SAT Math Practice Problems Hard

Subscribe to our ▶️ YouTube channel 🔴 for the latest videos, updates, and tips.

Problem 1 :

Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is 58 square centimeters cm (cm²), and the surface area of right rectangular prism Y is 1,450 cm². The volume of right rectangular prism Y is 1,250 cubic centimeters (cm³ ). What is the sum of the volumes, in cm³, of right rectangular prism X and right rectangular prism Y?

Solution :

Problem 2 :

In the given equation, k is a constant. The equation has exactly one real solution. What is the value of 4k?

Solution :

Problem 3 :

In triangle ABC, ∠B is right angle, measure of ∠A is 30°, the length of hypotenus is 20√3 units and the area of triangle can be written as a√b, where a and b are positive integers and also not perfect squares. What is the value of a + b?

Solution :

Problem 4 :

72x² – (18p + 9q)x + 64p = –32q

In the given equation, p and q are positive constants. The product of the solutions to the given equation is n(2p + q), where n is a constant. What is the value of n?

Solution :

Problem 5 :

In the xy-plane, the graph of y = g(x) is the result of translating the graph of y = f(x) down 3 units and to the left by 5 units. If the function f is defined by f(x) = (x – 3)² + 1, which equation defines the function g?

A) g(x) = (x – 8)² – 2

B) g(x) = (x + 2)² – 2

C) g(x) = (x – 8)² + 4

D) g(x) = (x + 2)² + 4

Solution :

Problem 6 :

In the figure shown, point B lies on line segment DF, point E lies on line segment AC, line segments AB and DE intersect at point W, and line segments BC and EF intersect at point X. The measure of angle ∠A is 40°, the measure of angle ∠C is 60°, the measure of angle ∠D is 68°, and the measure of angle ∠FBC is 72°. What is the measure, in degrees, ∠AED?

Solution :

Problem 7 :

A clothing store sells a certain jacket for 20% more than its suggested retail price. Using a rewards card, a customer is able to purchase the jacket for 25% less than what the store sells it for. The customer's purchase price is what percent of the jacket's suggested retail price?

A) 80%

B) 84%

C) 90%

D) 95%

Solution :

Problem 8 :

If p : q is equal to √(p - x²) : √(q - x²), then which of the following is equal to x²?

A) p/(p + q)

B) q/(p + q)

C) pq/(p - q)

D) pq/(p + q)

Solution :

Problem 9 :

If 5^a = 6^b = 30^–c, what is the value of 1/a + 1/b + 1/c?

Solution :

Problem 10 :

The ratio of radii of two circles is 2 : 5 and the area of the first circle is 36π. Which of the following is the area of the second circle?

A) 5π

B) 25π

C) 125π

D) 225π

Solution :

Problem 11 :

A table that weighs 348 pounds puts equal amounts of pressure on each of its 4 identical legs. If the pressure that each leg exerts on the floor is 12 pounds per square inch, what is the area, in square inches, of the bottom of each leg that is touching the floor?

Solution :

Problem 12 :

In the figure below, lines a and b are parallel. What is the value of x + y?

Solution :

Problem 13 :

The functions f and g are defined by the given equations, where x ≥ 0. Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where x ≥ 0?

I. f(x) = 232(0.4)^x^{+ 2}

II. g(x) = 232(0.4)(0.4)(0.4)^x^{- 2}

A) I only

B) II only

C) I and II

D) Neither I nor II

Solution :

Problem 14 :

An equilateral triangle with side lengths of k is inscribed in a circle. Which of the following defines the radius, r, in terms of k?

A) k/2

B) k√3/6

C) k√3/3

D) k√3/2

Solution :

Problem 15 :

If 2a/b = 4.25 and a/bn = 8.5, what is the value of n?

Solution :

Problem 16 :

An object’s speed is increasing at a rate of 4.80 meters per second squared. What is the rate in miles per minute squared, rounded to the nearest tenth? (Use 1 mile = 1609 meters).

Solution :

Problem 17 :

An acceptable noise criterion rating for the background noise in a school classroom is 30. For a noise criterion rating of 30, the equation y = 19(0.978)^{x -}⁵⁰ + 37 gives the estimated sound pressure level, y in decibels, as a function of the octave band center frequency, x in hertz, where x ≥ 50. Which of the following is the best interpretation of 37 in this context?

A) 37 is 19 less than the estimated sound pressure level, in decibels, at an octave band frequency of 50.

B) 37 is 19 less than the estimated sound pressure level, in decibels, at an octave band frequency of 0.

C) 37 is the estimated sound pressure level, in decibels, at an octave band frequency of 50.

D) 37 is the estimated sound pressure level, in decibels, at an octave band frequency of 0.

Solution :

Problem 18 :

The graph of the quadratic function y = f(x) in the xy-plane intersects the x-axis when x = 74 and x = k, where k is a constant. The maximum value of y = f(x) occurs at the point (17, m), where m is a constant. What is the value of k?