**Hardest SAT Math Questions Listed and Solved**

The math portion of the SAT is tricky for many students, covering both the math you practice across all high school math courses (solving linear equations, for example) and the math you come across less frequently – sorry, what’s an imaginary number? This post breaks down the hardest SAT math question types and how to solve them with real examples from recent SAT tests. Keep reading to master even the trickiest math on the SAT!

**About SAT Math**

Math on the SAT is divided into two sections – math, no-calculator, and math, calculator allowed. The no-calculator math section will ask you to complete various addition, subtraction, multiplication, and division problems on your own, while the calculator allowed section will ask for more complex mathematical functions that you can complete on your calculator – see acceptable calculator types here.

Both math sections will require a strong understanding of Algebra I, Geometry, Algebra II, and Pre-Calculus (sometimes called Analysis). You’ll have access to a formula sheet with a few key geometrical formulas and these relationships:

- The number of degrees of arc in a circle is 360
- The sum of the measures in degrees of the angles of a triangle is 180
- The number of radians of arc in a circle is 2π

In addition, to earn a strong score on the test, you should understand how to use the formulas outlined in this post.

**Topics Covered On The SAT**

The chart below summarizes the various topics you’ll come across on the math portion of the SAT.

Category | Subjects |

Heart of Algebra | solving linear equations and linear inequalities; interpreting linear functions; linear equation/linear inequality word problems; graphing linear equations; linear function word problems; systems of linear equalities/systems of linear equations word problems; solving systems of linear equations |

Passport to Advanced Mathematics | solving quadratic equations; interpreting nonlinear expressions; quadratic and exponential word problems; manipulating quadratic and exponential expressions; radical and rational exponents/equations; operation with rational expressions/polynomials; polynomial factors and graphs; nonlinear equation graphs; linear and quadratic systems; structure in expressions; isolating quantities; function notation |

Problem Solving and Data Analysis | ratios, rates, and proportions; percents; units; table data; scatterplots; features of graphs; linear and exponential growth; data inferences; center, spread, and shape of distributions; data collection and conclusions |

**Why Learn About The Hardest SAT Math Questions**

When studying for the SAT, you’ll find that a lot of your review involves taking concepts you already understand and practicing those concepts until you can answer questions quickly and confidently. Yet when test day rolls around, you might stumble across a question that requires an obscure formula you can’t remember, or involves multiple steps – including a step you’ve never done before in high school math classes.

A great study tip is to learn about the hardest problems on the SAT math sections, even if you haven’t mastered every “simple” or “intermediate” concept yet. That way, when you come across the 5-10 hardest problems on your specific SAT test, you’ve at least seen the problem type solved before!

**Hardest SAT Math Questions To Be Prepared For**

Below, find 15 challenging math questions from real SAT tests, with answers given and the problem solving process explained.

**Question Type 15: The Quadratic Formula (No Calculator)**

- Answer: 11

Explanation: This question is tricky because you’ll need to use the quadratic formula to find two solutions for x. The quadratic formula looks like this:

For any given quadratic equation, you find your a, b, and c values like this: ax^{2} + bx + c. Here, your a = 1, your b = -10, and your c = 14.

When you plug in your values for a, b, and c and then simplify, you’ll find two solutions: x = 5 + √11 and x = 5 – √11. Because the problem asks for the x = 5 + √*n*, your answer will be 11. (Although 11 is the answer in either solution!)

**Question Type 14: Problems With ***No Solution*** or ***Infinite Solutions*** (No Calculator)**

*No Solution*

*Infinite Solutions*

- Answer: D

Explanation: There are different types of solutions when solving equations: *one *solution, as with a linear equation, *two or more* solutions, as with quadratics or polynomials, *infinite *solutions, when x = x, or *no *solution, when an equation is untrue – for example, 1 = 2. (It’s not!)

To solve this problem, you’ll need to evaluate the answer choices to determine when the system of equations would render no solution. Because they tell us that our first equation is 2x + 2y = 2, we can see that answer choice D, 2x + 2y = 3, would give us that “no solution” option. Because the variables on the left side of either equation are the same (2x + 2y), that means the numbers of the right side of the equation have to be equal – yet 2 is *not *equal to 3.

**Question Type 13: Exponential Growth Equations (No Calculator)**

- Answer: D

Explanation: Representing exponential growth looks like X_{T} = X_{0}(1+R)^{t }, where X_{0 }is your initial value, R is your rate of change, and t is your time in intervals.

In an exponential growth equation, the initial value (here, 10 phlox plants) should be to the right of the = sign. This narrows the solution set down to answer C and answer D.

Next, we’ll represent the amount of growth occurring each year. A growth rate of 140% means that the initial value is changing from 100% to 240% – essentially, that you’re adding 140% for each increase in *t*.

Option C shows growth of **1.14**, which is only an increase of 14% per year.

Option D, the correct answer, represents that 240% growth with **2.4**, a simplified version of the equation X_{T} = 10(1+1.4)^{t}.

**Question Type 12: Congruence and Similarity (No Calculator)**

- Answer: B

Explanation: Triangles are congruent if they have identical angle measures and identical side lengths. For the two triangles in the problem, we already know that all corresponding angle measures are the same – if two angles are 35 degrees, the third angle *must *be 110 degrees, because angle measures always add to 180 degrees for triangles. This rules out answer choice A, because it’s a given.

When deciding between answer choice B and answer choice C, you simply have to decide which sides are corresponding. The answer to the question is B, because lengths BC and EF are corresponding sides. (Remember: when two angles on a triangle are the same, you have an *isosceles *triangle – two side lengths are the same as well.)

**Question Type 11: Volume Problems (No Calculator)**

- Answer: 3

Explanation: Per the formula sheet given on the SAT, you know that the volume of a rectangular prism is V = lwh, or volume = length x width x height. The problem tells us that the prisms are similar, meaning the relationship between each side is consistent. Because we know the length of prism A is 6, we know the width and height must multiply to 16. (Such that the total volume is 96). To solve the problem, we take the square root of 16 (4) and make this number the assumed width and height of prism A.

Next, we can assign values to the length, width, and height of prism B, looking for a volume of 12 (per the chart) and a consistent relationship to the length, width, and height of prism A.

When we try a length of 3 (½ of 6) and a length/height of 2 (½ of 4), we find a volume of 12: 3 x 2 x 2 = 12!

**Question Type 10: Representing Equations (Calculator Allowed)**

- Answer: C

Explanation: Although this question looks confusing at first, it ultimately wants you to understand that when dealing with a %, you’re thinking in terms of parts of 100. The answer is C because it represents x as = 100%, given that 3/p x 100 = 100% of x.

**Question Type 9: Complex Exponents (Calculator Allowed)**

- Answer: B

Explanation: To solve this problem, you begin by distributing the -2(x + 1) to look like (-2x – 2). Next, you need to know that 3^{-2} is 1/9. (Brush up on your exponent rules if you aren’t familiar with how to handle negative exponents!)

After you pull out the 1/9, you’re left with 1/9(3^{-2x}). To simplify further, you simply pull out the negative, leaving you with 1/9(1/3)^{2x}.

**Question Type 8: Understanding Statistics (Calculator Allowed)**

- Answer: B

Explanation: To answer this problem, you need to know that a larger sample size yields a *smaller *margin of error, because a larger sample size indicates the conclusion is more likely to be accurate.

Answer choice B indicates that Sample B included *more *visitors than Sample A, meaning the margin of error would be smaller.

**Question Type 7: Finding Intercepts (Calculator Allowed) **

- Answer: ⅘ or 0.8

Explanation: To find an x intercept, you substitute y for 0. The problem is rewritten 5/4x = 1; when you divide both sides by 5/4 to isolate x, your answer becomes x = ⅘, or x = 0.8.

**Question Type 6: Absolute Value Problems (Calculator Allowed)**

- Answer: A

Explanation: To solve this problem, you’ll need to rewrite the equation twice per absolute value problem rules: x – 1 = 8 and x – 1 = -8. This gets you two answers: x = 9 and x = -7. However, this problem has two steps: you’ll then test your two answers per this second equation, x – 1. When you test the x = 9 answer, you get 8 as a solution, but 8 is *not *one of the answer choices. When you test -7, you get -8 as your solution, which is answer choice A!

**Question Type 5: Circle Equation (Calculator Allowed)**

- Answer: D

Explanation: For this problem, you need to know the equation for a circle: (x – h)^{2} + (y – k)^{2} = r^{2}, where the center of the circle is at (h, k). Knowing this, you can eliminate answer choice A, which would give you a radius of √10, not 10.

Next, you’ll need to decide which circle passes through the origin with a radius of 10: answer B with a center at (5, -5), answer C with a center at (10, 10), or answer D with a center at (5√2, -5√2). Use the distance formula to figure out which answer choice has a distance of 10 from the origin, (0, 0), and find that the correct answer is D!

Distance formula:

**Question Type 4: Equations With Multiple Steps (Calculator Allowed)**

- Answer: B

Explanation: To solve this problem, begin by adding 5 to both sides of the equation, so you have 3(m + 2) = m + 7. Next, distribute the 3 on the left side, to give you 3m + 6 = m + 7. Subtract 6 from both sides and subtract 1m from both sides; now you have 2m = 1. Divide by two, and you find that m = 0.5.

You’re not done yet! Don’t forget that the equation is asking for m + 2, which gives you 2.5 or answer choice B.

**Question Type 3: Modeling Change (Calculator Allowed)**

- Answer: C

Explanation: Since one complete orbit occurs at 310 years, understanding how much of an orbit has occurred (or how many total) involves modeling the number of years *out of *310 – or answer choice C. After 1 year, 1/310 of an orbit will have occurred; after 620 years, 2 orbits, or 620/310.

**Question Type 2: Converting Between Exponents and Radicals (Calculator Allowed) **

- Answer: A

Explanation: This problem requires two steps: first, simplifying the exponents using exponent rules. Start with the (y^{3/4})^{3/2} and multiply across to get y^{9/8}. Next, remember that y^{1/8}(y^{9/8}) is basically y^{1/8} x y^{9/8}, meaning you will *add *these exponents to get y^{10/8}, or y^{5/4}. (Memorizing exponent rules will make this a lot easier!) Lastly, you’ll need to remember this rule:

Now you see why the answer is A!

**Question Type 1: Systems of Equations With 3 Variables (Calculator Allowed)**

- Answer: 22

Explanation: You’ll tackle this problem in two parts. First, decide when any quadratic equation with a 10x would give you only *one *real solution – put another way, when you’ll be dealing with (x + 5) and (x + 5), so that -5 is your only solution for x. (You’ll find that the equation should look like y = x^{2} + 10x + 25.)

Next, rewrite the equations to look like -3 = x^{2} + 10x + a, and then simply to 0 = x^{2} + 10x + a + 3. For what value of a will your equation be x^{2} + 10x + 25?

This is how you find a = 22!

**What Do The Hardest SAT Math Problems Have In Common?**

On the SAT, you’ll notice trends in the questions that you find extra challenging and take longer to solve.

**Cover Several Mathematical Concepts At Once**

The hardest questions often take multiple topics and apply them to the same problem. For example, you may be working with a systems of equations word problem that *also *requires you know that a sales increase of 1.83 is an 83% increase in sales.

**Require Multiple Steps**

These problems are tricky because you aren’t “done” after the first step – for example, you may be looking to find the circle equation, but *first *you have to find the radius using the distance formula. These questions are time consuming, and you may benefit from making a note by these problems and coming back to them at the end if you have extra time.

**Test The Concepts You Are Less Familiar With**

In school, you don’t spend a lot of time on every math concept – some problem types, like using imaginary numbers or finding the arc length of a circle, only come up in a single unit. On the SAT, you’ll find a few questions that require you remember the topics you only learned once, making studying for the SAT essential!

**Are Explained In An Unusual Way**

These problems may not always require complex math, but they do present information in a way that makes it hard to know where to begin. For these questions, underline key information and try to figure out what the question is asking you to solve – are you looking for x? Are you simplifying a fraction? When you know what the problem is asking you for, you’re much more likely to answer the question correctly!

**Use Many Different Variables**

When problems go beyond solving for x (or even solving for x and y, as in a system of equations problems), things can get complicated quickly. Practicing a few problems where you’re working with three variables – x, y, and z, for example – will be a big help if you’re faced with this type of problem on test day.

**Conclusion**

Prepping for the very hardest math questions on the SAT may seem counterintuitive – after all, if you aren’t going to get every single ‘easy’ question right, why jump into the hardest ones? Yet preparing for the hardest questions is a great study method, helping you identify the gaps in your knowledge. Maybe working through the questions in this post made you realize you *don’t *have the quadratic formula memorized, or that you need to take another look at exponents rules. Likewise, learning the most challenging problems helps you tackle simpler problems with ease – if you can complete steps A, B, and C for a really hard problem, you can definitely complete step A by itself for an easier one!

As you prepare for the SAT, don’t skip the hardest material because it seems daunting; instead, tackle it with the mindset that every hard problem solved equips you to solve two or three simple problems with no stress.

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## About the Author

Renée Flory studies professional & creative writing and English literature at Johns Hopkins University. She scored a 1570 on the SAT, and in her free time enjoys writing novels.