SAT math practice questions, with step-by-step answers

A. Correct. So C(d) = 70d + 45.

1. The per-day rate, $70, is the slope: it multiplies the number of days.
2. The one-time fee, $45, is the starting amount: the constant term.
3. So C(d) = 70d + 45.

Why the others are wrong: B This adds the fee into the rate, so the fee would grow with every day. It is paid only once. C This charges the one-time fee every day. The fee is paid once, so it is the constant term; the per-day rate multiplies d. D Multiplying out gives 70d + 3150: the fee would be $3150, not $45.

C. Correct. Here 5 multiplies d, so it is the rate; 80 is the value at d = 0.

1. In y = b + mx, the constant b is the value when x = 0 and m is the change for each unit of x.
2. Here 5 multiplies d, so it is the rate; 80 is the value at d = 0.

Why the others are wrong: A That would be the value of H at a particular time, found by substituting. The number itself is the starting value. B That describes the rate, 5. The number 80 is the value when d = 0. D 80 is not a time. It is a starting amount.

C. Correct. So 4a + 6b ≤ 95.

1. Money spent on tickets for students: 4a. On tickets for adults: 6b.
2. "At most $95" means the total is less than or equal to 95.
3. So 4a + 6b ≤ 95.

Why the others are wrong: A The prices are switched: each of the tickets for students (a) costs $4. B This limits the number of items, not the money spent. Each item has to be multiplied by its price. D "At most" means the total cannot go over the budget, so it is ≤, not ≥.

D. Correct. They cross at (4, −3).

1. The solution of a system is the point on both graphs: where the lines cross.
2. They cross at (4, −3).

Why the others are wrong: A The coordinates are switched: the lines cross 4 right of the y-axis, so x = 4. B Check the sign of y: the crossing is below the x-axis. C Check the sign of x: the crossing is right of the y-axis.

A. Correct. Last: −9 × −9 = 81.

1. First: 5x × 4x = 20x².
2. Outer plus inner: −45x + −36x = −81x.
3. Last: −9 × −9 = 81.

Why the others are wrong: B One of the middle products has the wrong sign. Check each sign as you multiply. C This multiplies only the first terms and the last terms. The outer and inner products make the middle term. D The constant term is the product of the last terms, −9 × −9 = 81, not their sum.

B. Correct. After t years: 3,600 × 0.9^t.

1. A 10% decrease each year multiplies the amount by 1 − 0.1 = 0.9.
2. After t years: 3,600 × 0.9^t.

Why the others are wrong: A This grows by 10% a year. Decay of 10% multiplies by 0.9 each year. C This changes by the same amount each year (linear). A percent change each year is exponential: the change grows with the amount. D Multiplying by 0.1 keeps only 10% each year. The yearly multiplier is 0.9.

C. Correct. x² = 8: positive, two solutions.

1. y = x² + 3 opens up with its lowest point at y = 3.
2. x² = 8: positive, two solutions.

Why the others are wrong: A The parabola's lowest point is at y = 3. The line y = 11 is above it, so it crosses twice. B A line and a parabola can never be the same graph. D The parabola's lowest point is at y = 3. The line y = 11 is above it, so it crosses twice.

D. Correct. Substitute x = 19 into the line: 4(19) + 19 = 95.

1. Substitute x = 19 into the line: 4(19) + 19 = 95.

Why the others are wrong: A This is the prediction for x = 21. B This leaves out the y-intercept, 19. C The slope and the intercept are switched.

A. Correct. Probability: 28 ÷ 61.

1. The person comes from the "Ninth grade" row: 61 people.
2. Of those, 28 are in the "No pet" column.
3. Probability: 28 ÷ 61.

Why the others are wrong: B This divides by the "No pet" total. The person is chosen from the "Ninth grade" group, so that row's total, 61, is the denominator. C This divides by everyone surveyed. The choice is only among the 61 people in the "Ninth grade" group. D This is the probability that a person from the whole survey is in the "Ninth grade" group.

D. Correct. Here that is all visitors to that state park.

1. Results from a random sample can be generalized to the population the sample was drawn from, and no further.
2. Here that is all visitors to that state park.

Why the others are wrong: A A random sample of this size is large enough to estimate the group it was drawn from. B The sample was drawn from one place only, so it represents that place, not a wider population. C Because the sample was random, the results can be generalized to the whole group it was drawn from.

D. Correct. With the same method and population, a larger random sample gives a smaller margin of error.

1. With the same method and population, a larger random sample gives a smaller margin of error.

Why the others are wrong: A More data pins down the estimate more tightly, so the margin of error shrinks as the sample grows. B A smaller sample gives a less precise estimate, so its margin of error is larger. C Both being random makes both fair, but the larger one is more precise.

A. Correct. $9.36 ÷ 24 = 39 cents per ounce, the lowest.

1. Divide each price by its number of ounces.
2. The lowest is 39 cents per ounce: 24 ounces for $9.36.

Why the others are wrong: B $5.52 ÷ 12 = 46 cents per ounce. C $3.60 ÷ 8 = 45 cents per ounce. D $6.40 ÷ 16 = 40 cents per ounce.

D. Correct. The mean rises by about 9.0, the median by 3.

1. Median before: (30 + 36)/2 = 33. After: 36.
2. Mean before: 32.67. After: 41.71.
3. A value far from the rest pulls the mean much more than the median.

Why the others are wrong: A With 7 values, the median becomes 36, up from 33. B The median moves only to the next middle value; the large value pulls the mean much more. C 96 is larger than every value, so the mean goes up.

A. Correct. So x = 180 − 89 = 91.

1. The angle in the matching position at line m is 89°; x° sits beside it on a straight line.
2. So x = 180 − 89 = 91.

Why the others are wrong: B Angles around a point add to 360°, but here the pair adds to 180° or is equal. C No right angle is shown. Use parallel-line angle pairs. D These angles sit on a straight line with an angle equal to the one marked, so they add to 180°.

A. Correct. Hypotenuse: 2 × 11 = 22.

1. In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2 (opposite 30°, 60°, 90°).
2. Hypotenuse: 2 × 11 = 22.

Why the others are wrong: B 11√3 is the other leg. The hypotenuse is twice the short leg. C The hypotenuse is twice the side opposite 30°. D √2 belongs to a 45-45-90 triangle.

A. Correct. (x − h)² + (y − k)² = r²: (x + 1)² + (y − 2)² = 9.

1. Center: (−1, 2). Radius: 3 (count from the center to the circle).
2. (x − h)² + (y − k)² = r²: (x + 1)² + (y − 2)² = 9.

Why the others are wrong: B The center (−1, 2) appears as (x − h) and (y − k), so the signs inside are the opposite of the center's. C The center is (−1, 2): x goes with −1. D The right side is the radius squared: 3² = 9.