notes |
|---|---|---|
| Linear systems (substitution/elimination) | Yes, heavily | Desmos makes graphing solutions trivial — but you still need to set them up |
| Slope and y-intercept from a graph or verbal rate | Yes, heavily | Same skill |
| Absolute-value equations | Yes, lightly | Usually 1 per test |
| Rearranging a formula with a squared term | Yes | Same skill |
| Counting solutions of a linear equation | Yes | “Infinitely many” and “no solution” are staple answer choices |
| Interpreting x-intercepts as f(a) = 0 | Yes | Same skill |
| Factored form of a parabola with a labeled vertex | Yes | Same skill |
| Isosceles right triangle perimeter from hypotenuse | Yes | 45-45-90 side ratio is on the Digital SAT reference sheet |
| Tangent-to-circle slope (perpendicular to radius) | Yes | Same skill |
| Exponential decay by fixed percent per step | Yes | Same skill |
| Radical simplification with mixed indices | Yes, lightly | 1 per test at most |
| Slope of a parallel line | Yes | Same skill |
| Monomial rational expression simplification | Yes | Same skill |
| Cylinder-vs-sphere volume comparison | Yes | Volume formulas are on the reference sheet |
| Sum-and-product of quadratic roots (Vieta) | Yes | Framed as a grid-in on both formats |
4. 20 original SOMATH No-Calculator problems with solutions
Original SOMATH problems — not reprints. Same skills the retired May 2023 Section 3 tested. Click any question to show the answer and full step-by-step work.
Q1 — Reading a scatterplot for a linear-combination word problem
A quiz has only 3-point questions and 5-point questions and is worth 60 points total. A scatterplot shows every valid combination as the ordered pairs (number of 3-point questions, number of 5-point questions): (0, 12), (5, 9), (10, 6), (15, 3), (20, 0). If a student's plan includes exactly 5 of the 3-point questions, how many 5-point questions does the plan include?
- A) 3
- B) 6
- C) 9
- D) 12
Answer: C) 9.
Read the plotted point at x = 5. The y-value is 9. Check: 5(3) + 9(5) = 15 + 45 = 60. ✔
Q2 — Equation of a linear function from its graph
A linear function g passes through the points (0, −2) and (1, 1). Which equation defines g?
- A) g(x) = 3x − 2
- B) g(x) = 3x + 2
- C) g(x) = −3x − 2
- D) g(x) = −3x + 2
Answer: A) g(x) = 3x − 2.
Slope = (1 − (−2)) / (1 − 0) = 3/1 = 3. The point (0, −2) is the y-intercept, so b = −2. That gives g(x) = 3x − 2.
Q3 — Interpreting x-intercepts of a polynomial
The polynomial function h has x-intercepts at (−4, 0) and (7, 0). Which statement MUST be true?
- A) h(−4) = 0
- B) h(7) = −4
- C) h(0) = 7
- D) h(−4) = 7
Answer: A) h(−4) = 0.
An x-intercept (a, 0) means the graph crosses the x-axis at x = a, which by definition means h(a) = 0. So h(−4) = 0 and h(7) = 0 are the only two facts we know for sure. B, C, D make claims about values that are not determined by the intercepts alone.
Q4 — Two-equation linear system
Solve the system: y = 2x + 5 and 3x − y = 4. What is the solution (x, y)?
- A) (−9, −13)
- B) (−1, 3)
- C) (1, 7)
- D) (9, 23)
Answer: D) (9, 23).
3x − (2x + 5) = 4 x − 5 = 4 x = 9 y = 2(9) + 5 = 23Check both original equations: 23 = 2(9) + 5 ✔ and 3(9) − 23 = 27 − 23 = 4 ✔.
Q5 — Absolute-value equation
What are all real solutions to |x + 7| = 0 ?
- A) −7
- B) 0
- C) 7
- D) −7 and 7
Answer: A) −7.
|E| = 0 is a special case: the ONLY way an absolute value equals zero is if the inside equals zero. So x + 7 = 0, giving x = −7. There is exactly one solution, not two. This is a common trap — students see “absolute value” and reflexively write ±, but that only applies when the right side is positive.
Q6 — Rearranging a formula with a squared term
The equation p = t(w − 3)² relates positive numbers p, t, and w. Which equation gives w in terms of p and t when w > 3 ?
- A) w = 3 + √(p / t)
- B) w = 3 + √p / t
- C) w = −3 + √(p / t)
- D) w = −3 − √(p / t)
Answer: A) w = 3 + √(p / t).
p = t(w − 3)² p / t = (w − 3)² √(p / t) = |w − 3| Since w > 3, w − 3 > 0, so |w − 3| = w − 3. w − 3 = √(p / t) w = 3 + √(p / t)The condition w > 3 is what tells you to take the positive root, not the negative one. Without that condition you would write w = 3 ± √(p / t).
Q7 — Linear equation from a verbal rate and y-intercept
In a relationship between x and y, each increase of 1 in the value of x decreases the value of y by 4. When x = 0, y = 9. Which equation represents this relationship?
- A) y = −4x + 9
- B) y = −4x − 9
- C) y = 4x + 9
- D) y = −(1/4)x + 9
Answer: A) y = −4x + 9.
“Each increase of 1 in x decreases y by 4” is the definition of slope: slope = −4. “When x = 0, y = 9” is the y-intercept: b = 9. Slope-intercept form: y = −4x + 9.
Q8 — Factored-form parabola with a labeled vertex
A downward-opening parabola crosses the x-axis at x = 0 and x = 10 and reaches a maximum height of 25. It can be modeled by y = −x(x − 10) / k for some constant k. What is the value of k ?
- A) 25
- B) 5
- C) 1
- D) 1/5
Answer: C) 1.
Vertex x-coordinate is halfway between the roots: x = (0 + 10) / 2 = 5 At x = 5: y = −(5)(5 − 10) / k y = −(5)(−5) / k y = 25 / k Set equal to the maximum height 25: 25 / k = 25 k = 1The vertex of a parabola in factored form y = a(x − r₁)(x − r₂) is always halfway between the two roots. Plug that x-value in, set y equal to the labeled max/min, and solve for the constant.
Q9 — Isosceles right triangle perimeter
An isosceles right triangle has a hypotenuse of length 6. What is its perimeter?
- A) 3√2
- B) 6√2
- C) 6 + 6√2
- D) 6 + 12√2
Answer: C) 6 + 6√2.
45-45-90 side ratio: leg : leg : hypotenuse = 1 : 1 : √2 So each leg = 6 / √2 = 6√2 / 2 = 3√2 Perimeter = leg + leg + hypotenuse = 3√2 + 3√2 + 6 = 6√2 + 6Rationalize 6/√2 by multiplying top and bottom by √2. This is the same 45-45-90 pattern that appears on the Digital SAT reference sheet.
Q10 — Counting solutions of a linear equation
How many solutions does the equation 5(x − 3) = 2x + 3(x − 4) have?
- A) Zero
- B) Exactly one
- C) Exactly two
- D) Infinitely many
Answer: A) Zero.
Left side: 5(x − 3) = 5x − 15 Right side: 2x + 3(x − 4) = 2x + 3x − 12 = 5x − 12 Equation reduces to: 5x − 15 = 5x − 12 Subtract 5x from both sides: −15 = −12 That is a contradiction (false statement), so there are no solutions.Three outcomes are possible after fully simplifying a linear equation: a real number equation like x = 4 (exactly one solution), a true identity like 0 = 0 (infinitely many), or a contradiction like −15 = −12 (zero).
Q11 — Tangent-to-circle slope
Circle C has center (4, 3). Line m is tangent to circle C at the point A = (7, 7). What is the slope of line m ?
- A) −3/4
- B) −4/3
- C) 3/4
- D) 4/3
Answer: A) −3/4.
Slope of radius CA = (7 − 3) / (7 − 4) = 4/3 A tangent line is perpendicular to the radius at the point of tangency. Perpendicular slope = negative reciprocal of 4/3 = −3/4Perpendicular lines have slopes whose product is −1. Flip the fraction, flip the sign.
Q12 — Function-input interpretation in context
The function P(n) = 500 − 120(1.6)−0.2n predicts a runner's finish time (in seconds) after n weeks of training. Which expression represents the runner's predicted finish time after 0 weeks of training?
- A) P(−0.2)
- B) P(0)
- C) P(120)
- D) P(500)
Answer: B) P(0).
The variable n is defined as “number of weeks of training.” “After 0 weeks” means n = 0, so the expression is P(0). This is a pure function-notation reading question — you do not need to evaluate.
Q13 — Exponential decay by fixed percent per step
A machine's cycle time is 80 seconds on trial 1. On each following trial, the cycle time decreases by 25% from the previous trial. Approximately how many seconds is the cycle time on trial 4 ?
- A) 20
- B) 34
- C) 45
- D) 60
Answer: B) 34.
Each trial the time is multiplied by (1 − 0.25) = 0.75 Trial 1: 80 Trial 2: 80 · 0.75 = 60 Trial 3: 60 · 0.75 = 45 Trial 4: 45 · 0.75 = 33.75 ≈ 34General formula: trial n has time = 80 · (0.75)n−1. For n = 4: 80 · (0.75)³ = 80 · 0.421875 = 33.75.
Q14 — System of two inequalities: pick the valid table
Which of the following tables lists (x, y) pairs where every row satisfies BOTH y < (1/3)x + 2 AND y > (1/2)x − 4 ?
- A) (−3, −1), (0, 5), (6, 6)
- B) (−3, −6), (0, 1), (6, −1)
- C) (−3, 0), (0, 2), (6, −3)
- D) (−3, −2), (0, 0), (6, 0)
Answer: D) (−3, −2), (0, 0), (6, 0).
For each x-value, compute the two boundary y-values: x = −3: y < 1 (from first) AND y > −5.5 (from second) x = 0: y < 2 AND y > −4 x = 6: y < 4 AND y > −1 Check option D: (−3, −2): −2 < 1 ✔ and −2 > −5.5 ✔ (0, 0): 0 < 2 ✔ and 0 > −4 ✔ (6, 0): 0 < 4 ✔ and 0 > −1 ✔ All three rows satisfy both inequalities. Options A, B, C each have at least one row that fails one of the two inequalities.Strategy for these “which table” problems: don't try to graph. For each option, evaluate both inequalities at each row. The first row that fails knocks the whole table out.
Q15 — Radicals with different indices
Which of the following is equivalent to √50 · ∛8 ?
- A) 10√2
- B) 10 · ∛4
- C) 25√2
- D) 100 · ∛2
Answer: A) 10√2.
√50 = √(25 · 2) = 5√2 ∛8 = 2 (since 2³ = 8) Product: 5√2 · 2 = 10√2Trick: don't try to combine the two roots directly (different indices). Simplify each one separately first. The cube root evaluates cleanly to an integer, then you just multiply.
Q16 — Slope of a parallel line
Line k passes through (0, 1) and (6, 3). Line j is parallel to line k. What is the slope of line j ?
- A) −3
- B) −1/3
- C) 1/3
- D) 3
Answer: C) 1/3.
Slope of k = (3 − 1) / (6 − 0) = 2 / 6 = 1/3 Parallel lines have equal slopes. Slope of j = 1/3Parallel ⇒ same slope. Perpendicular ⇒ negative reciprocal. Memorize both directions.
Q17 — Monomial rational expression
The expression 45x⁸ / (9x²) is equivalent to c · xd, where x > 0 and c and d are constants. What is the value of c + d ?
- A) 5
- B) 6
- C) 11
- D) 30
Answer: C) 11.
45x⁸ / (9x²) = (45/9) · (x⁸ / x²) = 5 · x^(8−2) = 5x⁶ So c = 5 and d = 6. c + d = 5 + 6 = 11Exponent rule: xm / xn = xm−n. Coefficient divides normally.
Q18 — Solve a linear equation
What value of h satisfies the equation 3.4(h + 2) = 4.4h + 3.4 ?
- A) 1.7
- B) 3.4
- C) 6.8
- D) 10.2
Answer: B) 3.4.
3.4(h + 2) = 4.4h + 3.4 3.4h + 6.8 = 4.4h + 3.4 6.8 − 3.4 = 4.4h − 3.4h 3.4 = hDecimals in an SAT No-Calc problem look scary but almost always cancel or simplify. Distribute, collect like terms, isolate.
Q19 — Cylinder vs. sphere volume
A cylinder and a sphere both have the same radius r, where r > 0. The cylinder has a height of 12. The volume of the sphere is one-third the volume of the cylinder. What is the value of r ?
- A) 2
- B) 3
- C) 4
- D) 6
Answer: B) 3.
Cylinder volume: V_cyl = πr² · 12 = 12πr² Sphere volume: V_sph = (4/3)πr³ Given V_sph = (1/3) · V_cyl: (4/3)πr³ = (1/3)(12πr²) (4/3)πr³ = 4πr² Divide both sides by 4πr² (safe since r > 0): (1/3)r = 1 r = 3Both volume formulas are on the Digital SAT reference sheet, so you never need to memorize them — but you do need to know how to set up the equation.
Q20 — Sum and product of quadratic roots
For the equation x² + bx + c = 0 with real constants b and c, the two roots (given by the quadratic formula) satisfy −b + √(b² − 4c) = 14 and −b − √(b² − 4c) = −2. What is one possible value of x that solves x² + bx + c = 0 ?
- A) −7
- B) −1
- C) 1
- D) 7
Answer: B) −1 or D) 7 (either is accepted).
Quadratic formula: x = (−b ± √(b² − 4c)) / 2 The two given expressions are 2x (numerator of the quadratic formula, not x itself): −b + √(b² − 4c) = 14 ⇒ x₁ = 14 / 2 = 7 −b − √(b² − 4c) = −2 ⇒ x₂ = −2 / 2 = −1 Either root solves the equation.On the real SAT this appeared as a grid-in (no answer choices) so either 7 or −1 was accepted. In multiple choice we've included both correct options so you can confirm you spotted the ÷2 step in the quadratic formula. If you skipped the divide, you'd answer 14 or −2, which are the two most common wrong answers on this type.
5. Answer key summary
| Q | Answer | Skill | Q | Answer | Skill |
|---|---|---|---|---|---|
| 1 | C | Read scatterplot | 11 | A | Tangent-to-circle slope |
| 2 | A | Linear from graph | 12 | B | Function input in context |
| 3 | A | x-intercept = f(a) = 0 | 13 | B | Exponential decay 25% |
| 4 | D | Linear system | 14 | D | Two inequalities |
| 5 | A | |E| = 0 special case | 15 | A | Radical simplification |
| 6 | A | Rearrange with root | 16 | C | Parallel-line slope |
| 7 | A | Linear from words | 17 | C | Monomial rational |
| 8 | C | Factored-form vertex | 18 | B | Solve linear equation |
| 9 | C | 45-45-90 perimeter | 19 | B | Cylinder vs. sphere |
| 10 | A | Count solutions | 20 | B or D | Sum/product of roots |
6. Next steps for Digital SAT prep in NYC
If you're preparing for the current Digital SAT and you found any of the 20 concepts above shaky, that's exactly what SOMATH's 12-week Digital SAT track fixes. We meet in person at 226 W 79th St on the Upper West Side.
- Digital SAT tutoring in NYC — 2026 structure, dates & scores
- SAT prep on the Upper West Side, 2026 — what changed and how to prep
- SAT math vs. school math — why strong Algebra II students still miss SAT questions
- Digital SAT Practice Test 11 — Reading & Writing Module 1 walkthrough
- Book a free 30-minute SAT evaluation at SOMATH
7. Frequently asked questions
What was on the May 2023 paper SAT No-Calculator math section?
The May 2023 US paper SAT No-Calculator section (Section 3) had 20 questions in 25 minutes: 15 multiple-choice and 5 grid-in. The concept spread covered linear equations and functions (slope, y-intercept, systems), quadratic and polynomial features (x-intercepts, factored form, sum-and-product of roots), radicals and exponent rules, one absolute-value equation, and one geometry set (isosceles right triangle, tangent-to-circle, cylinder-vs-sphere volume). It was the final May sitting before College Board retired the paper SAT for the Digital SAT in spring 2024.
Are the practice problems in this post the actual questions from the May 2023 SAT?
No. Every problem in this post is an original SOMATH problem written by our teaching team. They test the same underlying math skills as the retired May 2023 No-Calculator section, but the wording, numbers, contexts, and answer choices are new. This is a deliberate choice: we do not host or reprint College Board test material because it is copyrighted. The official released paper SATs are available for free from College Board's practice archive if you want to see the real exam.
Is a retired paper SAT still useful practice for the Digital SAT?
Yes, for the math specifically. About 80% of the No-Calculator skill list carried over: linear systems, factoring, quadratic features, radicals and exponents, function notation, geometry with right triangles and circles, and exponential growth or decay. The Digital SAT compressed the layout (shorter modules, no-calculator restriction removed, adaptive by module), but the underlying algebra and geometry are almost identical. Where the paper SAT is weaker as prep: no adaptive routing, no Bluebook-style interface, no built-in Desmos. Use it as a concept-drill supplement, not as a full mock.
How is the Digital SAT math different from the paper SAT math?
Four real differences. (1) No separate No-Calculator section — Desmos is available on every math question. (2) Two adaptive modules of 22 questions each in 35 minutes each, versus paper's 20-question 25-minute No-Calc plus 38-question 55-minute Calculator. (3) The second module gets harder or easier based on how you did on the first. (4) The score scale is the same 200–800 but built from different item-response calibration. The math content is nearly the same — everything in this post is still fair game on the Digital SAT.
How does SOMATH prep students for the Digital SAT in NYC?
SOMATH runs small-group and one-on-one SAT prep on the Upper West Side at 226 W 79th St. Our Digital SAT track is 12 weeks: 4 weeks of concept rebuild (linear, quadratic, exponent, geometry, data), 4 weeks of Bluebook-native practice using the official College Board practice tests, and 4 weeks of full-length adaptive mocks with detailed error review. Students typically move 100–180 points from their baseline diagnostic. Call (646) 668-6151 or book a free 30-minute evaluation to see if the program is a fit.
Where can I find the actual released paper SATs to practice on the real thing?
College Board removed most paper SAT practice tests from their site after the Digital SAT launch in 2024, but the four official practice tests they released for the paper SAT era are still archived across tutoring sites and on the Internet Archive. For current SAT prep the College Board Bluebook app is the source of truth — it has 6+ full-length Digital SAT practice tests, all free, with authentic scoring. That is where our students do their timed mocks.