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ACT Section Two: Math
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Question: 263
If
and a and c are doubled, what happens to the value of d?
Answer: A
Explanation:
If a and c are doubled, the fraction on the left side of the equation becomes
. The fraction has been multiplied by 2/2 which is equal to 1. Multiplying a fraction by 1 does not change its value:
.
The value of d remains the same.
Question: 264
Balloons are sold according to the chart above. If a customer buys one balloon at a time, the cost is $1.00 per balloon. If a customer buys ten balloons at a time, the cost is $0.90 per balloon. If Carlos wants to buy 2,000 balloons, how much money does he save by buying 1,000 balloons at a time rather than ten balloons at a time?
Answer: D
Explanation:
If Carlos buys ten balloons, he will pay 10 x $0.90 = $9. In order to total 2,000 balloons, Carlos will have to make this purchase 2,000 ÷ 10 = 200 times. It will cost him a total of 200 x $9 = $1,800. If Carlos buys 1,000 balloons, he will pay 1,000 x $0.60) = $600. In order to total 2,000 balloons, Carlos will have to make this purchase 2,000 ÷ 1,000 = 2 times. It will cost him a total of 2 x $600 = $1,200. It will save Carlos $1,800 – $1,200 = $600 to buy the balloons 1,000 at a time.
Question: 265
Given the following figure with one tangent and one secant drawn to the circle, what is the measure of
?
Answer: E
Explanation:
The measure of an angle in the exterior of a circle formed by a tangent and a secant is equal to half the difference of the intercepted arcs. The two intercepted arcs are
which is 60°, and
which is 110°. Find half of the difference of the two arcs: 1/2 x (110 – 60) = 1/2 x 50 = 25º.
Question: 266
In the diagram above, what is the sum of the measures of the
,
and
?
Answer: B
Explanation:
There are 180º in a line: (x + (supplement of
)) + (y + (supplement of
)) + (z + (supplement of
)) = 540. The supplement of
, the supplement of
, and the supplement of
are the interior angles of a triangle. There are 180º in a triangle, so those supplements sum to 180. Therefore, x + y + z + 180 = 540, and x + y + z = 360.
Question: 267
If the surface area of a cube is 384 cm2 , what is the volume of the cube?
Answer: C
Explanation:
The surface area of a cube is equal to 6 x e2, where e is the length of one edge of the cube; 6 x e2 = 384 cm, e3 = 64, e = 8 cm. The volume of a cube is equal to e3; (8 cm)3 = 512 cm3.
Question: 268
Greg has nine paintings. The Hickory Museum has enough space to display three of them. From how many different sets of three paintings does Greg have to choose?
Answer: C
Explanation:
Be careful not to count the same set of three paintings more than once – order is not important. A nine-choose-three combination is equal to
.
Question: 269
It takes eight people 12 hours to clean an office. How long would it take six people to clean the office?
Answer: C,
Explanation:
There is an inverse relationship between the number of people and the time needed to clean the office. Multiply the number of people by the hours needed to clean the office: 8 x 12 = 96. Divide the total number of hours by the new number of people, 6: 96 ÷ 6 = 16. It takes six people 16 hours to clean the office.
Question: 270
Lindsay grows only roses and tulips in her garden. The ratio of roses to tulips in her garden is 5:6. If there are 242 total flowers in her garden, how many of them are tulips?
Answer: E
Explanation:
The number of roses, 5x, plus the number of tulips, 6x, is equal to 242 total flowers: 5x + 6x = 242, 11x = 242, x = 22. There are 5 x 22 = 110 roses and 6 x 22 = 132 tulips in Lindsay’s garden.
Question: 271
The point (2, 1) is the midpoint of a line with endpoints at (-5, 3) and:
Answer: D
Explanation:
The midpoint of a line is equal to the average x-coordinates and the average y-coordinates of the line’s endpoints: (-5 + x) / 2 = 2, -5 + x = 4, x = 9 (3 + y) / 2 = 1, 3 + y = 2, y = -1The other endpoint of this line is at (9, -1).
Question: 272
The expression
is equivalent to:
Answer: C
Explanation:
Factor the numerator and denominator and cancel like factors: (x2 + 2x – 15) = (x + 5) x (x – 3) (x2 + 4x – 21) = (x + 7) x (x – 3) Cancel the (x – 3) term from the numerator and the denominator. The fraction reduces to (x + 5) / (x + 7).
Question: 273
In the diagram above, lines EF and GH are parallel, and line AB is perpendicular to lines EF and GH. What is the length of line AB?
Answer: C
Explanation:
Line AB is perpendicular to line BC, which makes triangle ABC a right triangle. and
are alternating angles, i.e. angles made by a pair of parallel lines cut by a transversal.
, therefore,
.
form a line. There are 180º in a line, so the measure of of
. Triangle ABC is a 30-60-90 right triangle, which means that the length of the hypotenuse, AC, is equal to twice the length ofthe leg opposite the 30- degree angle, BC. Therefore, the length of BC is 10/2, or 5. The length of the leg opposite the 60-degree angle, AB, is v3 times the length of the other leg, BC. Therefore, the length of AB is 5√3.
Question: 274
The statement “Raphael runs every Sunday” is always true. Which of the following statements is also true?
Answer: A
Explanation:
The statement “Raphael runs every Sunday” is equivalent to “If it is Sunday, Raphael runs.” The contra positive of a true statement is also true. The contra positive of “If it is Sunday, Raphael runs” is “If Raphael does not run, it is not Sunday.”
Question: 275
Rob has six songs on his portable music player. How many different four-song orderings can Rob create?
Answer: D
Explanation:
The order of the four songs is important. The orderings A, B, C, D and A, C, B, D contain the same four songs, but in different orders. Both orderings must be counted. The number of six-choose-four orderings is equal to 6 x 5 x 4 x 3 = 360.
Question: 276
A dormitory now houses 30 men and allows 42 square feet of space per man. If five more men are put into this dormitory, how much less space will each man have?
Answer: B
Explanation:
30 men x 42 square feet = 1260 square feet of space; 1260 square feet ÷ 35 men = 36 square feet; 42 – 36 = 6, so each man will have 6 less square feet of space.
Question: 277
If 30% of r is equal to 75% of s, what is 50% of s if r = 30?
Answer: B
Explanation:
If r = 30, 30% of r = 0.30 x 30 = 9. 9 is equal to 75% of s. If 0.75s = 9, then s = 12. 50% of s = 0.50 x 12 = 6.
Question: 278
Line y = 2/3x – 5 is perpendicular to line:
Answer: E
Explanation:
Perpendicular lines have slopes that are negative reciprocals of each other. The slope of the line given is 2/3. The negative reciprocal of 2/3 is 3/2. Every line with a slope of -3/2 is perpendicular to the given line; y = -3/2x + 5 is perpendicular to y = 2/3x – 5.
Question: 279
In the graph above, ABCD is a square. What are the coordinates of point B?
Answer: B
Explanation:
Point B is the same distance from the y-axis as point A, so the x-coordinate of point B is the same as the x-coordinate of point A (-1). Point B is the
same distance from the x-axis as point C, so the y-coordinate of point B is the same as the y-coordinate of point C (4). The coordinates of point B are (- 1, 4).
Question: 280
If the expression
, then one possible value of x could be:
Answer: A
Explanation:
Cross multiply and solve for x:
3 x 2x = (2 + x) x (x – 5) 6x = x2 – 3x – 10 x2 – 9x – 10 = 0 (x – 10) x (x + 1) x = 10, x = -1