Digital SAT Math Practice Questions
Practice SAT Math by Skill
Build Digital SAT Math skills with a targeted diagnostic, answer explanations, calculator strategy, and practice by topic. Start with the diagnostic to identify weak areas, then review Algebra, Advanced Math, Problem-Solving and Data Analysis, Geometry and Trigonometry, or Calculator Strategy.
Review Your Diagnostic
Your Results
Domain & Skill Analysis
Choose a Math Skill to Review
Algebra
Linear equations, inequalities, systems, slope, and linear models.
Linear Equations and Expression Targets
- one-variable linear equations
- fractions and signs
- expression targets
- parameter values
- recognizing when solving for x is unnecessary
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Easyexpression targetMultiple choice
1. If 5x + 7 = 32, what is the value of 10x + 14?
- 50
- 57
- 64
- 71
Show solution
Choice C is correct because 10x + 14 is exactly 2(5x + 7). Since 5x + 7 = 32, doubling gives 64.
Double the given expression: 2(32)=64.
Mental
The target expression is a multiple of the expression already given.
Not needed.
Solving for x first, which works but takes longer.
C. 64
MediumfractionsStudent-produced response
2. If (2x − 5)/3 + (x + 4)/2 = 7, what is the value of x?
Show solution
Multiplying every term by 6 gives 2(2x − 5) + 3(x + 4) = 42, which simplifies to 7x + 2 = 42.
Clear denominators first, then solve: 7x = 40, so x = 40/7.
Algebra
Clearing fractions turns the equation into a standard linear equation.
Not needed.
Multiplying only the fractions by 6 instead of every term.
40/7
Very Hardno solutionStudent-produced response
3. The equation p(3x − 2) − q(x + 5) = 20x − 14 has no solution. If p − q = 6, what is q?
Show solution
The correct answer is 1. Expanding the left side gives an x-coefficient of 3p − q. For no solution, that coefficient must equal 20, while the constants are different.
Use 3p − q = 20 and p − q = 6. Substitute p=q+6: 3(q+6)−q=20, so 2q+18=20 and q=1.
Algebra
Coefficient matching is faster than trying to solve a parameterized equation for x.
Not needed.
Forgetting that no solution requires equal x-coefficients but different constants.
1
Linear Inequalities and Constraint Ranges
- solving one-variable inequalities
- compound inequalities
- integer solution counts
- inclusive vs. exclusive endpoints
- least/greatest integer satisfying a condition
- parameter values tied to a boundary
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Mediumcompound intervalStudent-produced response
1. How many integer values of x satisfy −3 ≤ (x + 5)/2 < 6?
Show solution
The inequality simplifies to −11 ≤ x < 7, so the integers are −11 through 6.
Multiply all parts by 2, then subtract 5: −6 ≤ x+5 < 12, so −11 ≤ x < 7. Count from −11 to 6.
Algebra
Solving the interval first makes the integer count direct.
Not needed.
Including 7 even though the right endpoint is open.
18
Hardparameter boundaryStudent-produced response
2. For what integer value of a is x = 4 not a solution of 2x + a ≥ 3a − 1, but x = 5 is a solution?
Show solution
The correct answer is 5. Testing the two boundary values gives a > 4.5 and a ≤ 5.5, so the only integer is 5.
Plug in x = 4 and require the inequality to be false. Plug in x = 5 and require it to be true. Combine the restrictions.
Boundary value
Testing the named values is faster than solving for the entire solution set first.
Not needed.
Treating “not a solution” as if the inequality should still hold.
5
Hardboundary valueMultiple choice
3. For what value of k is the solution set of 7 − 3x ≥ kx + 1 equal to x ≤ 2?
- −2
- 0
- 2
- 4
Show solution
If the solution set is x ≤ 2, then x=2 is the boundary where equality holds. Substituting gives k=0.
Use the equality at the boundary: 7−3(2)=2k+1. This gives 1=2k+1, so k=0.
Boundary value / Algebra
The endpoint gives the parameter without solving the full inequality first.
Not useful here.
Forgetting that the boundary value must make both sides equal.
B. 0
Linear Functions, Slope, and Intercepts
- slope from two points
- y-intercept and x-intercept
- writing a linear function
- interpreting rate of change
- stepping from a known point instead of building a full equation
- parallel and perpendicular slope
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Basicslope from pointsMultiple choice
1. A line passes through (−3, 7) and (5, −9). What is the slope of the line?
- −4
- −2
- 1/2
- 2
Show solution
The change in y is −9 − 7 = −16 and the change in x is 5 − (−3) = 8. The slope is −16/8 = −2.
Use rise over run directly from the two points.
Mental / Algebra
The coordinate differences are small enough to compute quickly.
Not needed.
Subtracting coordinates in inconsistent order.
B. −2
Mediumrate of changeMultiple choice
2. A linear function h satisfies h(3)=14 and h(9)=38. What is h(12)?
- 42
- 46
- 50
- 54
Show solution
Choice C is correct. The output increases by 24 when the input increases by 6, so the rate is 4. From 9 to 12 is 3 input units, so the output increases by 12.
Compute the rate: (38−14)/(9−3)=4. Then h(12)=38+4(3)=50.
Mental
Stepping from a known point avoids writing the full equation.
Not needed.
Using the rate once instead of for each input step.
C. 50
Very Hardhidden intersectionStudent-produced response
3. In the xy-plane, the line y = mx + b passes through (4, 1). It intersects y = 2x − 5 at a point whose x-coordinate is twice its y-coordinate. What is m?
Show solution
The correct answer is −1. The intersection point is (10/3, 5/3), and the slope through that point and (4,1) is −1.
Use x = 2y, so y = x/2. Set x/2 = 2x − 5 to get x=10/3 and y=5/3. Then compute the slope from (4,1) to that point.
Algebra
Finding the intersection first keeps the problem linear and exact.
Not needed.
Trying to use m before finding the actual intersection point.
−1
Systems of Linear Equations
- elimination
- substitution
- expression targets
- no solution
- infinitely many solutions
- system interpretation
- parallel distinct lines
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
MediumeliminationMultiple choice
1. If 3x + y = 16 and x − y = 4, what is 2x + y?
- 9
- 11
- 13
- 16
Show solution
Choice B is correct. Adding the equations gives 4x=20, so x=5. Then y=1 and 2x+y=11.
Add the equations to eliminate y immediately, then substitute into either equation.
Algebra
The y-terms cancel without rearranging either equation.
Not needed.
Stopping after finding x instead of answering the requested expression.
B. 11
Mediumparallel linesMultiple choice
2. For what value of k does the system kx + 4y = 7 and 6x + 8y = 20 have no solution?
- 2
- 3
- 4
- 6
Show solution
To make the y-coefficients match, double the first equation. The x-coefficient must also double, so 2k=6 and k=3. The constants become 14 and 20, so the lines are distinct.
Match coefficients and check that the constants do not match.
Coefficient comparison
No-solution systems usually reduce to parallel but distinct lines.
Not needed.
Forgetting to check that the constants are different.
B. 3
Very Hardno solution parameterStudent-produced response
3. For some constant k, the system below has no solution: (2k − 1)x + 3y = 7
4x + (k + 2)y = 10 What is the sum of all possible values of k?
Show solution
The correct answer is −3/2. No solution requires the coefficient rows to be proportional, so the determinant must be 0. The resulting quadratic has root sum −3/2.
Set the determinant to 0: (2k−1)(k+2)−12=0, which simplifies to 2k2+3k−14=0. The sum of roots is −b/a = −3/2.
Algebra
Vieta’s sum avoids solving for both k-values separately.
Not needed.
Solving for one k-value and forgetting the question asks for the sum of all possible values.
−3/2
Very Hardinfinite solutionsStudent-produced response
4. A system of two linear equations has infinitely many solutions. One equation is 3x − 2y = 12. The other equation is ax − 8y = b. What is the value of a + b?
Show solution
For infinitely many solutions, the second equation must be an exact multiple of the first equation.
Since −8y is 4 times −2y, multiply the entire first equation by 4: 12x − 8y = 48. Therefore, a=12, b=48, and a+b=60.
Coefficient matching
Infinite-solution systems require every coefficient and constant to scale by the same factor.
Not useful for symbolic parameters.
Matching only the y-coefficient and forgetting to scale the constant term too.
60
Linear Models and Word Problems
- model setup
- fixed fee and rate
- comparing two plans
- break-even points
- least/greatest whole number
- linear projection over time
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Easyfixed feeStudent-produced response
1. A printing company charges a fixed fee plus a constant amount per poster. The cost is $115 for 20 posters and $175 for 35 posters. What is the fixed fee, in dollars?
Show solution
The correct answer is 35. The cost increases $60 for 15 posters, so the rate is $4 per poster. Using 20 posters gives a fixed fee of $35.
Compute the rate: (175−115)/(35−20)=4. Then 115 = fixed fee + 4(20), so the fixed fee is 35.
Algebra
Two points reveal the rate quickly, then one point gives the intercept.
Not needed.
Using 115 as the fixed fee instead of subtracting the variable cost.
35
Easyfixed fee and rateMultiple choice
2. A service charges a setup fee of $18 and $3.50 for each unit. If the total charge is $74, how many units were purchased?
- 14
- 15
- 16
- 18
Show solution
The model is 74 = 18 + 3.50u. Subtract 18 to get 56, then divide by 3.50 to get 16.
Remove the fixed fee first, then divide by the unit rate.
Algebra
The setup fee happens once, so subtracting it isolates the repeated cost.
Not needed.
Multiplying the setup fee by the number of units.
C. 16
Hardthreshold inequalityStudent-produced response
3. Plan A charges $50 plus $0.10 per minute. Plan B charges $26 plus $0.16 per minute. What is the least whole number of minutes for which Plan A costs less than Plan B?
Show solution
The plans are equal at 400 minutes, so Plan A is less expensive only for whole-number minutes greater than 400.
Solve 50 + 0.10m < 26 + 0.16m. This gives 24 < 0.06m, so 400 < m. The least whole number is 401.
Algebra
This is a threshold question; solve the inequality and round up.
Graphing can verify the break-even point, but hand-solving is faster.
Answering 400, where the costs are equal.
401
Very Hardlinear projectionStudent-produced response
4. A company had 1,240 members in 2022 and 1,720 members in 2026. The number of members increased linearly. According to this model, in what year will the company first have at least 2,000 members?
Show solution
The correct answer is 2029. The increase is 480 members over 4 years, or 120 per year. The model first reaches at least 2,000 members 7 years after 2022.
Use 1240 + 120t ≥ 2000. Then 120t ≥ 760, so t ≥ 19/3. The first whole year is t=7, which is 2029.
Algebra
Threshold questions require rounding up to the first whole input that satisfies the inequality.
Not needed.
Rounding 6.33 down and choosing 2028, which is still below 2,000.
2029
Advanced Math
Quadratics, nonlinear equations, functions, equivalent expressions, and exponential/rational patterns.
Quadratics: Zeros, Vertex, and Discriminant
- factoring
- intercept form
- vertex form
- axis of symmetry
- discriminant
- minimum/maximum value
- Vieta’s formulas
- parameter values producing one/two/no real solutions
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
BasicVieta sumMultiple choice
1. For the equation 3x2 − 18x + 5 = 0, what is the sum of the two solutions?
- −6
- −5/3
- 6
- 18
Show solution
For ax2 + bx + c = 0, the sum of roots is −b/a. Here a=3 and b=−18, so the sum is 6.
Use Vieta’s formula instead of solving the quadratic.
Mental / Algebra
The question asks only for the sum, not the roots.
Not needed.
Using c/a, which gives the product, not the sum.
C. 6
Hardintercept formMultiple choice
2. A quadratic function has zeros 2 and 10 and passes through (0, 80). What is its minimum value?
- −80
- −64
- −48
- 64
Show solution
Choice B is correct. The function is f(x)=4(x−2)(x−10). The axis is x=6 and f(6)=−64.
Use intercept form. Since 80=a(−2)(−10)=20a, a=4. Evaluate the midpoint of the zeros: x=6.
Algebra
Zeros make intercept form and symmetry faster than expanding.
Not needed.
Using one zero as the vertex instead of the midpoint between the zeros.
B. −64
Hardintercept form and vertexMultiple choice
3. A quadratic function has zeros −1 and 5 and passes through (0, −15). What is its minimum value?
- −36
- −27
- −18
- 27
Show solution
The function is f(x)=3(x+1)(x−5). The axis is halfway between the zeros at x=2, and f(2)=−27.
Use intercept form and find a: −15=a(1)(−5), so a=3. Then evaluate at the midpoint of −1 and 5.
Algebra
The zeros immediately reveal the useful form.
Desmos can verify, but exact algebra is cleaner.
Using the y-intercept as the minimum value.
B. −27
Very Hardsymmetry and valueStudent-produced response
4. A quadratic function f satisfies f(4)=0, has axis of symmetry x = 5/2, and f(2)=6. If f(x)=ax2+bx+c, what is c?
Show solution
The correct answer is −12. The root 4 reflects across the axis x=5/2 to root 1, so f(x)=a(x−4)(x−1).
Use f(2)=6: 6=a(−2)(1), so a=−3. Then c=f(0)=−3(−4)(−1)=−12.
Algebra
The axis reveals the second zero immediately.
Not needed.
Trying to expand before using symmetry.
−12
Very Hardvertex formStudent-produced response
5. A quadratic function has vertex (3, −8) and passes through (5, 4). If the function is written as f(x)=a(x−3)2−8, what is the value of a?
Show solution
The vertex form is already given, so substitute the point (5,4).
Use 4=a(5−3)2−8. Then 4=4a−8, so 12=4a and a=3.
Algebra
Vertex form lets you substitute directly without expanding.
Not needed.
Forgetting to add 8 before dividing by 4.
3
Equivalent Expressions and Structure
- factoring
- expanding
- simplifying rational expressions
- equivalent forms
- restrictions
- coefficient matching
- forms that reveal features
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Basicdifference of squaresMultiple choice
1. Which expression is equivalent to (x − 5)2 − 9?
- (x − 8)(x − 2)
- (x − 14)(x + 4)
- x2 − 25
- x2 − 10x + 34
Show solution
This is a difference of squares: (x−5)2 − 32. It factors to (x−5−3)(x−5+3).
Use the difference-of-squares pattern before expanding.
Mental / Algebra
Factoring the structure is faster than full expansion.
Not needed.
Expanding first and making a sign error.
A. (x − 8)(x − 2)
Mediumrational simplificationMultiple choice
2. For x ≠ −2, the expression (x2 − 4)/(x2 + 4x + 4) is evaluated at x = 8. What is the value?
- 1/3
- 3/5
- 5/3
- 5
Show solution
Factoring simplifies the expression to (x−2)/(x+2). At x=8, this is 6/10 = 3/5.
Factor first: x2−4=(x−2)(x+2) and x2+4x+4=(x+2)2. Then substitute.
Algebra
Factoring before substituting avoids larger arithmetic.
Not needed.
Canceling terms instead of factors.
B. 3/5
Mediumrational simplificationMultiple choice
3. For x ≠ −4, which expression is equivalent to (x2 + 7x + 12)/(x + 4)?
- x + 3
- x + 4
- x + 7
- x2 + 3
Show solution
The numerator factors as (x+3)(x+4). Since x≠−4, the factor x+4 can be canceled.
Factor first, cancel only the common factor, and keep the restriction in mind.
Algebra
Factoring exposes the cancellation immediately.
Not needed.
Canceling terms instead of factors.
A. x + 3
Very Hardcoefficient matchingStudent-produced response
4. For x ≠ −3, (x2 + (a + 3)x + 3a)/(x + 3) = x + 5 for all allowed values of x. What is a?
Show solution
The correct answer is 5. For the quotient to be x+5, the numerator must equal (x+3)(x+5).
Expand (x+3)(x+5)=x2+8x+15. Compare with x2+(a+3)x+3a. Then a+3=8 and 3a=15, so a=5.
Algebra
Matching the target structure is faster than long division.
Not needed.
Canceling without making the numerator match the denominator times the quotient.
5
Nonlinear Equations and Systems
- solving quadratic equations
- radical equations
- rational equations
- extraneous solutions
- number of intersections
- tangent/repeated-root conditions
- nonlinear systems
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Easyradical equationStudent-produced response
1. If x + 5 = x − 1, what is x?
Show solution
Squaring gives x+5=(x−1)2, which simplifies to (x−4)(x+1)=0. Only x=4 satisfies the original equation.
Check the domain, square both sides, then verify the candidates.
Algebra
The radical is isolated, so squaring is direct.
Not needed.
Keeping the extraneous solution x = −1.
4
Mediumrational equationStudent-produced response
2. The positive solution to (x + 6)/x = 5 is r. What is r2 − r?
Show solution
Multiply by x to get x+6=5x, so x=3/2. Then r2−r = 9/4 − 6/4 = 3/4.
Solve for r first, then compute the requested expression.
Algebra
The equation becomes linear after clearing the denominator.
Not needed.
Returning r instead of r² − r.
3/4
Hardradical equationStudent-produced response
3. If x + 16 − x = 2 and x > 0, what is x?
Show solution
The correct answer is 9 because 25 − 9 = 5−3 = 2.
Use Desmos or isolate and square carefully, then verify in the original equation.
Desmos Graph or Algebra
Graphing avoids common radical-squaring errors and quickly confirms the solution.
Graph y = sqrt(x + 16) - sqrt(x) and y = 2; read the x-coordinate of the intersection.
Squaring without checking the result in the original equation.
9
HarddiscriminantMultiple choice
4. The graphs of y = x2 + kx + 9 and y = 4x + 5 intersect at exactly one point. What is the product of all possible values of k?
- −16
- 0
- 8
- 16
Show solution
Setting the equations equal gives x2 + (k−4)x + 4 = 0. Exactly one solution means the discriminant is 0, which gives possible values k=0 and k=8. Their product is 0.
Use (k−4)2−16=0, so k−4=±4.
Algebra
A tangent line to a parabola becomes a quadratic with discriminant 0.
Not needed.
Thinking exactly one intersection means there is only one possible k-value.
B. 0
Nonlinear Functions: Features, Graphs, and Models
- nonlinear function features
- zeros
- intercepts
- minimum and maximum values
- interpreting constants
- comparing graph/table/equation
- exponential growth/decay
- function features in context
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Mediumgrowth modelStudent-produced response
1. A population is modeled by P(t)=600(1.25)t, where t is years after 2020. What population does the model predict for 2022?
Show solution
The correct answer is 937.5. Since 2022 is 2 years after 2020, evaluate P(2).
Compute 600(1.25)2 = 600(1.5625)=937.5.
Algebra
Substituting the correct input is faster than making a table.
Not needed.
Using t = 2022 instead of t = 2.
937.5
Mediumvertex formMultiple choice
2. What is the minimum value of f(x) = (x − 3)2 + 7?
- 3
- 7
- 9
- 16
Show solution
The squared term (x−3)2 is never negative. Its smallest value is 0, so the smallest output is 7.
Read the vertex form directly.
Mental
No expansion or graphing is needed when the vertex is visible.
Not needed.
Reporting the x-value 3 instead of the minimum y-value 7.
B. 7
Mediumexponential modelStudent-produced response
3. A quantity is modeled by P(t)=800(1.06)t. By what percent does it increase each time t increases by 1?
Show solution
The multiplier 1.06 means 1 + 0.06, so the growth rate is 6%.
Subtract 1 from the growth factor and convert to a percent.
Mental
The growth factor directly encodes the percent increase.
Not needed.
Saying 106% instead of a 6% increase.
6
Very Hardminimum distanceMultiple choice
4. For positive t, the point (t, 20/t) lies on xy = 20. For what value of t is the distance from this point to the origin minimized?
- 10
- 4
- 25
- 52
Show solution
The distance is minimized at the same t-value that minimizes t2 + 400/t2. This occurs when t = 20/t, so t2=20.
Graph y = x2 + (20/x)2 for x > 0 and read the x-coordinate of the minimum, or reason by symmetry.
Desmos Graph
The squared-distance expression has the same minimizing input as distance.
Graph y = x^2 + (20/x)^2 with x > 0; read the x-coordinate of the minimum.
Reporting the minimum y-value instead of the x-value.
C. 25
Exponents, Radicals, and Rational Expressions
- exponent rules
- common-base equations
- radical equations
- rational expressions
- restrictions
- equivalent forms
- interpreting growth/decay factors
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Mediumcommon baseMultiple choice
1. If 4x + 2 = 82x − 1, what is x?
- 5/4
- 7/4
- 9/4
- 3
Show solution
Rewrite both sides with base 2: 22x+4 = 26x−3, so 2x+4=6x−3.
Solve 7=4x, so x=7/4.
Algebra
Common-base rewriting turns the exponential equation into a linear equation.
Not needed.
Multiplying the exponent on 8 incorrectly.
B. 7/4
Mediumcommon baseMultiple choice
2. If 16x−1 = 8x+2, what is x?
- 6
- 8
- 10
- 12
Show solution
Rewrite both sides with base 2: 24x−4 = 23x+6. Therefore 4x−4=3x+6 and x=10.
Use common bases, then equate exponents.
Algebra
The bases are powers of 2, so the equation becomes linear.
Not necessary.
Forgetting to distribute the exponent across the base conversion.
C. 10
Mediumrational function valueStudent-produced response
3. If h(x)=((x−2)2 + 5)/(x + 3), what is h(7)?
Show solution
Substitute x=7: the numerator is (5)2+5=30 and the denominator is 10, so h(7)=3.
Substitute 7 with grouped numerator and denominator.
Algebra or Desmos Table
Only one input value is needed.
A table works well if the expression is entered with parentheses around the whole numerator and denominator.
Typing the expression without grouping the fraction.
3
Hardrational equationStudent-produced response
4. If x ≠ 3 and (2x + 5)/(x − 3) = 4, what is x?
Show solution
Multiply both sides by x−3, which is allowed because x ≠ 3.
2x+5=4(x−3), so 2x+5=4x−12. Then 17=2x, so x=17/2.
Algebra
Clearing the denominator turns the rational equation into a linear equation.
Not needed.
Multiplying only one side by the denominator or ignoring the restriction.
17/2
Hardradical equationStudent-produced response
5. If x + 25 − x = 1 and x > 0, what is x?
Show solution
x=144 works because 169 − 144 = 13 − 12 = 1.
Use Desmos or square carefully after isolating one radical, then verify.
Desmos Graph or Algebra
Graphing is efficient for radical equations with a clean numeric intersection.
Graph y = sqrt(x + 25) - sqrt(x) and y = 1; read the x-coordinate of the intersection.
Squaring and keeping an extraneous value.
144
Function Notation, Composition, and Transformations
- f(a)
- f(g(x))
- shifted inputs such as f(x+k)
- transformed function values
- graph shifts
- reflections
- max/min changes after vertical transformations
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
MediumcompositionStudent-produced response
1. If f(x)=x2−4 and g(x)=3x+1, what is f(g(2))?
Show solution
First g(2)=7. Then f(7)=72−4=45.
Work from the inside out.
Mental / Algebra
Composition becomes two simple evaluations.
Not needed.
Plugging 2 directly into f instead of into g first.
45
Hardshifted inputStudent-produced response
2. For all x, f(3x + 2) = x2 − x What is f(14)?
Show solution
To find f(14), the input 3x+2 must equal 14, so x=4. Then x2−x=16−4=12.
Set 3x+2=14, solve for x=4, and substitute into the right side.
Algebra
Matching the input of f prevents substituting 14 into the wrong expression.
Not needed.
Calculating 14² − 14 as if 14 were x.
12
Hardtransformation extremumStudent-produced response
3. The minimum point of y = f(x) is (−1, 5). The function g is defined by g(x) = −2f(x − 4) + 3 What is the maximum y-value of g?
Show solution
The correct answer is −7. The minimum output of f is 5. The outside transformation sends that output to −2(5)+3=−7. Because the multiplier is negative, the original minimum becomes a maximum.
Ignore the horizontal shift for the y-value. Transform the output 5: −2(5)+3=−7.
Mental / Transformation reasoning
The input shift changes where the point occurs, not the transformed output value.
Not needed.
Calling −7 the minimum instead of the maximum.
−7
Hardshifted inputStudent-produced response
4. For all x, f(2x + 3) = x2 + 2x. What is f(11)?
Show solution
To get input 11, solve 2x+3=11, so x=4. Then the output is 42+2(4)=24.
Match the input of f first; do not plug 11 directly into the right side.
Algebra
The input-matching step prevents a wrong substitution.
Not needed.
Using 11 as x instead of as the input to f.
24
Hardtransformed maximumStudent-produced response
5. The maximum point of y=f(x) is (2,9). The function h is defined by h(x)=3f(x+1)−4. What is the maximum y-value of h?
Show solution
The maximum output of f is 9. The outside transformation changes output values by multiplying by 3 and subtracting 4.
Compute 3(9)−4=23. The input shift x+1 changes where the maximum occurs, not the maximum y-value.
Mental / Transformation reasoning
The question asks for the y-value only, so the horizontal shift can be ignored.
Not needed.
Trying to change the x-coordinate even though only the maximum y-value is requested.
23
Problem-Solving and Data Analysis
Ratios, percentages, data, probability, sampling, and statistical claims.
Ratios, Rates, and Units
- ratios
- proportions
- unit rates
- conversions
- density/speed
- compound units
- scale from part-to-total
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
EasyscaleMultiple choice
1. On a map, 1 inch represents 25 miles. What distance, in miles, is represented by 3.6 inches?
- 72
- 90
- 96
- 125
Show solution
Multiply the map distance by the scale: 3.6 × 25 = 90 miles.
Use direct proportion because the scale is linear.
Mental / Algebra
The units already match the requested output.
Not needed.
Dividing by 25 instead of multiplying.
B. 90
Mediumunit rateMultiple choice
2. A machine fills 3/5 of a tank in 18 minutes. At the same constant rate, how many minutes will it take to fill 5/6 of the tank?
- 20
- 24
- 25
- 30
Show solution
The machine fills 1/30 of a tank per minute, so filling 5/6 takes 25 minutes.
Find the unit rate: 3/5 ÷ 18 = 1/30. Then 5/6 ÷ 1/30 = 25.
Unit rate
The unit rate keeps the proportion organized.
Not needed.
Multiplying by 18 instead of dividing by the rate.
C. 25
Very Hardcompound unitsStudent-produced response
3. A chemical is added to water at a rate of 2.4 milliliters per liter. How many liters of water can be treated with 3.6 liters of chemical? Use 1 liter = 1,000 milliliters.
Show solution
The correct answer is 1500. The chemical amount is 3,600 mL, and each liter of water requires 2.4 mL.
Convert first: 3.6 L = 3600 mL. Then divide: 3600 ÷ 2.4 = 1500.
Unit cancellation
Converting to milliliters first makes the required division direct.
Not needed.
Using 3.6 instead of 3,600 after the unit conversion.
1500
Percentages and Growth Factors
- percent increase/decrease
- reverse percentages
- discounts
- growth factors
- weighted percentages
- consecutive percent changes
- percentage points vs percent change
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Easypercent decreaseMultiple choice
1. A jacket originally priced at $80 is discounted to $68. What is the percent decrease in the price?
- 12%
- 15%
- 17%
- 20%
Show solution
The decrease is $12, and 12/80 = 0.15 = 15%.
Subtract to find the decrease, then divide by the original price.
Mental / Algebra
The original price is the percent-change base.
Not needed.
Dividing by the discounted price instead of the original price.
B. 15%
Mediumweighted percentageMultiple choice
2. In a school district, 70% of students attend later-start schools and 30% attend earlier-start schools. If 80% of later-start students and 50% of earlier-start students prefer later start times, what percentage of all students should be estimated to prefer later start times?
- 65%
- 68%
- 71%
- 80%
Show solution
The groups have different sizes, so use a weighted average: 0.70(80)+0.30(50)=71.
Use 100 students. Later-start: 70 students, and 80% is 56. Earlier-start: 30 students, and 50% is 15. Total: 71.
100-student model
The 100-student model makes the weights visible.
Not needed.
Averaging 80% and 50% without using the group sizes.
C. 71%
Hardconsecutive percent changesStudent-produced response
3. A store increases the price of an item by p%, then decreases the new price by 25%. The final price is 5% greater than the original price. What is p?
Show solution
The total multiplier must be 1.05, so 0.75(1+p/100)=1.05.
Divide by 0.75: 1+p/100=1.40. Therefore p=40.
Multipliers
Multipliers avoid percent-change base errors.
Not needed.
Adding and subtracting percents directly.
40
One-Variable Data: Mean, Median, Range, and Spread
- mean
- median
- range
- IQR/spread intuition
- effect of outliers
- adding/removing/replacing values
- comparing distributions
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Mediummean median rangeMultiple choice
1. The data set is 3, 5, 7, 9, 11. If the value 11 is replaced with 19, which statement is true?
- The mean and median increase, but the range stays the same.
- The mean and range increase, but the median stays the same.
- The median and range increase, but the mean stays the same.
- The mean, median, and range all increase.
Show solution
The total increases, so the mean increases. The middle value is still 7, so the median stays the same. The range increases from 8 to 16.
Compare the old and new lists: 3,5,7,9,11 and 3,5,7,9,19. No exact mean calculation is needed.
Mental
Direction of change is enough; exact means would waste time.
Not needed.
Assuming the median changes because the largest value changed.
B
Mediummean median rangeMultiple choice
2. The data set is 4, 6, 8, 10, 20. If 20 is replaced with 12, which statement is true?
- The mean and median decrease, but the range stays the same.
- The mean and range decrease, but the median stays the same.
- The median and range decrease, but the mean stays the same.
- The mean, median, and range all decrease.
Show solution
The total decreases, so the mean decreases. The middle value remains 8, so the median stays the same. The range decreases from 16 to 8.
Compare the old and new ordered lists without calculating exact means.
Mental
Direction is enough; exact means are unnecessary.
Not needed.
Assuming the median changes because the largest value changed.
B. The mean and range decrease, but the median stays the same.
Mediummissing valueStudent-produced response
3. Five numbers have mean 14. Four of the numbers are 8, 11, 15, and 20. What is the fifth number?
Show solution
The total must be 5 × 14 = 70. The four known numbers sum to 54, so the missing number is 16.
Use total = mean × count, then subtract the known values.
Mental / Algebra
The mean formula gives the total in one step.
Not needed.
Dividing the known sum by 5 before finding the missing value.
16
Very Hardmean totalStudent-produced response
4. A set of 12 numbers has mean 18. Two numbers, a and b, are added, and the new mean is 21. If a = 2b, what is a?
Show solution
The correct answer is 52. The original total is 216 and the new total is 294, so the added sum is 78. Since a=2b, a=52.
Original total: 12(18)=216. New total: 14(21)=294. Added sum: 78. With a=2b, 3b=78, so a=52.
Algebra / Totals
Mean problems often become total problems.
Not needed.
Averaging 18 and 21 or forgetting the count changes from 12 to 14.
52
Two-Variable Data: Tables, Scatterplots, Lines of Best Fit, and Residuals
- scatterplot trend
- line of best fit
- residual = actual − predicted
- prediction from model
- slope/intercept interpretation
- association vs causation
- table-to-model reasoning
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Mediummodel predictionStudent-produced response
1. A line of best fit relating temperature T to energy use E is E = −1.8T + 246. The model predicts energy use of 156. What temperature does the model predict?
Show solution
The correct answer is 50. Set E=156 and solve for T.
Use 156 = −1.8T + 246. Then −90 = −1.8T, so T=50.
Algebra
The model is already given, so substitution is direct.
Not needed.
Substituting 156 for T instead of for E.
50
MediumresidualMultiple choice
2. A model predicts that a plant will be 32 centimeters tall. The actual height is 38 centimeters. What is the residual, in centimeters?
- −6
- 6
- 32
- 38
Show solution
Residual = actual − predicted = 38 − 32 = 6.
Use residual = actual − predicted.
Mental
Only subtraction is needed once the terms are identified.
Not needed.
Reversing the subtraction and getting −6.
B. 6
HardresidualMultiple choice
3. A line of best fit predicts that a plant will be 31.2 centimeters tall after 18 days. The plant’s actual height after 18 days is 34.7 centimeters. What is the residual, in centimeters?
- −3.5
- −2.5
- 3.5
- 65.9
Show solution
Choice C is correct. Residual equals actual minus predicted: 34.7−31.2=3.5.
Use residual = actual − predicted.
Mental / Algebra
The formula gives the sign and value in one step.
Not needed.
Reversing the subtraction and getting −3.5.
C. 3.5
Hardassociation languageMultiple choice
4. A scatterplot shows that students who spend more hours on a practice app tend to have higher quiz scores. Which conclusion is best supported?
- Using the app definitely causes every student to score higher.
- There is an association between app time and quiz score.
- The app lowers quiz scores for students who study less.
- Quiz scores determine how many hours students use the app.
Show solution
A scatterplot can show association, but it does not by itself prove causation.
Choose the conclusion that matches the evidence without overclaiming.
Conceptual elimination
Study-design wording is usually about what can and cannot be concluded.
Not applicable.
Turning association into causation.
B. There is an association between app time and quiz score.
Probability and Conditional Probability
- simple probability
- conditional probability
- table probability
- complement probability
- without-replacement probability
- inclusion-exclusion
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Mediumconditional denominatorStudent-produced response
1. In a group of students, 36 juniors and 24 seniors take calculus. If one calculus student is selected at random, what is the probability that the student is a junior?
Show solution
Since the selected student is known to take calculus, the total is only calculus students: 36+24=60.
Restrict the denominator first. Probability = 36/60 = 3/5.
Restrict the total
Conditional probability becomes easy once the correct denominator is identified.
Not needed.
Using all students as the denominator instead of only calculus students.
3/5
Hardwithout replacementMultiple choice
2. A bag contains 3 red marbles and 5 blue marbles. If two marbles are selected without replacement, what is the probability that both are red?
- 3/64
- 3/28
- 9/64
- 3/8
Show solution
The probability of red first is 3/8. Then 2 red marbles remain out of 7 total, so multiply: 3/8·2/7=3/28.
Update the counts after the first selection.
Algebra
The without-replacement structure has only two steps.
Not needed.
Using 3/8 twice as if the first marble were replaced.
B. 3/28
Very Hardconditional probabilityStudent-produced response
3. A table classifies 200 voters by age group and candidate preference. Of 90 voters under 30, 54 prefer Candidate A. Of 110 voters age 30 or older, 44 prefer Candidate A. If a voter who prefers Candidate A is selected at random, what is the probability that the voter is under 30?
Show solution
The correct answer is 27/49. The condition is that the voter prefers Candidate A, so the denominator is 54+44=98.
Use the restricted denominator: 54/98=27/49.
Restrict the total
The condition tells you which row/column becomes the denominator.
Not needed.
Using 90 or 200 as the denominator.
27/49
Very Hardconditional tableStudent-produced response
4. In a survey, 72 students play an instrument, 45 students play a sport, and 18 students play both. If a student who plays an instrument is selected at random, what is the probability that the student also plays a sport?
Show solution
The condition is that the selected student plays an instrument, so the denominator is 72.
Use the restricted denominator: 18/72=1/4.
Restrict the total
Conditional probability becomes direct once the condition sets the denominator.
Not needed.
Using all surveyed students or all sport players as the denominator.
1/4
Sampling, Margin of Error, and Inference
- margin-of-error intervals
- sample vs population
- sample statistic vs true population value
- whether a majority is guaranteed
- overlapping intervals
- random sampling and generalization
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Mediummargin intervalMultiple choice
1. A random sample of 500 city residents found that 49% support a proposal. The margin of error is ±5 percentage points. Which conclusion is best supported?
- A majority definitely supports the proposal.
- The true percentage is exactly 49%.
- A majority is possible, but not guaranteed.
- The margin of error proves support is below 50%.
Show solution
The interval is 44% to 54%, so values above and below 50% are both possible.
Compute the interval: 49−5=44 and 49+5=54. Since the interval crosses 50, a majority is not guaranteed.
Interval reasoning
Margin-of-error questions usually become interval questions.
Not needed.
Treating 49% as exact or treating 54% as guaranteed.
C
MediumgeneralizationMultiple choice
2. A random sample of 1,000 voters in a state is surveyed about a ballot measure. Which population can the results most reasonably be generalized to?
- All voters in the state
- All voters in the country
- Only the 1,000 surveyed voters
- All people in the state, including nonvoters
Show solution
Because the sample was randomly selected from voters in the state, the results can reasonably generalize to that population.
Identify the population from which the random sample was taken.
Conceptual elimination
The population in the sampling frame controls the valid generalization.
Not applicable.
Generalizing beyond the sampled population.
A. All voters in the state
Very Hardoverlapping intervalsMultiple choice
3. Poll A reports 47% ± 3% support. Poll B reports 53% ± 4% support. Can we conclude that Poll B’s population support is definitely higher than Poll A’s?
- Yes, because 53 is greater than 47.
- Yes, because both polls have margins of error.
- No, because random samples are never useful.
- No, because the intervals overlap.
Show solution
Choice D is correct. Poll A could be as high as 50%, and Poll B could be as low as 49%, so B is not definitely higher.
Build the intervals: Poll A is 44% to 50%; Poll B is 49% to 57%. They overlap from 49% to 50%.
Interval reasoning
Interval overlap quickly rules out a definite comparison.
Not needed.
Comparing only the point estimates 47 and 53.
D
Statistical Claims and Study Design
- observational study vs experiment
- random assignment
- random sample
- causation vs association
- generalization to a population
- supports vs proves wording
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Mediumobservational studyMultiple choice
1. An observational study finds that students who eat breakfast tend to have higher math scores than students who do not. Which conclusion is best supported?
- Eating breakfast definitely causes higher math scores.
- There is an association between eating breakfast and math scores.
- Not eating breakfast prevents learning math.
- Breakfast has no relationship to math scores.
Show solution
An observational study can support association, but it does not prove causation without random assignment or experimental control.
Choose the conclusion that does not overclaim causation.
Conceptual elimination
The study type directly limits the conclusion.
Not applicable.
Treating an observational relationship as proof of cause and effect.
B. There is an association between eating breakfast and math scores.
Hardrandom assignmentMultiple choice
2. Researchers randomly assigned 150 volunteers to use either Method X or Method Y for four weeks. The Method X group answered an average of 6 more review questions correctly. Which conclusion is best supported?
- For these volunteers, Method X caused higher average performance.
- Method X will cause every student to answer exactly 6 more correctly.
- Method X is proven better for all students.
- Method Y prevents students from learning.
Show solution
Random assignment supports cause-and-effect for the volunteers, but volunteers are not automatically representative of all students.
Look for random assignment and population. Random assignment supports causation for the study participants only.
Conceptual elimination
The design feature tells you the strength of the conclusion.
Not needed.
Confusing random assignment with random sampling.
A
Very Hardsampling plus assignmentMultiple choice
3. A researcher randomly selected 600 students from a district, then randomly assigned them to use either Method A or Method B. Students using Method A scored higher on average. Which conclusion is best supported?
- Method A caused every student to score higher.
- The result cannot be generalized at all.
- Only association is supported, not causation.
- Method A caused higher average scores for the sample, and the result may generalize to the district.
Show solution
Choice D is correct. Random selection supports generalization to the district, and random assignment supports causal comparison.
Identify both design features: random selection plus random assignment.
Conceptual elimination
Both features together allow a stronger conclusion than either one alone.
Not needed.
Saying the result applies to every student or guarantees individual improvement.
D
Very Hardobservational claimMultiple choice
4. A city collects data and finds that neighborhoods with more bike lanes tend to have lower car-accident rates. The data were observational and did not involve random assignment. Which conclusion is best supported?
- Adding bike lanes definitely causes every neighborhood to have fewer accidents.
- There is an association between bike-lane availability and car-accident rates.
- Bike lanes have no relationship to accident rates.
- Car accidents cause cities to remove bike lanes.
Show solution
Observational data can support an association, but it does not by itself prove cause and effect.
Identify the study type. Without random assignment, avoid causal wording.
Conceptual elimination
Study-design questions often depend on whether the conclusion overclaims causation.
Not applicable.
Turning an observed relationship into a definite causal claim.
B
Geometry and Trigonometry
Area, volume, triangles, similarity, trigonometry, circles, and coordinate geometry.
Area, Perimeter, Volume, and Scale
- area
- perimeter
- volume
- surface area
- algebraic dimensions
- scale factors
- length vs area vs volume units
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Basicrectangle areaMultiple choice
1. A rectangle has length 12 and width 7. What is its area?
- 19
- 38
- 84
- 144
Show solution
Area of a rectangle is length × width, so 12 × 7 = 84.
Multiply length by width.
Mental
This is a one-step formula question.
Not needed.
Finding perimeter instead of area.
C. 84
Mediumarea scaleMultiple choice
2. Two similar triangles have corresponding side lengths in a ratio of 3:5. If the area of the larger triangle is 100, what is the area of the smaller triangle?
- 36
- 45
- 60
- 64
Show solution
Choice A is correct. The area ratio is the square of the side ratio, so smaller-to-larger area is 9:25.
Compute 100 · 9/25 = 36.
Geometry theorem
Squaring the scale factor gives the area relationship immediately.
Not needed.
Using the side ratio 3:5 directly for area.
A. 36
Mediumscale factorStudent-produced response
3. Two similar solids have side lengths in a ratio of 2:5. If the smaller solid has volume 24, what is the volume of the larger solid?
Show solution
The volume scale factor is (5/2)3 = 125/8. The larger volume is 24·125/8=375.
Cube the side-length scale factor before scaling volume.
Algebra
Volume uses cubic scaling, not linear scaling.
Not needed.
Multiplying by 5/2 instead of cubing the factor.
375
Hardrectangular prismMultiple choice
4. A rectangular prism has volume 1,080. Its length is 10, its width is 2x, and its height is x + 3. What is the positive value of x?
- 4
- 5
- 6
- 9
Show solution
The equation is 1080=10(2x)(x+3), which simplifies to 54=x(x+3). Testing x=6 gives 6(9)=54.
Simplify first, then test the choices.
Use Choices or Algebra
Testing small choices after simplification is faster than factoring a quadratic.
Not needed.
Forgetting that the width is 2x, not x.
C. 6
Lines, Angles, and Triangles
- triangle angle sum
- exterior angles
- right triangles
- Pythagorean theorem
- special right triangles
- altitude to hypotenuse theorem
- algebraic side/angle expressions
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Easyangle sumStudent-produced response
1. Two angles of a triangle measure 47° and 68°. What is the measure, in degrees, of the third angle?
Show solution
The angles in a triangle sum to 180°, so the third angle is 180 − 47 − 68 = 65.
Subtract the two known angles from 180.
Mental
This is a direct triangle-sum question.
Not needed.
Forgetting to subtract both angles.
65
Mediumexterior angleMultiple choice
2. In a triangle, two interior angles measure 48° and 67°. What is the measure of the exterior angle adjacent to the third interior angle?
- 65°
- 77°
- 105°
- 115°
Show solution
Choice D is correct. The exterior angle equals the sum of the two remote interior angles: 48+67=115.
Use the exterior angle theorem instead of finding the third angle first.
Mental theorem
The theorem gives the exterior angle in one addition.
Not needed.
Giving the third interior angle, 65°, instead of the exterior angle.
D. 115°
Mediumspecial right triangleMultiple choice
3. In a 30°-60°-90° triangle, the shortest side has length 6. What is the length of the hypotenuse?
- 62
- 63
- 12
- 18
Show solution
In a 30°-60°-90° triangle, the sides are in the ratio 1 : √3 : 2. The hypotenuse is twice the shortest side, so it is 12.
Use the special-right-triangle ratio.
Mental
The ratio avoids the Pythagorean theorem.
Not needed.
Multiplying by √3 instead of 2.
C. 12
Hardaltitude theoremMultiple choice
4. In right triangle ABC, angle C = 90°. The altitude from C to hypotenuse AB meets AB at D. If AD = 5 and DB = 20, what is AC?
- 5
- 55
- 105
- 25
Show solution
The full hypotenuse is 25. The theorem gives AC2 = AD · AB = 5·25=125, so AC=55.
Use the altitude-to-hypotenuse theorem: leg² = adjacent hypotenuse segment × whole hypotenuse.
Geometry theorem
Recognizing the theorem avoids reconstructing the whole triangle.
Not needed.
Using DB instead of AD for side AC.
B. 55
Hardaltitude to hypotenuseMultiple choice
5. In right triangle ABC, angle C = 90°. The altitude from C to hypotenuse AB meets AB at D. If AD=9 and DB=16, what is AC?
- 12
- 15
- 20
- 25
Show solution
The full hypotenuse is AB=25. The altitude theorem gives AC2 = AD·AB = 9·25 = 225, so AC=15.
Recognize the right-triangle altitude theorem.
Geometry theorem
The theorem avoids reconstructing the whole triangle.
Not useful.
Using DB instead of AD for side AC.
B. 15
Similarity and Right Triangle Trigonometry
- similar triangles
- side ratios
- area scale factor
- volume scale factor
- sine/cosine/tangent
- complementary trig
- Pythagorean triples
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Mediumtriangle similarityStudent-produced response
1. Triangle A is similar to Triangle B. A side of Triangle A has length 18, and the corresponding side of Triangle B has length 30. If another side of Triangle A has length 24, what is the corresponding side length in Triangle B?
Show solution
The scale factor from Triangle A to Triangle B is 30/18 = 5/3.
Multiply the corresponding side by the scale factor: 24 · 5/3 = 40.
Ratio scaling
Similar figures use the same scale factor for all corresponding side lengths.
Not needed.
Using an area or volume scale factor instead of a side-length scale factor.
40
Mediumarea scale factorMultiple choice
2. Two similar triangles have corresponding side lengths in a ratio of 4:7. If the area of the smaller triangle is 48, what is the area of the larger triangle?
- 84
- 112
- 147
- 196
Show solution
The side-length scale factor from smaller to larger is 7/4, so the area scale factor is 49/16.
Compute 48 · 49/16 = 3 · 49 = 147.
Algebra / Scaling
Area scales by the square of the side-length scale factor.
Not needed.
Multiplying by 7/4 instead of squaring the scale factor.
C. 147
Mediumcomplementary trigStudent-produced response
3. Angles A and B are complementary acute angles in a right triangle. If sin A = 5/13, what is cos B?
Show solution
For complementary acute angles in a right triangle, sin A = cos B. Therefore cos B = 5/13.
Use the complementary-angle identity.
Mental
No side lengths need to be calculated.
Not needed.
Treating sin A and cos B as unrelated values.
5/13
Very Hardtrig ratioStudent-produced response
4. In a right triangle, sin A = 5/13 and the hypotenuse is 39. What is the area of the triangle?
Show solution
The ratio is a 5-12-13 triangle scaled by 3, so the legs are 15 and 36.
Since the hypotenuse 13 scales to 39, the scale factor is 3. Area = 1/2(15)(36)=270.
Mental / Ratio scaling
Recognizing the Pythagorean triple avoids solving for the missing leg from scratch.
Not needed.
Using 5 and 13 as actual side lengths instead of ratio parts.
270
Very Hardtrig areaStudent-produced response
5. In a right triangle, sin A = 3/5, and the hypotenuse is 40. What is the area of the triangle?
Show solution
The triangle is a 3-4-5 triangle scaled by 8, so the legs are 24 and 32.
Recognize the 3-4-5 triple. Area = 1/2(24)(32)=384.
Mental / Algebra
The Pythagorean triple avoids solving for the missing leg from scratch.
Not needed.
Using 40 as a leg instead of the hypotenuse.
384
Circles: Arcs, Sectors, Equations, Chords, and Tangents
- radius vs diameter
- circumference and area
- arc length
- sector area
- circle equation
- chord length from a line
- completing the square
- tangent-radius perpendicular relationship
- distance from center to tangent line
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Hardsector areaStudent-produced response
1. A sector of a circle has arc length 8π and area 40π. What is the radius of the circle?
Show solution
Use A = 1/2rs.
Substitute: 40π = 1/2(r)(8π)=4πr, so r=10.
Formula shortcut
The sector shortcut avoids central-angle formulas.
Not needed.
Using full-circle area and circumference formulas separately.
10
Hardsector shortcutStudent-produced response
2. A sector of a circle has arc length 15π and area 60π. What is the radius of the circle?
Show solution
Use A=1/2rs. Then 60π = 1/2(r)(15π) = 7.5πr, so r=8.
Use the sector area shortcut with arc length.
Formula recognition
The shortcut avoids central-angle formulas.
Not needed.
Using full-circle formulas separately.
8
Very Hardcircle chordMultiple choice
3. The line y = 0 intersects the circle x2 + y2 − 4x + 6y − 5 = 0 at two points. What is the distance between those two points?
- 4
- 5
- 6
- 10
Show solution
Setting y=0 gives x2−4x−5=0, with roots −1 and 5. The distance is 6.
Substitute y=0 immediately, solve for the two x-values, and subtract them.
Algebra
There is no need to complete the square for the whole circle when the chord lies on y=0.
Not needed.
Finding the radius instead of the chord length.
C. 6
Very Hardtangent lineMultiple choice
4. In the xy-plane, the line y = x + c is tangent to the circle x2 + y2 = 18. What is the positive value of c?
- 3
- 32
- 6
- 62
Show solution
The radius is 18=32. The distance from (0,0) to x−y+c=0 is |c|/√2, so c=6.
Set distance from center to line equal to radius: |c|/√2 = 32.
Distance from point to line
Tangency means the distance from the center to the tangent line equals the radius.
Not needed.
Using 18 as the radius instead of √18.
C. 6
Very Hardcircle chordMultiple choice
5. The line y=0 intersects the circle x2 + y2 − 8x + 2y − 5 = 0 at two points. What is the distance between those two points?
- 25
- 221
- 45
- 62
Show solution
Set y=0 to get x2 − 8x − 5 = 0. The roots are 4 ± 21, so the distance between them is 221.
Substitute y = 0 immediately instead of completing the square for the whole circle.
Algebra
The line gives a one-variable quadratic right away.
Graphing can verify the chord, but algebra is cleaner for the exact answer.
Giving the radius instead of the chord length.
B. 221
Coordinate Geometry
- midpoint
- distance
- slope
- coordinate signs
- perpendicular/parallel slopes
- unknown coordinate from distance
- geometric facts on the coordinate plane
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
EasymidpointMultiple choice
1. What is the midpoint of the segment with endpoints (−2,5) and (8,1)?
- (3,3)
- (5,4)
- (6,−4)
- (10,6)
Show solution
Average the x-coordinates and y-coordinates: (−2+8)/2=3 and (5+1)/2=3.
Average the coordinates component by component.
Mental
The numbers are small and direct.
Not needed.
Adding coordinates without dividing by 2.
A. (3,3)
MediummidpointStudent-produced response
2. The midpoint of (a, 4) and (2, 10) is (5, 7). What is a?
Show solution
The correct answer is 8. The x-coordinate of the midpoint is the average of a and 2.
Set (a+2)/2 = 5. Then a+2=10 and a=8.
Algebra
The y-coordinates already confirm the midpoint, so only the x-coordinate matters.
Not needed.
Averaging all four coordinates together.
8
MediumdistanceStudent-produced response
3. What is the distance between (−1,4) and (7,−2)?
Show solution
The horizontal difference is 8 and the vertical difference is 6. The distance is the hypotenuse of a 6-8-10 right triangle, so it is 10.
Use the right-triangle structure instead of writing the full formula.
Mental / Geometry
Recognizing the 6-8-10 triple saves calculation.
Not needed.
Forgetting to square both coordinate differences if using the formula.
10
Very Harddistance structureStudent-produced response
4. Point P(a, 4) is 13 units from Q(−5, −1). If a > −5, what is a?
Show solution
The correct answer is 7. The vertical difference is 5, and the distance is 13, so the horizontal difference must be 12.
Use the 5-12-13 right triangle. Since a is greater than −5, a = −5 + 12 = 7.
Mental / Distance
Recognizing the Pythagorean triple avoids the full distance formula.
Not needed.
Using the other possible x-coordinate, −17, even though a > −5.
7
Very Hardcircle distanceStudent-produced response
5. Point R(3,b) is on a circle centered at (−1,2) with radius 5. If b > 2, what is b?
Show solution
The horizontal distance from (−1,2) to (3,b) is 4. The radius is 5, so the vertical distance must be 3.
Use the 3-4-5 right triangle. Since b > 2, use b=2+3=5.
Geometry / Distance
Recognizing the right-triangle structure avoids the full distance formula.
Not needed.
Using b=−1, the lower point on the circle, despite the condition b > 2.
5
Calculator Strategy: Desmos, Tables, Graphs.
Strategy section · not an official SAT Math domainDesmos Graphing: Intersections, Zeros, and Extrema
- graph two sides separately
- tap intersections
- read the requested coordinate
- graph distance squared instead of distance when minimizing
- use restrictions like x > 0 when needed
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Lesson Checkapproximate rootLesson Check
1. The equation 0.45x² + 1.8x = 11.7 has a positive solution. Which value is closest to that solution?
- 2.8
- 3.5
- 4.2
- 5.1
Show solution
The positive solution is about 3.48, so 3.5 is closest.
Graph y=0.45x2+1.8x and y=11.7, or table-check the choices.
Desmos Graph
The equation has decimals and asks for a closest value, so approximation is efficient.
Graph both sides separately and tap the positive intersection.
Rounding before reading the intersection carefully.
3.5
Lesson Checkrounded zeroLesson Check
2. The equation x² − 7.4x + 10 = 0 has two solutions. What is the greater solution, rounded to the nearest tenth?
Show solution
The greater zero is about 5.62, so the rounded answer is 5.6.
Graph y=x2−7.4x+10 and y=0. Read the greater x-intercept.
Desmos Graph
Graphing is faster because the answer is rounded.
Use the x-intercepts; choose the greater x-value.
Reading the smaller zero or reading the y-coordinate.
5.6
Desmos Tables and Function Values
- evaluating h(7)
- matching tables to equations
- testing answer choices
- checking a parameter at one input
- reading the requested value carefully
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Lesson Checkfunction valueLesson Check
1. Let h(x) = ((x − 2)2 + 5)/(x + 3). What is h(7)?
Show solution
The correct answer is 3. At x=7, the numerator is (5)2+5=30 and the denominator is 10.
Use a Desmos table or substitute directly: 30/10=3.
Desmos Table or Algebra
The input is specific, so a table gives the value immediately.
Enter h(x)=((x-2)^2+5)/(x+3), then add a table and read x=7.
Entering (x−2)2 + 5/x + 3 without grouping the fraction.
3
Lesson Checkcompare functionsLesson Check
2. At x=4, how much greater is p(x)=2x2−1 than q(x)=x+23?
Show solution
The correct answer is 4. p(4)=31 and q(4)=27, so the difference is 4.
Use a table with both functions at x=4, then subtract 31−27.
Desmos Table
Tables are efficient when comparing function values at a named input.
Make a table for p(x) and q(x), then compare the row x=4.
Subtracting in the wrong order and getting −4.
4
Answer-Choice Strategy: Backsolving and Elimination
- test simple numeric choices
- use middle choices when ordered
- eliminate impossible values
- use bounds
- check original condition, not a distorted version
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Lesson CheckeliminationLesson Check
1. A weighted average is based on 70% of a group with value 80 and 30% with value 50. Before calculating exactly, which range must the answer be in?
Show solution
The answer must be between the two group values and closer to 80 because the 80 group is larger.
Eliminate choices outside 50 to 80 and choices closer to 50. Then compute if needed: 0.70(80)+0.30(50)=71.
Eliminate, then Mental
Reasonableness can remove bad answers before arithmetic.
Not needed.
Averaging 80 and 50 without using weights.
Between 50 and 80, closer to 80
Lesson Checkuse choicesLesson Check
2. For 48 = x(x + 2), the answer choices are 4, 5, 6, and 8. Which choice works?
- 4
- 5
- 6
- 8
Show solution
The correct answer is 6 because 6(8)=48.
Test the choices in the simplified equation.
Use Choices
Small integer choices make direct testing faster than factoring.
Not needed.
Testing in an unsimplified equation and making arithmetic harder.
6
Calculator Entry Accuracy and When Not to Use Desmos
- fractions need grouped numerator and denominator
- negative inputs need parentheses
- binomials need parentheses before squaring
- read x vs y from graph
- skip Desmos when algebra is simpler
Practice this skill
These questions build from foundation to SAT-style difficulty. Try each one before revealing the solution.
Lesson Checkentry accuracyLesson Check
1. Which Desmos entry correctly represents (x + 5)/(x − 2)?
- x+5/x-2
- (x+5)/(x-2)
- x+(5/x)-2
- (x+5)/x-2
Show solution
Choice B is correct because it groups the full numerator and the full denominator.
Use parentheses around both groups: (x+5)/(x−2).
Calculator accuracy
Grouping prevents Desmos from dividing only part of the expression.
The second entry is correct.
Typing x+5/x−2, which means x + 5/x − 2, not the intended fraction.
B. (x+5)/(x−2)
Lesson Checkwhen not to use DesmosLesson Check
2. If 5x + 7 = 32, what is the fastest route to find 10x + 14?
Show solution
The fastest route is mental because 10x+14 = 2(5x+7).
Double the given value: 2(32)=64.
Mental
The expression target is obvious, so graphing or solving for x is unnecessary.
Not worth it.
Opening Desmos for a one-step structure question.
64
Official Digital SAT Math Resources
Use these tools together: full tests in Bluebook, score review in My Practice, targeted question filters in the Student Question Bank, and official-aligned lessons in Khan Academy.
