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4 Hardest SAT Math Questions

Oct 20, 2022

We’ve compiled a list of the 4 hardest SAT math questions on the current SAT, along with techniques and answer explanations for each. Because these are all difficult SAT Math problem questions from College Board SAT exams, comprehending them is one of the greatest methods to prepare for those of you looking for excellence.

The SAT’s third and fourth components will always be math sections. The first math component does not enable you to use a calculator, however, the second math portion does.


But don’t be too concerned about the no-calculator section. If you’re not permitted to use a calculator on a question, it implies you don’t need one to answer it.

Question 1: Grid-in response, no calculator allowed

Question 1
Subject: Functions
3/2 is the answer.


Explanation of the Answer:
First, recognize that g(x) is the function g’s output and x is its input. a, on the other hand, indicates a specific input that was entered into the equation.
The output is g(a) = a^2

To obtain the value of a, we must first find an alternate value for g(a), resulting in a system of equations. We get the alternative value for g(a) by filling ‘a’ into the equation for function g, g(x).
We get, g(a) = (a – 3)^2


We can now set (a – 3)^2 = a^2 because they both constitute the same value, g(a). Then, by foiling the left side, we solve for a first:
(a – 3)^2 = a^2
a^2 – 6a + 9 = a^2

Because it is a quadratic equation, we must shift everything to one side to solve it. By shifting the a2 to the left and simplifying, we get the following:
a^2 – 6a + 9 – a^2 = 0
-6a + 9 = 0


We now have a two-step equation to solve. 6a added to both sides, then divided by 6 on both sides yields a:
9 = 6a
a = 9/6 = 3/2

Question 2: Multiple choice, no calculator allowed

math question 2


math question 3
Subject: Quadratics
Answer is 7

Explanation of the Answer:

In this question type, we are told that one expression equals another. As a result, we may begin by making them equal:
(2x + 6)(ax – 5) – x^2 + 30 = bx


We want to remove the parenthesis to view all of the coefficients. As a result, our next step will be to FOIL:
2ax^2 – 10x + 6ax – 30 – x^2 + 30 = bx

Then, taking the terms with x^2 on the left, we can combine them:
2ax2 – x^2


We know there is no x^2 on the right, hence the following must be true for the two equations in our issue to be equal:
2ax^2 – x^2 = 0x^2

We may then find a by dividing the x^2 into both sides:
2a – 1 = 0

Then, on both sides, add 1 and divide by 2:
2a = 1
a = 1/2

To obtain b, we can combine the words with x on the left as follows:
-10x + 6ax

We know there is a bx on the right, hence the following must be true for the two expressions in our issue to be equal:
-10x + 6ax = bx

By dividing the x into both sides, we get the following:
-10 + 6a = b

Then we may swap 1/2 for a, yielding
-10 + 6(1/2) = -10 + 3 = -7

Question 3: Multiple choice, no calculator allowed

question 3

Subject: Systems of Equations
x is y plus 1/6 is the answer

Response Explanation: You’ll see that just x and y are left in the answer selections. So, using the elimination procedure, we may begin by removing r and s.

First, substitute r with s + 1/2 so now all we have to do is get rid of s:
x + s + 1/2 = 4x – 9
y + s = 4y – 9

Then we may multiply the bottom equation by -1 and add the equations to remove the s.
x + s + 1/2 = 4x – 9
-y – s = -4y + 9

Adding the equations yields the following results:
x – y +1/2 = 4x – 4y

Finally, we’d want to modify the remaining equation above. The response options suggest that we wish to isolate the x variable. To begin, remove x and add 4y on both sides to move the x’s to the right and the y’s to the left:
-y + 4y + 1/2 = 4x – x

Simplifying results in the following:
3y + 1/2 = 3x

Then, to get x on its own, divide both sides by 3:
x = y + 1/6

Question 4: Multiple choice with the use of a calculator is permissible

Question 4
Subject: Polygons
Answer: 14.5
Answer Explanation:
This issue is difficult since it lacks a visual and connects two advanced topics: polygons and trigonometry. Begin by drawing a circle with a radius of 10 cm inscribed in a regular pentagon:

Diagram 1

We may construct a right triangle with the radius, half of the pentagon side, and a hypotenuse going from the pentagon corner to the center:

Diagram 2

Then we can use the formula for the sum of a polygon’s interior angles to find the measure of each angle in the pentagon. The sum of a polygon’s interior angles is (n- 2) * 180, where n is the number of sides.

Because a pentagon has 5 sides, the total of its interior angles is (5 – 2) * 180 = 540. Then each angle is 540 / 5 = 108 degrees.

Because our right triangle’s hypotenuse divides that 108-degree angle, one of the triangle’s angles is 108 / 2 = 54 degrees. This provides us with the following information about our right triangle:

Diagram 3With right triangles, we have a few alternatives for determining x, which represents half of the pentagon side. There is the Pythagorean Theorem, however, it requires two known sides to be used.

There are also unique right triangles, but we’d need a 30-60-90 or 45-45-90 triangle to use them. SohCahToa is another alternative.

We would use the tan function and write because the sides we are looking at are opposite and next to the 54-degree angle.
tan(54) = 10/x

Solving for x gives us x = 10 / tan(54). We multiply that by 2 to give us the length of the full pentagon side: 2 x 10 / tan(54) = 14.5.

Ace Exponents And Ace Expressions Combined

You should brush up on your exponents before taking the SAT. You will be asked to add, multiply, or divide exponents such that they are equal and then factor them into an appropriate equation.

Denominators With Complex Numbers Rationalized

You will also be required to rationalize denominators with complex numbers, denoted by a+bi, on the SAT Math part.
7+3i, for example, is a complicated number.

On your test, you may be asked to find the value of a given fraction and the value of i, requiring you to rationalize the denominator to answer the issue. Working with radical expressions like these may be difficult. Therefore you must know how to work with complex numbers.

Decay Exponential
When taking the hardest SAT Math questions, you will encounter “story” problems that require you to answer exponential decay issues that are frequently related to population density or radioactive chemical decay.

To solve these SAT math practice questions successfully,

you must be able to substitute abstract figures with tangibles from the problem (for example, replacing t in the equation with the appropriate value of years; using people or grams to represent the population or the remaining amount of a substance), as well as use exponential decay models.

Expressions That Are Equivalent
Another sort of hardest SAT math question on the SAT Math exam is one that requires you to determine an equivalent phrase. These questions essentially need you to look at an expression and determine which multiple-choice option is equal to the original word.

Tasks Are Split
Splitting task questions feature two variables and need you to assess how much each variable is present in a specific narrative problem or circumstance.

Only Use Coefficients To Solve Problems
You will be asked to answer equations using coefficients rather than simplifying the problem on the SAT Math portion. To answer these questions, you must understand when and how to set coefficients equal.

Tables With Sap

Some SAT Math practice questions will require Substitute Answers in the Problem (SAP) techniques utilizing a given table. For example, you may have a table with a column displaying x values and another column listing f values (x). Your task will solve the issue by substituting responses to determine which multiple-choice solution accurately defines f.

Graphs With Better Function

When taking the hardest SAT math questions, you may be presented with certain math problems that offer a graph of a specific function and ask you to determine the spots where a specified value intersects with the graphed function.

In addition, other difficulties may require you to utilize a graph to solve for a certain value. Many students are anxious about the SAT math part, but they shouldn’t be. Instead, continue to understand the fundamentals you’ll need to know and how to approach the most challenging questions you’ll face.

Hardest SAT Math Questions


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