ACT Form 73E Math Answer Explanation

2016 April(73E) Math Question 2

2. The length of a rectangle is 12 feet. The width of the rectangle is 1/2 the length. What is the perimeter of the rectangle, in feet?

Choice J

$$Width =\frac12\cdot Length = \frac12\cdot 12 =6 \\ $$ $$\Rightarrow Perimeter = 6+6+12+12=36$$

2016 April(73E) Math Question 3

- Simplify Algebraic Expressions

3. (9m — 4n) — (2n + 5m) is equivalent to:

Choice A

$$(9m-4n)-(2n+5m)=9m-4n-2n-5m=4m-6n$$

Number Types & Properties

2016 April(73E) Math Question 4

- Largest or Smallest Value

4. Which of the following numbers has the greatest value?

Choice F

$$0.\overline{3}(Choice\ F)=0.333333...> 0.3333(Choice\ K)$$

2016 April(73E) Math Question 5

- 2 Variable Addition

5. At a grocery store, Jo Ellen received $1.60 when she returned her cans, glass bottles, and plastic bottles. Jo Ellen received $0.05 for each can, $0.10 for each glass bottle, and $0.05 for each plastic bottle. She knew she had returned 6 cans and 8 glass bottles. How many plastic bottles did Jo Ellen return to the store?

Choice C

Let x be the number of plastic bottles Jo Ellen return to the store: $$$0.05\times6+$0.1\times8+$0.05\cdot x=$1.6 \\$$ $$\Rightarrow x = 10$$

2016 April(73E) Math Question 6

- Temperature

Choice K

$$change=12°C-(-4°C)=12+4=16$$

2016 April(73E) Math Question 7

- Fixed Cost + Variable Cost

7. Sienna will be paid $75, plus 25% of her total weekly sales, for the hours she is scheduled to work next week. Let w represent Sienna's total weekly sales, in dollars, for next week. Which of the following expressions gives Sienna's pay, in dollars, for the hours she is scheduled to work next week?

Choice A

$$Sienna's\ pay = $75+25\% \times Total \ weekly \ sales = $75+25\% \times w=0.25w+75 $$

2016 April(73E) Math Question 8

Matrices

- Addition & Subtraction

Choice G

A system of equations can be represented by an augmented matrix.

In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.

2016 April(73E) Math Question 9

- Solve Function Expressions

9. If g(x) = 2x^2 — 3x + 4, then g(-3) = ?

Choice E

$$g(-3)=2(-3)^2-3(-3)+4=31$$

- Collinear Points & Triangles

2016 April(73E) Math Question 10

Angles

10. The figure below shows lines AB and DC, line segments AT and BC, and 2 angle measures. What is the measure of ACB ?

Choice J

2016 April(73E) Math Question 11

- 1 Variable Equations

11. Marietta purchased a car that had a purchase price of $10,400, which included all other costs and tax. She paid $2,000 as a down payment and got a loan for the rest of the purchase price. Marietta paid off the loan by making 48 payments of $225 each. The total of all her payments, including the down payment, was how much more than the car's purchase price?

Choice B

$$ $2000+$225\times 48 - $10400=$2400 $$

2016 April(73E) Math Question 12

12. The dimensions of the rectangle shown below are given in inches. Which of the following expressions gives the area, in square inches, of the rectangle?

Choice J

$$Area =(x+2)(2x-1)=2x^2-x+4x-2=2x^2+3x-2$$

2016 April(73E) Math Question 13

- Solve Function Expressions

13. The population of a particular town is modeled by the equation P = 120,000(1.1)^t, where t is the number of years after January 1, 2011. Based on the model, which of the following numbers is closest to the population of the town on January 1, 2013 ?

Choice B

$$p=120000(1.1)^2=145200\approx 145000$$

2016 April(73E) Math Question 14

- Interpret Graphs or Figures

14. The phrases below represent 3 types of measurements as functions of time. I. The height of an object falling toward the ground II. The height of a plant growing at a constant rate III. The distance a car travels while increasing its velocity Graphs A, B, and C below each represent one of these functions. For all graphs, the x-axis represents time. Which graph is of which function?

Choice G

$$I. \ y(height)=h_{0}-\frac{1}{2}gx^2 \\$$ $$II. \ y=kx \\$$ $$III. y=a\cdot t\cdot t=at^2$$

2016 April(73E) Math Question 15

- Equations that Describe a Line

15. In the standard (x,y) coordinate plane below, 5 points are labeled on a parabola. Which of the following lines has the slope of least value?

Choice E

The flatter the line, the smaller the slope.

2016 April(73E) Math Question 16

Statistics

- Mean, Median

16. Jamal earned scores of 70, 75, 85, and 94 points on the first 4 history projects, and he has 1 more project to complete. What is the minimum score Jamal needs to earn on the 5th project so that the mean of his scores on all 5 projects is at least 2 points more than the mean of the scores he earned on the first 4 projects?

Choice J

$$Mean \ of \ the \ first \ 4 \ projects = \frac{70+75+85+94}{4}=81 \\$$ $$ Mean \ of \ the \ 5 \ projects = \frac{70+75+85+94+x}{5}\geq 81+2=83 \\ $$ $$\Rightarrow x \geq 91$$

2016 April(73E) Math Question 17

- Squares in a Shape

17. In the figure below, all of the small squares are equal in area, and the area of rectangle ABCD is 1 square unit. Which of the following expressions represents the area, in square units, of the shaded region?

Choice E

2016 April(73E) Math Question 18

Statistics

- Mean, Median

18. What is the median of the list of numbers below? 6, 3, 5, 7, 12, 9, 5, 5, 11

Choice G

From the smallest to the largest: $$3, 5, 5, 5, \underline{6}, 7, 9, 11, 12$$

2016 April(73E) Math Question 19

Probability

- More than One Event

19. A batch of 100 defective computer chips consisting of 2 types (I and II) and made by 2 companies (A and B) was selected, and it was determined how many of each type of chip was made by each company. The results are displayed in the table below.

Choice A

$$P=\frac{30}{100}$$

2016 April(73E) Math Question 20

20. Carpenters use the term pitch to describe the slope of a roof. For example, a roof with a pitch of 17-1 means the roof has 1 foot of vertical rise for every 4 feet of horizontal distance. The figure below shows a 13-foot-long roof with 5 feet of vertical rise and x feet of horizontal distance. What is the pitch of this roof?

Choice H

$$x=\sqrt{13^2-5^2}=12 \\$$ $$ \Rightarrow pitch = \frac{5}{12}$$

- Absolute Value Expressions

2016 April(73E) Math Question 21

Absolute Values

21. Given that x <= 2 and x + y >= 4, what is the LEAST value that y can have?

Choice D

When x = 2, the …

You are not logged in.

2016 April(73E) Math Question 22

- Distance & Midpoint

22. As shown below, A, B, C, and D are collinear, with B between A and C and with C between B and D. Given AC = BD = 12 cm and given BC = 3 cm, what is AD, in centimeters?

Choice J

$$AD=AB+BD=(AC-BC)+BD=12-3+12=21$$

You are not logged in.

2016 April(73E) Math Question 23

- Cost

23. Keanu bought a new laptop computer and paid a discount price that was 20% less than the $1,000 list price. He also paid tax on the laptop equal to 6% of the discount price. What is the total amount Keanu paid for the laptop computer?

Choice C

$$Total \ amouunt = $1000 …

You are not logged in.

2016 April(73E) Math Question 24

Angles

- Parallel Lines

24. In Middletown, Main Street and Market Street are parallel to each other. Patrick Street intersects Market Street to form a 76° angle at the northeast corner, as shown in the figure below. What is the measure of the angle formed at the southeast corner of Main Street and Patrick Street? (Note: Each street is straight and has the same uniform width.)

Choice H

You are not logged in.

2016 April(73E) Math Question 25

Probability

- More than One Event

25. Of the 900 students enrolled at Sierra Elementary School, 45% live south of Highway R. Of the students who live south of Highway R, 20% do NOT ride the bus to school. How many students who live south of Highway R ride the bus to school?
Back to top↑

Choice C

$$Number \ of \ students …

You are not logged in.

2016 April(73E) Math Question 26

- Equations that Describe a Line

26. Which of the following equations represents the line in the standard (x,y) coordinate plane that passes through (2,-3) and has a slope of —1/2 ?

Choice G

Let the line be: $$y …

You are not logged in.

2016 April(73E) Math Question 27

- Domain, Range, Asymptote, Undefined

27. The entire graph of the relation R of the ordered pairs (x,y) is shown in the standard (x,y) coordinate plane below. One of the following sets is the domain of the relation R. Which set is it?

Choice C

Definition: Function, domain, range

A …

You are not logged in.

2016 April(73E) Math Question 28

- Distance & Midpoint

28. On Friday night, the Morrison family set up camp in the Ocala National Forest. On Saturday morning they hiked to a wilderness area 3 miles due north and 4 miles due east of their campsite. The elevation of the wilderness area is the same as the elevation of the campsite. To the nearest 0.1 mile, what is the straight-line distance from the wilderness area to the Morrisons' campsite?

Choice H

$$Distance \ from \ Wilderness …

You are not logged in.

Ratios, Proportions, Percentages, Fractions

2016 April(73E) Math Question 29

- Percents & Fractions

29. What positive number when divided by its reciprocal has a result of 4/25 ?

Choice A

$$\frac{x}{\frac{1}{x}}=\frac{4}{25} \\$$$$ \Rightarrow x^2=\frac{4}{25} \\ …

You are not logged in.

2016 April(73E) Math Question 30

30. The base of an escalator in a store is 26 meters long and has a vertical lift of 10 meters as shown below. Which of the following expressions is closest to the angle of inclination between the base of the escalator and the horizontal floor?

Choice F

$$\sin x=\frac{10}{26} \\ \Rightarrow x=\sin^{-1}(\frac{10}{26})$$

You are not logged in.

2016 April(73E) Math Question 31

- Mixed Shapes

31. In the figure shown below, trapezoid ABCD is formed by AABC and AACD. The lengths are given in inches. What is the area of ABC, in square inches?

Choice E

$$Area \ of \ \triangle …

You are not logged in.

2016 April(73E) Math Question 32

32. Which of the following ratios is equal to cos B ? (Note: FG denotes the length of FG .)

Choice K

By definition, $$\cos \angle B …

You are not logged in.

2016 April(73E) Math Question 33

33. Suppose ABCD is placed in the standard (x,y) coordinate plane such that C is at (0,0), B is at (25,0), and A and D have positive x- and y-coordinates. What is the x-coordinate of D ?

Choice C

$$x=\sqrt{13^2-12^2}=5$$

You are not logged in.

2016 April(73E) Math Question 34

- 1 Variable Equations

34. A family will rent a picnic shelter for $200 for a reunion. The cost of the shelter will be distributed equally among the people who plan to attend. The current cost per person will decrease by $1 if 10 more people plan to attend the reunion. How many people are currently planning to attend the reunion?

Choice H

$$(\frac{200}{x}-1)\cdot(x+10)=200 \\ $$$$\Rightarrow x= 40$$

You are not logged in.

2016 April(73E) Math Question 35

35. The figure below shows 12 congruent line segments, each determined by a pair of adjacent points. The sum of the lengths of the 12 line segments is 36 centimeters. Each intersection of 4 of the line segments forms 4 right angles. What is the area, in square centimeters, of the shaded region?

Choice C

$$Length \ of \ each …

You are not logged in.

2016 April(73E) Math Question 36

- Systems of Equations

36. What is the value of b in the solution of the system of equations below?

Choice F

$$6a+3b=12 \ (I.) \\$$ $$ …

You are not logged in.

Similar Triangles & Types of Triangles

2016 April(73E) Math Question 37

- Similar Triangles

37. In the figure shown below, LABC ADEF, sides AB and BC are each 7.2 cm long, side DE is 10.8 cm long, and the measure of LE is 42°. What is the measure of C?

Choice E

$$\angle C=\frac{180°-42°}{2}=69°$$

You are not logged in.

2016 April(73E) Math Question 38

- Graphing Trig Functions

38. The function f(x) = 0.25 sin(4x) is graghed below for 0 <= x

Choice F

As seen from the graph, …

You are not logged in.

2016 April(73E) Math Question 39

- Distance & Midpoint

39. Point A lies at (2,6) and point B lies at (-5,10) in the standard (x,y) coordinate plane below. What is the length, in coordinate units, of AB ?

Choice B

$$AB=\sqrt{4^2+7^2}=\sqrt{65}$$

You are not logged in.

2016 April(73E) Math Question 40

- Write Algebraic Expressions

40. Each student's project in a history seminar is given a point score by the teacher and by each of the other students in the seminar. A student's project grade, g, is determined by the formula g = 3t+s / 3+n, where t is the score the teacher gives, s is the sum of the scores the students give, and n is the number of students in the seminar. What is t in terms of g, s, and n ?

Choice K

$$g=\frac{3t+s}{3+n} \\ $$$$\Rightarrow g(3+n)=3t+s \\ …

You are not logged in.

Ratios, Proportions, Percentages, Fractions

2016 April(73E) Math Question 41

- LCM, GCF, Prime Numbers

41. For all x not equal to —2, which of the following expressions is equal to (x2 + 5x + 6)/(x + 2) + x + 5 ?

Choice B

$$\frac{x^2+5x+6}{x+2}+x+5 = \frac{(x+2)(x+3)}{x+2}+x+5 =x+3+x+5=2x+8$$

You are not logged in.

2016 April(73E) Math Question 42

42. Two concentric circles are shown below. The radius of the larger circle is 10 feet and the radius of the smaller circle is 6 feet. What is the area, in square feet, of the shaded region bounded by the circles?

Choice J

$$Area \ of \ shaded …

You are not logged in.

Number Types & Properties

2016 April(73E) Math Question 43

- Complex Number

43. For i = square root of -1, (1 + 2i)^2 = ?

Choice A

$$(1+2i)^2=1+4i+(2i)^2=1+4i+4i^2=1+4i+4(-1)=-3+4i$$

You are not logged in.

2016 April(73E) Math Question 44

Probability

- One Event

44. A box contains a combination of solid-colored tickets: 1/10 of the tickets are green, 1/2 are red, 1/4 are blue, and the remaining 30 tickets are white. How many blue tickets are in the box?

Choice H

$$Number \ of \ blue …

You are not logged in.

2016 April(73E) Math Question 45

- Simplify Algebraic Expressions

45. Which of the following expressions is the greatest monomial factor of 80x^3y + 48x^2y^2 ?

Choice A

$$80x^3y+48x^2y^2=\underline{16x^2y}(5x+3y)$$

You are not logged in.

2016 April(73E) Math Question 46

- Interpret Graphs or Figures

46. The points graphed in the standard (x,y) coordinate plane below show the positions of 5 stars in a plane relative to a point represented by the origin, where each coordinate unit equals 1 light-year. A light-year is the distance that light travels in 1 year, and 1 light-year 5.9 x 1012 miles. The distance from Star A to Star D is approximately 11.4 light-years. Star A has a mass of 3 solar masses; and Stars B, C, D, and E each have a mass of 1 solar mass. Because Stars B, C, D, and E have the same mass, the position determined by the average of the x-coordinates and the average of the y-coordinates of those stars approximates the center of mass of those 4 stars. What are the coordinates of this position?

Choice K

The coordinates of the center …

You are not logged in.

2016 April(73E) Math Question 47

47. What is the tangent of the angle formed by CD and CE in the graph?

Choice D

$$\tan \angle DCE=\frac{5-2}{13-11}=\frac{3}{2}$$

You are not logged in.

Exponents, Radicals, & Scientific Notation

2016 April(73E) Math Question 48

- Scientific Notation

48. Which of the following values is closest to the number of miles between Stars A and D ?

Choice J

$$AAD=\sqrt{11^2+3^2}=11.4 \ light \ years …

You are not logged in.

2016 April(73E) Math Question 49

- Domain, Range, Asymptote, Undefined

49. In the standard (x,y) coordinate plane, for what value(s) of x, if any, is there NO value of y such that (x,y) is on the graph of y = (x - 3)/(x + 3)(x + 2)(x - 2)?

Choice A

The denominator of any fraction …

You are not logged in.

2016 April(73E) Math Question 50

- Simplify Algebraic Expressions

50. Which of the following number properties is illustrated in the statement below?

Choice G

The statement illustrates the commutative …

You are not logged in.

2016 April(73E) Math Question 51

- Other Shapes

51. The volume of a right circular cone with radius r and height h is 1/3 pi r^2h, where r and h have the same unit of measure. Cones A and B are both right circular cones. The radius of Cone B is 2 times the radius of Cone A. Cone B's height is 1/2 Cone A's height. Compared to the volume of Cone A, the volume of Cone B is:

Choice D

$$V_{Cone \ B}=\frac{1}{3} \pi r_{Cone …

You are not logged in.

2016 April(73E) Math Question 52

Angles

- Circles

52. From point A outside a circle and in the same plane as the circle, 2 rays are drawn tangent to the circle with the points of tangency labeled B and C, respectively. Segment BC is then drawn to form LABC. If A measures 70°, what is the measure of ABC ?

Choice B

Since the two lines are …

You are not logged in.

2016 April(73E) Math Question 53

Conics

- Ellipse

Choice D

What is the standard equation …

You are not logged in.

2016 April(73E) Math Question 54

54. Which of the following lists of numbers could be the side lengths, in inches, of a triangle?

Choice K

To be the side lengths …

You are not logged in.

2016 April(73E) Math Question 55

- Distance, Speed, Time

55. Carrie and Manuel are side by side when they begin to run at the same time in the same direction around a track. Carrie runs at a constant rate of 30 seconds per lap, while Manuel runs at a constant rate of 50 seconds per lap. How many seconds after beginning to run will Carrie have run exactly 1 more lap than Manuel?

Choice C

Let the number of seconds …

You are not logged in.

Exponents, Radicals, & Scientific Notation

2016 April(73E) Math Question 56

- Exponent Expressions

56. If a is a positive even integer and b is a positive odd integer, then [(-3)(+3)]^ab is:

Choice G

ab is a positive even …

You are not logged in.