ACT Practice Test: Math

A DVD player with a list price of $100 is marked down 30%. If John gets an employee discount of 20% off the sale price, how much does John pay for the DVD player ?

$56 is correct. 100(0.70) = 70 is the amount that would be paid if the DVD was marked down 30%, but there is another discount of 20%, so the price is going to be 80% of the marked-down price. The price will be 70(0.80) = 56.

The measure of ∠ABC in the figure is x°. Which of the following is an expression for β°

(180 - x)° The angles of the rectangular pieces of lumber measure 90°, so the sum of the measure of the angles at β is 360°. β + 90 + x + 90 = 360, or β = 180 - x.

If n = 8 and 16 • 2m = 4n - 8, then m = ?

-4 is the correct answer. When n = 8, 4n - 8 = 48 - 8 = 40 = 1, and 16 · 2m = 24 · 2m = 24 + m. So, 24 + m= 1, and any number to the zeroth power is 1, so 4 + m = 0, or m = -4.

What is the difference between 1.8 and 1.08? (repeating bar)

0.179 repeating bar is correct answer. The correct response is 0.719. Take 1.08 and repeat the pattern several times, then subtract that from 1.8. 1.8 - 1.08080808 ≈ 0.7191919. Realizing that the pattern should repeat, you can conclude that 0.719 is the correct answer.

Abandoned mines frequently fill with water. Before an abandoned mine can be reopened, the water must be pumped out. The size of pump required depends on the depth of the mine. If pumping out a mine that is D feet deep requires a pump that pumps a minimum of + 4D - 250 gallons per minute, pumping out a mine that is 150 feet deep would require a pump that pumps a minimum of how many gallons per minute? Individual Question

1,250 is the correct answer. The correct answer is D. If you substitute D with 150 in the expression, you get + 4(150) - 250 = + 600 - 250 = 1,250.

In quadrilateral PQRS below, sides PS and QR are parallel for what value of x ?

110 is the correct answer. The question states that PS and QR are parallel. If you treat PQ as a transversal, then ∠P and ∠Q are interior angles on the same side of a transversal, so their measures add up to 180°. Since the measure of ∠P is 70°, the measure of ∠Q is 180° - 70° = 110°.

Ding's Diner advertised this daily lunch special: "Choose 1 item from each column—only $4.95!" Thus, each daily lunch special consists of a salad, a soup, a sandwich, and a drink. How many different daily lunch specials are possible?

120 is the correct answer. By the fundamental counting principle, the number of different possible lunch specials is 3(2)(5)(4).

When x = 3 and y = 5, by how much does the value of 3x2 - 2y exceed the value of 2x2 - 3y ?

14 is the correct answer. When you use x = 3 and y = 5 in the given expressions, 3x2 - 2y = 3(3)2 - 2(5) = 27 - 10 = 17 and 2x2 - 3y = 2(3)2 - 3(5) = 18 - 15 = 3. Then subtract 3 from 17 to get 14.

Which of the following expressions is the closest approximation to the height h, in feet, of the roof truss shown below?

15 tan 30 YZ = XZ = 30) = 15. So, tan 20° = h/yz = h/15 . Then h = 15 tan 20°.

Points A, B, C, and D are on a line such that B is between A and C, and C is between B and D. The distance from A to B is 6 units. The distance from B to C is twice the distance from A to B, and the distance from C to D is twice the distance from B to C. What is the distance, in units, from the midpoint of to the midpoint of ?

18 is the correct answer. This is the correct response. BC = 2AB = 2(6) = 12 and CD = 2BC = 2(12) = 24. The distance between the midpoints of BC and CD is BC + CD = (12)+ (24) = 18

Taher plans to cut the 3 pieces of lumber for the flower bed border from a single piece of lumber. Each cut takes 1/8 inch of wood off the length of the piece of lumber. Among the following lengths, in inches, of pieces of lumber, which is the shortest piece that he can use to cut the pieces for the flower bed border?

181 The number of inches of wood needed if there were no cuts is 4 + 5 + 6 = 15 feet, or 180 inches. However, you need to add 2(<) for 2 cuts that are needed so that you have lumber for each of the 3 sides. Since 180 + 2() = 180 + , the minimum piece needed to construct the flower bed border including the 2 cuts would be 181 inches.

The lead of a screw is the distance that the screw advances in a straight line when the screw is turned 1 complete turn. If a screw is 2 1/2 inches long and has a lead of 1/8 inch, how many complete turns would get it all the way into a piece of wood?

20 is the correct answer. With every complete turn ⅛ inch of the screw goes into the wood. So after 8 complete turns, 1 inch of the screw would be in the wood. So, x(⅛) = 2½ . Multiplying by 8, x = 8(2½) = 8(5/2) = 20.

The geometric figure shown below consists of a square and 4 semicircles. The diameters of the semicircles are the sides of the square, and each diameter is 10 centimeters long. Which of the following is the closest approximation of the total area, in square centimeters, of this geometric figure?

260 You found the area of the square, the area of 4 semicircles (or the area of 2 full circles), and added them. 102 + 2(5)2 257. The closest answer is 260.

What is the degree measure of the acute angle formed by the hands of a 12-hour clock that reads exactly 1 o'clock?

30 degrees in the correct answer. One complete rotation of a clock hand is 360°, and there are 12 hourly markings on a clock. When the hands read exactly 1 o'clock, the degree measure of the angle formed by the clock hands is of a complete rotation, or (360°) = 30°.

For all nonzero real numbers p, t, x, and y such that x/y = 3p/2t , which of the following expressions is equivalent to t ?

3py/2x is correct. If you cross multiply, 2xt = 3py. Then dividing each side by 2x, you get t = 3py/2x.

A typical high school student consumes 67.5 pounds of sugar per year. As part of a new nutrition plan, each member of a track team plans to lower the sugar he or she consumes by at least 20% for the coming year. Assuming each track member had consumed sugar at the level of a typical high school student and will adhere to this plan for the coming year, what is the maximum number of pounds of sugar to be consumed by each track team member in the coming year?

54 is the correct answer. For each member of the track team to consume 20% less sugar, the track member will consume 100% - 20% = 80% of the level of a typical high school student. 80% of 67.5 = 0.80(67.5) = 54.

In the figure below, A, B, C, and D are collinear, FC is parallel to ED, BE is perpendicular to ED, and the measures of ∠FAB and ∠EBA are as marked. What is the measure of ∠FCB ?

57 degrees Since FC and ED are two parallel line segments cut by transversal BE, ∠E and ∠BGC are corresponding angles. So, the measure of ∠BGC is 90°. Since ∠ABG ∠GBC are supplementary angles, the measure of ∠GBC = 180° - 147° = 33°. Looking at ΔBGC, the sum of the measures of angles ∠GCB, ∠BGC, and ∠GBC is 180°. So, the measure of ∠GCB + 90° + 33° = 180°, or 180° - 90° - 33° = 57°.

How many irrational numbers are there between 1 and 6 ?

Infinitely many is the correct response. If you chose this answer, you know 1 and 6 are real numbers and that there are an infinite number of irrational numbers between any two real numbers.

Which of the following is a factor of the polynomial 2x² - 3x - 5 ?

The correct answer is (2x -5). 2x²- 3x - 5 = (x + 1)(2x - 5).

What is the slope of any line parallel to the line 9x + 4y = 7 ?

The correct answer is -9/4 . Since 4y = -9x + 7, y = -9/4x + 7/4. So the slope of this line is -9/4. Since parallel lines have the same slope, the slope of any parallel line must also be -9/4.Which of the following is a factor of the polynomial 2x2 - 3x - 5 ?

Sales for a business were 3 million dollars more the second year than the first, and sales for the third year were double the sales for the second year. If sales for the third year were 38 million dollars, what were sales, in millions of dollars, for the first year?

The correct answer is 16. This is the correct answer. If x = sales for the first year, then x + 3 = sales for the second year. Since sales for the third year were double the sales for the second year, sales for the third year = 2(x + 3). Sales for the third year were 38, so 2(x + 3) = 38. To solve this equation, you could first divide each side by 2 to get x + 3 = 19. Then, by subtracting 3 from both sides, x = 16.

The volume, V, of the right circular cone with radius r and height h, shown below, can be found using the formula V = r 2h. A cone-shaped paper cup has a volume of 142 cubic centimeters and a height of 8.5 centimeters. What is the radius, to the nearest centimeter, of the paper cup?

The correct answer is 4. 3V/pih = r². r = square root 3V/pih = pi 3(142)/8.5pi = 4.

In the figure below, ray was constructed starting from rays and . By using a compass D and G were marked equidistant from E on rays and . The compass was then used to locate a point F, distinct from E, so that F is equidistant from D and G. For all constructions defined by the above steps, the measures of ∠DEF and ∠GEF:

The correct answer is are equal. If you draw line segments DF and FG, you can show ΔDEF ≅ ΔGEF by SSS (side-side-side congruence). So, ∠DEF ≅ ∠GEF because corresponding parts of congruent triangles are congruent.

A boat departs Port Isabelle, Texas, traveling to an oil rig. The oil rig is located 9 miles east and 12 miles north of the boat's departure point. About how many miles is the oil rig from the departure point?

The correct response is 15. Using the Pythagorean theorem, 9² + 12² = c². So c = square root 9² + 12² = square root 81 + 144 = square root 225 = 15

Which of the following statements must be true whenever n, a, b, and c are positive integers such that n < a, c > a, and b > c ?

b - n > a -n is the correct answer. Since b > a, subtracting n from each side, b - n > a - n, will not change the relationship between b and a.

Which of the following is an equation of the circle with its center at (0,0) that passes through (3,4) in the standard (x,y) coordinate plane?

x² + y² = 25 The radius of the circle is the distance between (0,0) and (3,4), which is √((3-0)2+ (4-0)2) = 5. An equation of a circle where (h,k) is the center and r is the radius is (x - h)2 + (y - k)2 = r2. So (x - 0)2 + (y - 0)2 = 52 or x2 + y2 = 25.