25 Challenging SAT Problems - Review & practice

Aug 25, 2026

Uploaded by Patrick

A set of 25 highly challenging SAT math problems organized by topic and aligned with the progression of the corresponding study book. The problems span Algebra & Functions, Exponentials, Quadratics, Geometry & Trig, Statistics, plus scattered word problems, with many items centered on evaluating multiple options (I, II, III). The collection emphasizes advanced reasoning and deeper conceptual understanding of how SAT math relationships work under pressure.

LanguageEnglish

CategoryExam

FormatPDF

Pages15

Reading Time38 min

AccessFree to download

Chapter 1: Algebra & Functions

Chapter 2: Exponentials

Chapter 3: Quadratics

Word Problems

Chapter 4: Geometry & Trig

Chapter 5: Statistics

Final note: I, II, III option-evaluation tendency

25 Challenging SAT Problems [From learnsatmath.com](From learnsatmath.com) These questions are all variations of the hardest problems from the Fall 2024 and Spring 2025 SATs. The difficulty of these problems is comparable to the difficulty of the final 3 problems on the hardest module 2 . This set is also fairly comprehensive, meaning that it covers almost every challenging topic that you can expect to see. If you find these problems too challenging, checkout my book, SAT Math from the Ground Up at [learnsatmath.com. It will](learnsatmath.com. It will) teach you... you’ll never guess this one... the SAT math from the ground up. Rather than focusing on how to solve problems, it’ll teach you why the math really works, which is essential to tackle problems of this difficulty. These 25 problems follow the book’s ordering: • Chapter 1: Algebra & Functions-Q1-5 • Chapter 2: Exponentials-Q6-8 • Chapter 3: Quadratics-Q9-14 • Word Problems (scattered): Q15-16 • Chapter 4: Geometry & Trig-Q17-22 • Chapter 5: Statistics-Q23-25 Final footnote: You may notice that a disproportionate number of these problems have you evaluate 3 options (the I, II, III problems) . This is because: 1. These problems are harder/longer 2. These problems cover more concepts in less time Just know that on the actual SAT, you won’t be getting nearly as many of these. 1 Challenging Functions 􀔕(􀔧) = 􀔐 √􀔧 − 􀔑 + 12 √􀔧 − 􀔑 For the function 􀔕, 􀔐 and 􀔑 are constants, and 􀔕(4) = 0 . If 􀔕(5) > 0, which of the following could be a value for 􀔐 + 􀔑? A) -20 B) -12 C) -8 D) -4 2 Fraction Algebra 1-Constant Denominators 16 16 ~~ ~~ = ~~ ~~ 􀔐 􀔑 + 16 􀔒 Which of the following expresses 􀔐 in terms of 􀔑 and 􀔒 ? A) 􀔐 = 􀕽􀕾 B) 􀔐 = 􀕽􀕾 C) 􀔐 = 􀔑􀔒(􀔑 + 􀔒) D) 􀔐 = 16􀔑􀔒(􀔑 + 􀔒) 3 Fraction Algebra 2-Solving Coefficients 􀔐􀔧 3 + 􀔑􀔧 2 + 􀔒􀔧 + 􀔓 5 6􀔧 ~~ ~~ = ~~ ~~ + ~~ ~~ (􀔧 2 − 9)(􀔧 + 2) 􀔧 + 3 􀔧 − 3 What is the value of 􀔐 + 􀔑 + 􀔒? 4 Unintuitive Linear Models There are two lawn mowing companies. Company A charges a flat fee of $80 for the first hour, then an hourly rate of $60 for each additional hour. Company B charges no flat fee, but an hourly rate of $75 per hour. After how many minutes will company A’s total cost exceed company B’s? 5 Standard Form Solutions 56􀔧 + 􀔐􀔨 = 􀔑 1110􀔧 + 􀔒􀔨 = 2􀔓 If the system of equations above has infinite solutions, what must be the value of 􀕽􀕿? 6 Percentages 􀔐 is 540% more than 􀔑 and 􀔑 is 0 .02% of 􀔒 . If 􀔐 is 􀔚% of 􀔒 , then what must 􀔚 equal? 7 Exponential Functions 1-Fractional Exponents 􀔕(􀔣) = 􀔐(2) 21􀕙1 The given function 􀔕(􀔣) models the amount of money in an investment account 􀔣 years after the account is opened, where 􀔐 is a constant. How many months will it take for the initial investment to increase by 700%? 8 Exponential Functions 2-Constants & Coefficients 􀔕(􀔧) = 􀔚 3 (􀔚􀖓 ) + 􀔑 The function 􀔕 is defined by the equation above for 􀔧 ≥ 0 . 􀔚 and 􀔑 are positive constantsand 􀔚 < 1 . Which of the following must be true? I. The maximum value of the function is displayed as a constant or coefficient of the function. II. 􀔕(1) = 􀔕(0) ∗ 􀔚 III. 􀔕(􀔐) < 􀔑 for some 􀔐 > 0 . A) I only B) II and III only C) I and III only D) Neither I, II, nor III 9 Integer Factors 1-Maximizing Products 15􀔧 18 + 􀔚􀔧 9 + 24 The expression above has factors 􀔐􀔧 9 + 􀔑 and 􀔒􀔧 9 + 􀔓, where 􀔐 , 􀔑 , 􀔒 , and 􀔓 are all integer constants. What is the maximum value of 􀔚? 10 Integer Factors 2-Identifying Integers (ℎ􀔧 + 􀔚)(􀔠􀔧 + 􀔩) = 􀔐􀔧 2 + 􀔑􀔧 + 35 For the expression above, ℎ, 􀔚 , 􀔠 , and 􀔩 are all integer constants. Which of the following must be an integer? I. 35􀖆 II. 􀔑 III. − 􀕼􀖌 A) I and II only B) II and III only C) I and III only D) I, II, and III 11 Quadratics 1-Vertex Form 􀔕(􀔧) = −3􀔧 2 + 􀔑􀔧 + 􀔒 In the given quadratic function, 􀔑 and 􀔒 are constants. The graph of 􀔨 = 􀔕(􀔧) in the xyplane is a parabola that has a vertex at the point (ℎ, 􀔚), where ℎ and 􀔚 are

FAQs

How are the 25 SAT math problems organized in the set?

They follow the book’s ordering: Chapter 1 Algebra & Functions (Q1-5), Chapter 2 Exponentials (Q6-8), Chapter 3 Quadratics (Q9-14), scattered word problems (Q15-16), Chapter 4 Geometry & Trig (Q17-22), and Chapter 5 Statistics (Q23-25).

What makes these problems particularly difficult?

The set is based on variations of the hardest SAT problems, and many questions are harder and longer because they cover more concepts in less time, often requiring evaluation of three options (I, II, III).

How should a student respond if these problems are too challenging?

The document recommends using the book SAT Math from the Ground Up, which focuses on why the math works rather than only memorizing solution methods for this difficulty level.

Aug 25, 2026

Uploaded by Patrick