# 10 Hardest PSAT Math Questions (2023)

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## What’s Covered:

• Overview of PSAT
• How Will My PSAT Score Affect My College Chances?
• 10 Hardest PSAT Math Questions
• Final Tips

The PSAT is much more than a practice exam. While your performance on the PSAT will certainly help you prepare for the SAT by indicating which areas you need to improve upon, it can also make you eligible for national scholarships!

The Math Section of the PSAT tests student’s on various mathematical concepts, both with and without a calculator. Keep reading as we breakdown and explain some of the most difficult PSAT Math practice questions to prepare you for test day!

## Overview of PSAT

The Preliminary SAT/National Merit Scholarship Qualifying Test (PSAT/NMSQT), normally taken by high school juniors and sophomores, measures what you’ve learned in school and opens up doors for scholarship opportunities as well as the National Merit Scholarship Program.

The exam is typically three hours long and is divided into two sections: Evidence-Based Reading & Writing (which is further divided into the Reading Test and Writing Test), and the Math Section (which is divided into the Calculator and No Calculator portions). Unlike the SAT, the PSAT has an overall score of 1520, with the scores for each section ranging from 160-760.

As mentioned earlier, there are two separate portions of the Math Test: Calculator and No Calculator. The Math Test deeply focuses on three different areas of math, which include mastery of linear equations and systems, problem solving and data analysis, and the manipulation of complex equations. In addition to these main concepts, geometry and trigonometry concepts that are relevant to college are also tested.

83% of the Math Test is multiple choice questions, while 17% of the Math Test is grid-in questions, in which students are expected to enter their answers in the grids provided on the answer sheet.

## How Will My PSAT Score Affect My College Chances?

The PSAT is an optional exam that does not have an impact on your chances of getting into college. However, since the PSAT is very similar to the actual SAT in terms of scoring, format and content, being prepared for the PSAT can make you equally prepared for the SAT. Furthermore, your PSAT score can give you a good understanding of how to prepare for the SAT.

(Video) Solving the 46 Hardest SAT Math Problems from the Official Practice Tests

The PSAT is also used as a qualifying exam for the National Merit Scholarship, which awards different levels of recognition based on the PSAT score you receive. These titles include Commended Students and Semifinalists.

If you are selected as a Semifinalist, you can apply to compete in the National Merit Scholarship Program. If your application is outstanding, you may be chosen to be a Finalist. As a finalist, you earn the title of National Merit Scholarship recipient and can be awarded National Merit Scholarships to help fund your college tuition.

For more information on how your PSAT score can impact your college chances, feel free to use our free chancing engine. This tool will calculate your odds of acceptance at hundreds of schools across the country based on your tests scores, GPA, extracurriculars, and more.

## 10 Hardest PSAT Math Questions

The Math test, as mentioned below, tests on a multitude of topics taught over the course of high school. The PSAT is meant to test how well you are able to apply the math skills and knowledge you’ve learned.

While the difficulty of questions asked on the PSAT Math section range from easy to hard, in this post, we show you how to break down and solve 10 of the hardest PSAT math questions.

### Question 1: Calculator, Multiple Choice This is a math question that requires you to understand how percentages work. In this question, both the production of tomatoes and raspberries decrease by a percentage. In order to find the percentage by which the total yield of Boyd’s garden declined, we need to find out how many tomatoes and raspberries were lost in production.

Since the total amount of tomatoes declined by 20%, we know that the tomatoes declined in production by 0.20 × 140 = 28 pounds. Meanwhile, the raspberries declined in production by 0.5 × 60 = 30 pounds. The percent decline in the total yield can be found by dividing the decline in number of pounds of tomatoes and raspberries by the previous year’s production:

(28 + 30) / (140 + 60) = 29%

it is important to realize that the question is asking for the decline of the total yield. If the question were asking for the percentage of the garden that was retained this year, the calculation would be different. In this case, we would calculate the total tomatoes and raspberries produced this year (instead of the production that was lost) and divide that by the total production of last year (which would be 71%).

### Question 2: Calculator, Multiple Choice This question is quite tricky because, at first glance, it seems as if there is no minimum to the graph, only a maximum. So the initial reaction might be to assume that the minimum value of the function is negative infinity.

However, when reading carefully, the question asks for the minimum of the graph in the restricted domain of x between -4 and -6, inclusive. Therefore, the minimum value of the function occurs at the point (-4, -2), thereby making -2 the minimum value of the function.

### Question 3: Calculator, Multiple Choice This question requires us to both analyze data and utilize percentages as well as ratios to come to our solution – so there are quite a few steps in this problem!

First, we realize that there are a total of 61 + 48 = 109 students enrolled in the Propel program. Now, we know that the ratio of boys to girls is 2:3. This may seem like we can immediately multiply 109 by ⅔ to determine how many girls are in the program. However, we need to realize that a ratio of 2:3 implies that there are two boys and three girls for every group of 5 students. Therefore, we multiply 109 by ⅗ to determine how many girls are in the program, which is roughly 65 girls.

### Question 4: Calculator, Multiple Choice This problem combines both geometry and systems of equations together in one problem. Let us assume that S is the length of each side of the square sculpture while T is the length of each side of the equilateral triangle sculpture (both in inches). Since both sculptures are made of a rod of the same length we know that the perimeter of each shape is the same. The perimeter of the square sculpture is 4S while the perimeter of the triangle sculpture is 3T:

4S = 3T

We also know that each side of the triangle is 2 inches longer than each side of the square. This means that S + 2 = T. If we substitute the following equation in for the one we found above, we get:

4S = 3(S + 2)

4S = 3S + 6

S = 6

If the length of the side of the square sculpture is equal to 6 inches, that means the rod is 4 × 6 = 24 inches in length.

### Question 5: Calculator, Grid In One concept in math you should become very familiar with is graphs and their respective equations. The equation for a parabola is y = a(x-b)² + c where b is the x coordinate of the vertex, c is the y coordinate of the vertex, and a is a coefficient that can be found from the equation.

Since we know that the vertex of the graph is (4, 19), we can set up the parabola equation to be y = a(x-4)² + 19. Since we know that the parabola also passes through the point (0, 3), this means that a is equal to:

3 = a(0 – 4)² + 19

(Video) SAT® Math: Hard Questions EASY Hacks

-16 = a(-4)²

-16 = a(16)

a = -1

Now that we know the equation of the parabola is y = -(x-4)² + 19, we can determine the equation of the linear function knowing that the function passes through the points (0, -9) and (2, -1). The slope of the function, which we know is equal to change in y-coordinates over change in x-coordinates of the line, is [(-1) – (-9)] / [2 – 0] = 4. Knowing that the y-intercept of the linear function is -9, we can write the equation of the linear function to be y = 4x – 9.

With the two equations, we can now find the point at which the two graphs intersect through a set of equations. We can set the two equations equal to each other since both equations are solved for y:

y = 4x – 9 (Linear equation)

y = -(x-4)² + 19 (Parabola equation)

4x – 9 = -(x-4)² + 19

– 4x – 12 = 0

(x – 6) (x + 2) = 0

x = 6 and -2

Since (v,w) is on the positive side of the y-axis, v has to be equal to 6.

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(Video) Practice PSAT 25min Math section explained

### Question 6: Calculator, Grid In This is another system of equations question, however, this question is a bit more difficult since it is a word problem, so make sure to read the question carefully!

Let’s say that x = number of sections studied by a group of two students, while y = number of sections studied by a group of four students. We know that there are 78 total students, which means that 2x + 4y = 78. We also know that there are 24 total sections, which means that x + y = 24. Now you can use either the substitution method or the elimination method to determine what x and y are equal to (review both of these methods since they will be extremely useful when taking the math portion of the PSAT)!

Using the substitution method, we see that:

2x + 4y = 78

x + y = 24 → y = 24 – x

2x + 4(24 – x) = 78

2x + 96 – 4x = 78

-2x + 96 = 78

2x = 18

x = 9

Therefore, the total number of sections with two students is 9.

### Question 7: No Calculator, Multiple Choice This question is tricky since it requires you to remember how functions are written. We know that we are expected to solve the function for f(-3). Without initially reading the prompt, one may assume that x = -3 and incorrectly solve the problem.

We see in the problem statement that the equation is written in terms of f(x – 1). Therefore, we need to determine the actual value of x:

f(x – 1) = f(-3) = f(-2 – 1)

Therefore, x = -2.

Solving the equation for x = -2, we get:

2(-2) + 3 = -1

### Question 8: No Calculator, Multiple Choice Another important skill to remember when studying for the math section of the PSAT is exponential rules, so make sure to review these! For this particular question, we need to remember the power rule as well as the product rule (as shown below):

(x²y³)½ (x²y³) = xa/3ya/2

(x2 × ½ y3 × ½ )(x2 × y3 ×) = xa/3ya/2 (Power Rule)

(xy3/2) (x2/3y) = xa/3ya/2

(xy3/2) (x2/3y) = xa/3ya/2

(x1 + 2/3y1 + 3/2) = xa/3ya/2 (Product Rule)

(x5/3y5/2) = xa/3ya/2 (Simplification)

From the simplified equation, we can see that a = 5

(Video) Why did everyone miss this SAT Math question?

### Question 9: No Calculator, Multiple Choice One additional topic tested on the PSAT is geometry and trigonometry. This means you should review equations related to shapes, geometry rules and basic trig! In this example, we see that we have to remember quite a few triangle rules to answer this question.

In isosceles triangle ABC, we know that ∠B and ∠C are congruent (the same angle). Furthermore, ∠BED and ∠CFD are right angles which also makes them congruent. Since △BED and △CFD have angles that are corresponding, that means the two triangles are similar. A special rule of similar triangles is that all corresponding sides are proportional.

Therefore, we know that BD/DC = DE/DF. Given that DE/DF = 5/7, we know that BD/DC = 5/7. With this given ratio, let’s write the following equation as a result of the proportionality:

5x = BD and 7x = DC.

Other equations we know to be true from the problem statement are AB = AC and BC = 48. Knowing that BC = BD + DC, we know the following:

BC = BD + DC = 48

5x + 7x = 48

12x = 48

x = 4

Therefore, the length of DC = 7x = 7(4) = 28

### Question 10: No Calculator, Grid In This question requires us to manipulate expressions through algebra. This question might seem hard at first sight since we are given a variable instead of a value to factor out – but don’t worry, we’ll review how to approach this problem!

We know that x – 2 is a factor of the expression x²– bx + b. This means that x² – bx + b can also be written as (x – 2)(x – a) in which we need to solve for the value of a (notice that a is negative because the y-intercept of b is positive and two negative values make a positive).

Expanding (x – 2)(x – a) = x² – 2x – ax +2a. Let’s now equate the two expressions we have and compare like terms to find the missing values:

x² – 2x – ax +2a = x² – bx + b

Compare like terms:

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

2a = b

Since we know that 2a = b, we can substitute the value for b into our first equation to get:

-2 – a = -(2a)

-2 – a = -2a

-2 = -a

a = 2

Substituting a = 2 into 2a = b shows us that b = 4

## Final Tips

Now that we’ve covered 10 of the hardest PSAT math questions, here are some final tips to remember when preparing for and taking the PSAT!

### Start preparing early

As a high school sophomore or junior, preparing for the PSAT may seem like an additional stress you have to juggle in addition to AP/honors classes and extracurriculars. So remember, start early in terms of preparation and studying so that you will be more ready to take the PSAT in the fall! This way, you will be ready to take on the PSAT during the fall while still balancing the responsibilities school throws at you.

### Practice, Practice, Practice!

The best way to prepare for the test is to practice, practice, practice! There are multiple ways to prepare for the PSAT. CollegeBoard has released a number of PDFs of previously administered PSATs online for free (as a note, the PSAT and PSAT 10 are essentially the same, however the PSAT 10 is not used to determine National Merit Scholarships).

In addition, it is a good idea to review concepts of systems of equations, trigonometry, and algebra to prepare for the exam. Review concepts taught earlier in high school or even reach out to a math teacher to see if they have notes or prep material they can give to you. CollegeVine also offers numerous free, online PSAT resources that you might find helpful.

Make sure that you read every question carefully. Oftentimes, CollegeBoard writes questions that are meant to stump you if you read the question too quickly. You might even accidentally find an answer to something they are not asking if you do so. So, take your time when reading the questions and make sure to double check your work with any time you have remaining!

• How to Study for the PSAT: 6 Tips
• PSAT vs. SAT: Is the PSAT Easier?
• When is the PSAT? 2020 Dates and Deadlines

## FAQs

### What math questions are on the PSAT? ›

The PSAT Math questions focus on four main areas: heart of algebra; problem solving and data analysis; passport to advanced math, and additional topics in math, including limited geometry, trigonometry, and pre-calculus.

Is 700 a good PSAT score? ›

In contrast, an excellent score is one that's higher than the 90th percentile, or 90% of test takers. Based on that reasoning, a good PSAT score for a sophomore is a composite score higher than 1060, an OK score is one higher than 920, and an excellent score is anything higher than 1180.

What is the highest PSAT score for math? ›

An official PSAT/NMSQT score report contains seven parts.
• Total Score (on a scale of 320-1520)
• Evidence-Based Reading and Writing Score (on a scale of 160-760)
• Math Score (on a scale of 160-760)
• Subscores (on a scale of 1-15)
• Cross-test Scores (on a scale of 8-38)
• National Merit Scholarship Corporation Selection Index.

What are the 7 hardest math problems? ›

Clay “to increase and disseminate mathematical knowledge.” The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.

What is the hardest math question ever? ›

53 + 47 = 100 : simples? But those itching for their Good Will Hunting moment, the Guinness Book of Records puts Goldbach's Conjecture as the current longest-standing maths problem, which has been around for 257 years. It states that every even number is the sum of two prime numbers: for example, 53 + 47 = 100.

Is the math PSAT hard? ›

How Difficult is the PSAT? The PSAT focuses on math, reading, writing and language concepts that high school sophomores and juniors typically have already studied in their academic courses, so the test should not be difficult for well-prepared high school students.

Is PSAT harder than SAT? ›

The PSAT test is slightly easier than the SAT test. The PSAT test is meant to serve as practice for the SAT test. When you break down how much time you have per question for each section of the tests, you'll notice that you have a little more time to answer math questions on the PSAT test than you do on the SAT test.

What grade level math is on the PSAT? ›

The PSAT math sections cover up to high school geometry. No PSAT math section will include any math questions from Algebra II; however, Algebra II is covered in the SAT.

What PSAT score is required for Harvard? ›

But those averages can change depending on the college you're hoping to apply to. For example, the average PSAT score by Harvard admitted students was between 1420 and 1520 (or 210-238 on the old scale). That's a high goal to aim for, to be sure.

What is a good PSAT score for Ivy League? ›

In other words, if your main priority is getting into the Ivy League, you should aim for a composite PSAT score anywhere between 1280 and 1520. If you're looking at less selective institutions, then the 80th percentile is a good benchmark.

### Is 870 a good PSAT score for an 8th grader? ›

On the PSAT 8/9, 835 was the mean score for eighth grade test-takers, compared with 892 for their ninth grade peers, according to 2020-2021 data from the College Board. The mean score on the PSAT 10 and PSAT/NMSQT was 959 for sophomores and 1044 for juniors.

What's the lowest PSAT score? ›

Students receive a scaled score ranging from 160 to 760 for each of the sections, with 160 being the lowest possible score and 760 being the highest possible score. The two scaled scores are added together to get the student's total PSAT score, with a total score range from 320 to 1520.

Is 870 a good PSAT score for a freshman? ›

Anything above the 75th percentile is ranked as good. 1200 is a good PSAT score as a freshman. A composite score above 1170 is considered excellent.

Does a 1400 PSAT qualify for National Merit? ›

Practically speaking, this means that anyone who qualifies as a National Merit Scholarship semifinalist (PSAT scores of 1400/1520 and up, depending on the state of residence) should be able to easily achieve a confirming score (SAT scores as low as 1320/1600, and ACT scores as low as 26/36), especially given multiple ...

What is the 1 million dollar math problem? ›

The Riemann hypothesis – an unsolved problem in pure mathematics, the solution of which would have major implications in number theory and encryption – is one of the seven \$1 million Millennium Prize Problems. First proposed by Bernhard Riemann in 1859, the hypothesis relates to the distribution of prime numbers.

Has 3X 1 been solved? ›

After that, the 3X + 1 problem has appeared in various forms. It is one of the most infamous unsolved puzzles in the word. Prizes have been offered for its solution for more than forty years, but no one has completely and successfully solved it .

What are the 5 impossible math problems? ›

The problems consist of the Riemann hypothesis, Poincaré conjecture, Hodge conjecture, Swinnerton-Dyer Conjecture, solution of the Navier-Stokes equations, formulation of Yang-Mills theory, and determination of whether NP-problems are actually P-problems.

What math equation is impossible? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

What is the easiest math problem? ›

The Collatz Conjecture is the simplest math problem no one can solve — it is easy enough for almost anyone to understand but notoriously difficult to solve. So what is the Collatz Conjecture and what makes it so difficult? Veritasium investigates.

What is x3 y3 z3 K? ›

The equation x3+y3+z3=k is known as the sum of cubes problem.

### Is 970 a good PSAT score? ›

DECENT PSAT SCORE:

A decent score meets the benchmark for the College Board: 75% chance you won't flunk any classes freshman year of college. That lands typically around the 50th percentile on these tests. For the PSAT that's a 970 for juniors (510 Math / 460 Verbal) and a 910 for sophomores (480 Math / 430 Verbal).

How rare is a perfect PSAT score? ›

' " Acing the SAT or the PSAT, which is a preliminary version of the SAT, is like hitting the jackpot. Last year about one million students took the SAT, according to the College Board, the organization that administers the test. Only 545 got perfect scores, about one in 2,000.

Is 1050 a good PSAT score? ›

A good score places you around the 75th percentile, which is a composite score around 1050. An excellent score will place you in the 90th percentile, which equates with a composite score of around 1180.

Is 940 a good PSAT score for a junior? ›

According to this chart, a good PSAT score for a junior is a composite score higher than 1150, an OK score is one higher than 1000 or 1010, and an excellent score is anything higher than 1280.

Is 1490 a good PSAT score? ›

AN EXCELLENT SCORE:

An excellent PSAT score represents the top 1% of test takers, commensurate with at least a 1500-1550 SAT score. On the PSAT this would be between a 1460-1520 as a junior or 1370-1520 as a sophomore.

What is a 32 on SAT? ›

So you've taken both the ACT and SAT, and you want to know which score was better.
...
ACT to SAT Score Conversion Chart.
ACT ScoreSAT Score
331460
321430
311400
301370
24 more rows
Dec 2, 2020

Is 1420 a good PSAT score? ›

The average PSAT score is around 920 (460 in Math and 460 in Evidence-Based Reading and Writing), while an outstanding PSAT score (one that will qualify you as a National Merit Scholarship semi-finalist) is between 1420 and 1480.

Is 1350 a good PSAT score? ›

A VERY GOOD PSAT SCORE:

This is about the 95th percentile, or a 1350 on the PSAT for juniors or 1250 for sophomores.

What's the lowest GPA Harvard accepted? ›

To get to Harvard your GPA has to be at least a 4.0 and even then if you get in your lucky but they require at least a 4.18 GPA only .

Is a GPA of 4.7 good? ›

There are private schools that use a 4.5 scale or weight Honors as 1.0 and AP as 2.0. In cases like this, a grade point average could surpass a 5.0! Most of us know that a 4.78 gpa is very good.

### Do Ivy League schools look at PSAT? ›

Estimated PSAT Score Ranges

1420+ - Students who wish to attend highly selective schools, including schools like MIT, Stanford, and those in the Ivy League would want to score in the 99th percentile of the PSAT, which starts in the low 1400s.

What is harder PSAT or SAT? ›

Differences Between the PSAT and SAT

The SAT is scored on a total scale of 400-1600, with math and verbal scores of 200-800 each. Because the test is meant to cover an additional school year of material, the SAT is a bit harder than the PSAT, particularly in the math section.

Is the PSAT test difficult? ›

How Difficult is the PSAT? The PSAT focuses on math, reading, writing and language concepts that high school sophomores and juniors typically have already studied in their academic courses, so the test should not be difficult for well-prepared high school students.

How hard is the PSAT math? ›

How Hard Is the PSAT? The PSAT is a tad easier than its big brother, but the difference is pretty minimal. It's all toned down slightly, though. Questions that would be on the easy end of SAT math show up more frequently on the PSAT.

Is the PSAT 10 hard? ›

What Does the PSAT 10 Test? The content and format of the PSAT 10 is identical to that of the PSAT/NMSQT and very similar to that of the SAT. The only major difference is that the PSAT 10 is shorter than the SAT is, and its questions do not get as difficult since they're designed for students at the sophomore level.

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