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6 Hardest PSAT Math Questions

The PSAT is a standardized test that measures students’ readiness for college. Juniors typically take the PSAT in high school, and the scores help determine eligibility for National Merit Scholarships. PSAT practice questions are necessary to know to answer the PSAT math questions in exams.

Types of PSAT Math Questions

The PSAT is an optional test. It has no bearing on your likelihood of being admitted to college. But because the PSAT and the SAT are so similar in scoring, format, and content, being ready for the PSAT prepares you for the SAT too.

The PSAT questions are of four difficulty levels:
  • Basic questions
  • Intermediate-level questions
  • Advanced-level questions
  • Very Hard level questions

You can take the PSAT in the 9th grade to see how well you will do on the SAT and ACT. Your PSAT score will be used to decide which colleges you should apply to.

The article discusses some of the most popular and hardest PSAT questions and how to solve them.

How Will PSAT Score Affect Your College Chances?

Some people take the SAT or ACT instead of taking the PSAT because they want to focus on studying for one standardized test only.

Poor PSAT results won't have any immediate impact on your likelihood of being accepted into a college. However, a PSAT high score is still important because it is a strong indicator of college success.

6 Hardest PSAT Practice Questions

The hardest PSAT math questions require you to use your knowledge of algebra. These problems typically include a variable in an equation or a word problem.

1. If f(x - 1)=2x+3 for all values of x, what is the value of f(-3)?

A. -5

B. -1

C. -7

D. -3

Answer: B

Explanation: The equation is stated in terms of f(x – 1). So, we need to calculate the value of x:

f(x – 1) = f(-3) = f(-2 – 1)

Therefore, x = -2.

Solving for x = -2, we get:

2(-2) + 3 = -1

2. 78 students from a college archeology class are going to a dig site to find and study artifacts. The excavation site has been divided into 24 sections, with each section studied by a group of two or four students. How many sections will a group of two students study?

Answer: 9 sections

Explanation: Let x be the number of groups with 2 students and y be the number of groups with 4 students.

So, 2x + 4y = 78 ---> x + 2y = 39 ----(1)

x + y = 24 ----(2)

(1) - (2):

y = 15

Substitute 17 for y in (2).

x + 15 = 24

x = 9

3. (x2y3)1/2(x2y3)1/3 = xa/3ya/2

What is the value of a if the a-containing equation above holds true for all positive values of x and y?

A. 2

B. 3

C. 5

D. 6

Answer: C


(x2y3)1/2(x2y3)1/3 = xa/3ya/2

(x2y3)1/2 + 1/3 = xa/3ya/2

(x2y3)(3 + 2)/6 = xa/3ya/2

(x2y3)5/6 = xa/3ya/2

(x2)5/6(y3)5/6 = xa/3ya/2

x5/3y5/2 = xa/3ya/2

x5/3y5/2 = xa/3ya/2

Since the equation is true for all positive values of x and y, it follows that the corresponding exponents of x and y on both sides of the equation must be equal.

a/3 = 5/3

a = 5

4. If x-2 is a factor of x2 - bx + b, where b is constant, what is the value of b?

Answer: 4


x² – bx + b can also be written as (x – 2)(x – a) in which we need to solve for a.

Now, we need to expand:

(x – 2)(x – a) = x² – 2x – ax +2a.

Equate both expressions:

x² – 2x – ax +2a = x² – bx + b

Compare like terms:

-2x – ax = -bx → (cancel out the x) → -2 – a = -b

2a = b

Since 2a = b, substitute the value for b into the first equation:

-2 – a = -(2a)

-2 – a = -2a

-2 = -a

a = 2

Substituting a = 2 into 2a = b gives us b = 4.

5. A supermarket offers a brand of juice in both single bottles and packs of six bottles. The store sold 281 bottles of the brand of juice on a particular day, 29 of which were sold as single bottles. Which equation represents the number of bottles sold that day, p, in packs?

A. p = (281 - 29)/6

B. p = (281 + 29)/6

C. p = 281/6 - 29

D. p = 281/6 + 29

Answer: A


Out of the total 281 bottles sold, 29 were sold individually. The rest (281 - 29) were sold in packs of 6 bottles. So, the number of packs of bottles, p, sold that day in the store is p = (281 - 29)/6.

6. The equation y = 36 + 18x represents the relationship between the height, y, of a typical apple tree and the number of years, x, after it was planted. What does the y-intercept of the graph indicate if the equation is graphed in the xy plane?

A. The age of an apple tree when it is planted (in years).

B. The height of a typical apple tree when it is planted (in inches).

C. The number of years it takes an apple tree to grow.

D. The number of inches a typical apple tree grows each year.

Answer: B


y = 36 + 18x

In this equation, to get y-intercept, we need to substitute x = 0.

y = 36 + 18(0)

y = 36

When the number of years is 0, the height of the tree is 36 inches. Then, the height of the tree is 36 inches when it is planted.

Final Tip for Solving Hard PSAT Questions

The PSAT assesses how well you apply your math knowledge. If you want to find the answers to all PSAT math questions, download MathMaster now. It helps solve any task in 2 clicks and gives explanations with steps.