Examples of difficult ACT math problems
The math questions on this page are examples of ACT Math difficult questions. If you are interested in more ACT math problems, the following books are ideal for advanced math students that aim a perfect ACT score.
Barron’s ACT 36: Aiming for the Perfect Score
Out of all ACT test books, this one has the best reviews on Amazon.com. The practice questions are similar in format to those found on the actual test, and all ACT test topics are covered: English, mathematics, reading, science, and the optional Writing Test, which requires a brief essay.
Barron’s ACT 36: Aiming for the Perfect Score
The Real ACT Prep Guide: The Only Guide to Include 3Real ACT Tests
The Real ACT Prep Guideis the only book with insider testtaking tips and strategy, practice, and insight from the makers of the ACT.
The Real ACT Prep Guide: The Only Guide to Include 3Real ACT Tests

Question #1: What is the length of the transverse diameter of the following ellipse?
x^{2}/49 + y^{2}/19 = 1
(a) 7
(b) 14
(c) 49
(d) 2
(e) 19
Solution: For any x^{2}/a^{2} + y^{2}/b^{2} = 1 ellipse, the transverse diameter is 2a.
In our case a = 7, so the transverse diameter of the ellipse is 14.  Question #2: What is the real value of x in the equation log_{3}45 – log_{3}5 = log_{7}x?
(a) 49
(b) 16
(c) 5
(d) 7
(e) 81
Answer: log_{3}45 – log_{3}5 = log_{7}x
log_{3}(45/5) = log_{7}x
log_{3}9 = log_{7}x
log_{3}3^{2} = log_{7}x
2 = log_{7}x
x = 7^{2}
x = 49
 Question #4: Which of the following graphs could be the correct representation of 2 functions f(x) and g(x) that satisfy the equation g(x) = – f(x)?
(a)
(b)
(c)
(d)
Answer: (b) is the correct solution.
On the graph from solution (b), if we consider a random negative value for x, g(x) is negative.
f(x) will be positive and f(x) will be negative, same as g(x).
This is not the case with solutions(a),(c) or (d).  Question #8: A box contains 7 blue marbles, 4 red marbles and 6 green marbles. How many additional blue marbles must be added in the box so that the probability of randomly drawing a blue marble is 1/2?
(a) 2
(b) 3
(c) 4
(d) 5
(e) 6
Answer: If the probability of randomly drawing a blue marble is 1/2 after x blue marbles were added, the number of blue marbles must be half of the total numbers of marbles.
1/2 = (x + 7)/(x + 7 + 4 + 6)
x = 3
ACT Sample Questions
Examples of ACT geometry questions
Test your ACT math knowledge with our multiple choice tests.