ACT Math: Five Solutions (3/6/15)

Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (3/6/15). Below you’ll find the fully worked-out solutions to those problems.

11. B
The two numbers need to satisfy the equation xy + 8 = 44. Subtracting 8 from each side of the equation tells you that the product of x and y must be 36. Of the answer choices, the only one that doesn’t give you that product is B (8 x 4 = 32, not 36).

26. J
To solve an absolute value equation, you should write and solve two separate equations. If the absolute value of some quantity is 5, then the quantity must equal either 5 or -5.

x – 4 = 5       or       x – 4 = -5
x – 4 + 4 = 5 + 4        x – 4 + 4 = -5 + 4
x = 9        or        x = -1

Alternatively, you could have simply tried each of the answer choices to see which one worked. Beware of Choice G, however. Although 9 is a solution, it is not the only solution. After you determined that 9 was a solution, you would need to test any other answer choice that included x = 9.

35. A
Let’s first lay out what we have. We know that the third term is 18 and the sixth term is 39. The remaining terms can be represented with blanks.

____   _____  18  _____  _____  39

To go from the third term to the sixth term, we would need to add the common difference three times.  If we increased 21 units (39 – 18) in three additional terms, the difference between the terms must be 7 (21 ÷ 3). Now we can fill in the remaining terms.

4    7     18     25     32     39

The first term is 4.

 44. G
Let’s first use the perimeter formula for a rectangle to find the width of the rectangle (which is also the radius of the circle).

2L + 2W = P
2(10) + 2W = 28
20 + 2W = 28
2W = 8
W = 4

Now you can use the fact that the radius of the circle is 4 to find the area of the circle.

A = πr²
A = π(4)²
A = 16π

The shaded portion of the circle is a quarter circle because ∠CAB is an angle of the rectangle and, therefore, 90 degrees. So the shaded region is equal to 1/4(16π) = 4π.

53. E
The equation of a circle is (x – h)² + (y – k)² = r², where (h, k) is the center of the circle and r is the radius of the circle. The equation of our circle is (x + 4)² + (y + 2)² = 9, which can be thought of as (x – (-4))² + (y – (-2))² = 3². This tells you that the center of the circle is (-4, -2) and that the radius is 3. You can now sketch the circle in the standard xy-coordinate plane. Doing so will show you that the circle lies in Quadrant II and Quadrant III.

Five Solutions 3-6-5

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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