Math Review for ACT College Entrance Exam
Part I – The Review
by Elaine Ernst Schneider
1. If x + 6 = 9, then 3x + 1 = ?
Answer choices: (A) 3 (B) 9 (C) 10 (D) 34 (E) 46
Let’s think it through. Choice A is to tempt you into a quick (wrong) decision. When you solve for x in the first equation (x + 6 = 9) you get 3, which is choice A.
But that is not the question. You must go on to the second part of the question (3x + 1 = ?) Always be careful on two-part questions.
Substitute 3 for the x in the 2nd equation. 3x + 1 = ?. Therefore,
3(3) + 1 = 10.
Choice C is the correct answer.
2. If x > 1, which of the following decreases as x decreases?
I. x + x²
II. 2x² – x
x + 1
Answer choices: (F) I only (G) II only (H) III only (J) I & II only (K) II & III
The easiest way to solve this problem is to work from the answers, eliminating the wrong answers. Substitute a simple number you can work with quickly.
I. x + x² (substitute 2 for x)
2 + 2² = 6
x + x² (substitute 3 for x)
3 + 3² = 12
We know that I does what the question asks. We can see that x decreases as the equation decreases. However, the choices give us the possibility of I & II, so we must check out II as well.
II. 2 x² – x (substitute 2 for x)
2 (2² ) – 2 = 6
2 x² – x (substitute 3 for x)
3 (3² ) – 3 = 24
Again, this does what the questions asks. We can see that x decreases when x is decreased.
The correct answer is J: I & II only
3. Barney can mow the lawn in 5 hours, and Fred can mow the lawn in 4 hours. How long will it take them to mow the lawn together?
(A) 1 hour (B) 2 2/9 hour (C) 4 hours (D) 4 ½ hours (E) 5 hours
We must set up a formula that determines both Barney and Fred’s work for one hour. From this, we can then determine their work potential together. The formula for this problem would be set up as follows:
1/5 (Barney working 1 hour) + ¼ (Fred working 1 hour) = 1/x (Both working 1 hour)
Find a common denominator:
4/20 + 5/20 = 1/x
9/20 = 1/x
9x = 20
x = 2 2/9
Even if you weren’t able to figure out the appropriate formula, common sense would help you eliminate some of the answers. For example, if Fred can mow the lawn in 4 hours by himself, he would need less than 4 hours if someone helped him. That makes choices C, D, and E ridiculous. If Barney worked as quickly as Fred (4 hours) then together, it would take half as long, or 2 hours. But we know
that Barney is a little slower than Fred because it takes him 5 hours by himself. Logic tells us that together, it would take them a little more than 2 hours.
Therefore, 2 2/9 is a reasonable educated guess.
4. If a mixture is 3/7 alcohol by volume and 4/7 water by volume, what is the ratio of the volume of alcohol to the volume of water?
Answer choices: (F) 3/7 (G) 4/7 (H) ¾ (J) 4/3 (K) 7/4
A ratio is read aloud as “is to” something “as” something “is to” something else. When you see “as” or “is to,” it tells you to draw the line for the fraction. For example, 3 “is to” 7 means 3/7. So, let’s set up our equation, substituting math symbols for the words “as” and “is to.”
3 is to 7 = 3/7
4 is to 7 = 4/7
To solve this problem, divide 3/7 by 4/7.
To divide fractions, you must “flip” the second fraction and multiply.
Therefore, the problem would be set up as: (3/7) (7/4) = 21/28 = ¾
The answer is (H) ¾.
5. What is the maximum number of pieces of birthday cake size 4” X 4” that can be cut from a cake 20” X 20”?
Answer choices: (A) 5 (B) 10 (C) 16 (D) 20 (E) 25
Sketch this diagram in the margin of your test booklet. (Write in your test booklet, but NOT on your answer sheet.) Five pieces of cake (each one 4”) will fit down each side. Therefore 5 X 5 = 25. The answer is 25, E.
6. In the triangle given, AD is an angle bisector. <DAC is 30 degrees and angle ABC is a right angle. Find the measure of angle x.
(F) 30 degrees (G) 45 degrees (H) 60 degrees (J) 90 degrees (K) 120 degrees
The definition of a bisector is that it divides an angle evenly in half. Therefore, if <DAC is 30 degrees, then <BAD must be 30 degrees. <ABC is a right triangle. Right triangles measure 90 degrees. There are 180 degrees in a triangle.
If <BAD is 30, <ABC is 90, then to find <x, we must subtract 30 and 90 from 180.
When you take 180 and subtract 30, you get 150 degrees. Subtract 90 from 150, and the answer is 60. <x is 60 degrees, or answer H.
7. Estimate the value for:
|(.889 X 55)|
Estimate your answer to the nearest tenth.
Answer choices: (A) .5 (B) 4.63 (C) 4.9 (D) 7.7 (E) 49.1
You don’t want to take the time to work this out. Besides, the problems clearly indicates that the desired answer is an estimation.
Round up and approximate:
| .889 (55)|
8. Find the counting number that is less than 15 and when divided by 3, has a remainder of 1, and when divided by 4, has a remainder of 2.
Answer choices: (F) 5 (G) 8 (H) 10 (J) 12 (K) 13
Once again, the most expeditious way to work this problem is from the answers. Plug the different numbers in and see what works and what doesn’t.
Answer F is 5. Let’s try it. When you divide 5 by 3, you get 1 with the remainder of 2, not 1; so F isn’t the answer. Answer G is 8. When you divide 8 by 3, you get 2 with a remainder of 2; so G isn’t the answer either. Answer H is 10. 10 divided by three is 3 with 1 as the remainder. This is the answer.