# Hard Math Problem

This definitely eliminates B, and thinking logically, you could probably take a guess that the overall probability is going to be high. ANSWER: C This question is difficult primarily because you need to have some higher-level knowledge of the equations of circles and ellipses.

You should find that out of the first 11, 8 numbers are irrational.

This is a useful trick for problems like this on the ACT.

What’s important is that you visualize what an ellipse like this looks like: If we put a circle inside this shape, it can only have a diameter that is as wide as the ellipse is wide, otherwise it wouldn’t fit. Back to the ellipse equation: , or 9, in our answer choice for the radius of the circle, making our answer C. ANSWER: A Probably the most elegant way to solve this problem is to remember the Factor Theorem.

This means that all of the details in the question give you important clues that you need to solve the problem.

ANSWER: C There are two ways to solve this problem: the “math” way and the “test prep strategy” way. You see, this problem is a great candidate for plugging in numbers for . ANSWER: E The ACT rarely gives you any unnecessary information in a math word problem.

To master these problems, you’ll need to refresh your understanding of the concepts, then put them into practice. Check out the twelve ACT math challenge problems we’ve put together for you below. Point D is the midpoint of AC, which is 22 cm long. a=5 r=√5 Because every term is positive, we don’t have to consider the negative square root. 40 miles 30 miles so the Total Distance was 70 miles. The average speed in this problem is 14 mph, which is different from simply taking the mean of the two speeds. This option is in QII and QIII, so the part in QIII might go down low enough to contain the angle. Technically, this ray isn’t even included, because it’s the endpoint of an inequality, but we will just use this value.

As shown in the figure below, A is the center of the circle, and right triangle ABC intersects the circle at D and E. Here, we don’t know the common ratio, so let it be r. ANSWER: D Average Speed = Total Distance / Total Time. Notice how the test-maker has made this problem tricky!

This means that AD is the radius and should be half of 22 cm, or 11 cm.

In the case of this question, we know that AC is 22 cm, that D is the midpoint of AC, and that D serves as a point of intersection between the circle and the triangle.

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