ACT Math: Five Solutions (2/20/15)

Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (2/20/15). Below you’ll find the fully worked-out solutions to those problems.

9.  E
Simply expand the expression and multiply the two binomials (using the FOIL pattern). Finally, combine like terms.

(5x – 4)²
(5x – 4)(5x – 4)
25x² – 20x – 20x + 16
25x² – 40x + 16

20.  J
You can rewrite each of the bases (the numbers being raised to the powers) so that they are all powers of 3. From there you can apply the rules of exponents to find your answer.

(3^{x})(9)=27^{2}
(3^{x})(3^{2})=(3^{3})^{2}
3^{x+2}=3^{6}
x + 2 = 6
x = 4

This is the type of problem, however, where you could simply test the answer choices until the you got the correct answer. Start with Choice H (the middle number) and check to see if it makes the equation correct. You’ll end up with the left side of the equation being smaller than the right, so you know that x must be larger than 3.5. If you try x = 4, you’ll confirm that both sides are equal to 729.

33.  C
The three angles of the triangle must sum to 180 degrees. Because the central angle, ∠BAC, has a measure of 80 degrees, the other two angles must sum to 100 degrees. As you look at the triangle, you’ll realize that the sides AB and BC are radii of the circle and must have the same length. That means that the triangle is isosceles and its two base angles, ∠ABC and ∠ACB, have the same measure. Each of these angles is, therefore, 50 degrees.

46.  K
You first need to find the angle opposite d, the distance that we are trying to find. The north-south line forms a straight angle measuring 180 degrees. To find the angle you are looking for, subtract the two bearings from 180:  180 – 34 – 45 = 101 degrees.

Now you can simply plug into the Law of Cosines formula and find an expression for d.

c^{2} = a^{2}+b^{2}-2ab\cdot cosC
d^{2} = 80^{2}+100^{2}-2(80)(100)\cdot cos101
d = \sqrt{80^{2}+100^{2}-2(80)(100)\cdot cos101}

51.  B
Problems involving averages are often more about the sum of the numbers than the average. in this case, if five numbers have an average of 24, then their sum must be 5 x 24 = 120. The largest and smallest numbers have a sum of 43 + 10 = 53, so the remaining three numbers must have a sum of 120 – 53 = 67. Now divide 67 by 3 to get the average of those three numbers, which is 22 1/3.

If you have questions about these problems or anything else to do with the ACT, leave a comment below or send me an email at info@cardinalec.com.

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