ACT Form 68G Math Answer Explanation



2010 April(68G) Math Question 1

Angles

- Parallel Lines

1. On the map below, 2 parallel streets, North and South, intersect 2 other parallel streets, First and Second. The acute angle at which Second Street intersects North Street measures 55. What is the measure of the acute angle at which First Street intersects South Street?


Choice H

$$=8+20x+9-2x$$ $$=18x+17$$


Choice C

$$(-2,8)+(8,-2)=(6,6)$$


2010 April(68G) Math Question 4

Word Problems

- 2 Variable Addition

4.To attend an annual banquet, members pay S17 per ticket while nonmembers pay $20 per ticket. What is the total amount, in dollars, from the sale of 70 member tickets and n nonmember tickets?

Choice K

$$=17\times70+20\times n$$


2010 April(68G) Math Question 5

Word Problems

- Cost

5. A coat originally priced at $80 is discounted to $60
What is the percent of discount on this coat?

Choice C

$$\frac{60}{80}=0.75$$


2010 April(68G) Math Question 6

Angles

- Collinear Points & Triangles

6.In△ABC,∠A and ∠ C are congruent, and the measure of∠B is 114°. What is the measure of ∠A?

Choice F

$$180-114=66$$ $$66\div2=33$$


2010 April(68G) Math Question 7

Probability

- One Event

7. Bella will pick 1 jelly bean at random out of a bag containing 28 jelly beans that are in the colors and quantities shown in the table below. Each of the jell beans is 1 color only.

Choice E

$$P=\frac{blue+yellow}{28}$$ $$=\frac{8+4}{28}$$ $$=\frac{12}{28}$$ $$=\frac{3}{7}$$


2010 April(68G) Math Question 8

Word Problems

- Distance, Speed, Time

8. What is an automobile's average speed, in miles per hour, if it travels 60 miles in 1½ hours?

Choice G

$$\frac{60}{1\frac{1}{2}}$$ $$=\frac{60}{\frac{3}{2}}$$ $$=60\times\frac{2}{3}$$ $$=40$$


2010 April(68G) Math Question 9

Statistics

- Mean, Median

9. To determine a student's overall test score for the semester, Ms. Ackerman deletes the lowest test score and calculates the average of the remaining test scores.
Niels took all 5 tests and earned the following test scores in Ms. Ackermans class this semester;
62,78,83,86and 93. What overall test score did Niels earn in Ms. Ackermans class this semester?

Choice D

$$62\ is\ the\ lowest\ score.$$ $$(78+83+86+93)\div4=85$$


Choice F

$$5y=m-n$$ $$y=\frac{m-n}{5}$$


Choice D

$$5+2x=19$$ $$2x=19-5=14$$ $$x=7$$ $$3x=21$$


Choice H

$$\frac{24}{4}=6$$ $$6\times6=36$$


Choice C

$$7\times10^{-2}\times4.58\times 10^{6}$$ $$=32.06\times10^{4}$$ $$=320600$$


2010 April(68G) Math Question 14

Quadratic Equations

- Factoring

14. Which of the following expressions is a factor of the expression x²-6x +8?

Choice G

$$=(x-4)(x-2)$$


Choice D

$$c\lt d,\ e\lt c,\ e\gt b,\ b\gt a$$ $$\Rightarrow a\lt b \lt e\lt c \lt d$$


Choice F

$$=\frac{2\times4+1\times6}{2\times1-4\times6}$$ $$=\frac{8+6}{2-24}$$ $$=\frac{14}{-22}$$ $$=\frac{7}{-11}$$


2010 April(68G) Math Question 17

Word Problems

- 1 Variable Equations

17.A high school band needs to make 3 sizes of flag small, medium, and large-for an upcoming halftime show. The 3 sizes of flags are made by combining I-foot squares of material that come in 2 colors Four band members have agreed to make the flags. The information they will need is given to them in the tables below. The table with colors tells how many I-foot squares of each color are needed for each size flag, and the table with names tells how many flags of h size that each person is to make.
How many l-foot squares of blue material does Lance need to make his flags?

Choice B

$$red:\ r,\ blue:\ b$$ $$Lance=3\ medium+3\ large$$ $$medium=5r+3b$$ $$large=10r+5b$$ $$Lance=3(5r+3b)+3(10r+5b)$$ $$=15r+9b+30r+15b$$ $$=45r+24b$$ $$\Rightarrow 24\ blue$$


Choice J

$$80=2^{4}\times5$$ $$70=2\times5\times7$$ $$30=2\times3\times5$$ $$[L.C.M]=2^{4}\times3\times5\times7=1680$$


2010 April(68G) Math Question 19

Trigonometry

- Pythagorean Theorem & sin cos tan

19. The figure below shows Mark's, Cals, and Lakecia's houses at the 3 vertices of the right triangle formed by
3 roads. The distances given in the figure are in miles Mark and Lakecia each make a trip from Mark's house to Lakecia's house. Mark takes Grapevine Road.
Lakecia takes Harbor Road to pick up Cal, and then takes Lott Road. How many miles shorter is Mark,'s trip than Lakecia's trip?

Choice B

$$It's\ a\ 5-12-13\ triangle.$$ $$(5+12)-13=4$$


Choice K

$$(a^{8})^{24}$$ $$=a^{8\times24}$$ $$=a^{192}$$


Choice D

$$h(-3)$$ $$=4(-3)^{2}-3(-3)$$ $$=36+9$$ $$=45$$


2010 April(68G) Math Question 22

Word Problems

- Cost

22. The Carousel Clothes Shop is advertising a sale featuring 30% off the marked price on any item. which of the following gives the sale price, in dollars, of an item with a marked price of p dollars?

Choice K

$$\Rightarrow p-0.3p$$


Choice B

$$1.29\times\frac{2}{5}=0.516$$


2010 April(68G) Math Question 24

Graphing Properties

- Create Graphs or Figures

24. Johnny is selling lemonade. He paid $5.00 for his supplies and charges $0.50 per glass of lemonade.
Johnny's profit is found by subtracting his expenses from his income. Which of the following graphs represents his profit as a function of the number of glasses of lemonade he sells?

Choice K

Its cost is $5, so …


2010 April(68G) Math Question 25

Probability

- Factorials

25. The positive integer n! is defined as the product of all the positive integers less than or equal to n. For example,3!=1(2)(3)=6. What is the value of the expression  6!/ 3!2! ?

Choice D

$$6!=1\times2\times3\times4\times5\times6$$ $$3!=1\times2\times3$$ $$2!=1\times2$$ $$\Rightarrow \frac{1\times2\times3\times4\times5\times6}{1\times2\times3\times1\times2}$$ …


2010 April(68G) Math Question 26

Word Problems

- 2 Variable Addition

26. Ms. Johnson purchased 1,000 prizes for the school carnival for $90. Each prize costs either $0.05
$0. 25. How many of the less expensive prizes did she buy?

Choice K

$$x+y=1000---(1)$$ $$0.05x+0.25y=90---(2)$$ $$(2)\times20$$ $$\Rightarrow x+5y=1800---(3)$$ …


2010 April(68G) Math Question 27

Perimeter, Area, Volume

- Mixed Shapes

27. A circle with a radius of 1 inch is inscribed in a square as shown below.
What is the area of the shaded region, in square inches?

Choice C

68g

$$\Box= 2\times2=4$$ $$4-1\times1\times\pi$$ $$=4-\pi$$


Choice K

$$4:7=20:x$$ $$4x=140$$ $$x=35 $$


2010 April(68G) Math Question 29

Trigonometry

- Pythagorean Theorem & sin cos tan

29. For the polygon below, which of the following represents the length, in meters, of AE?

Choice C

68g

$$\frac{20}{x}=\tan 50°$$ $$x=\frac{20}{\tan 50°}$$


2010 April(68G) Math Question 30

Perimeter, Area, Volume

- Other Shapes

30. The perimeter of a parallelogram is 76 inches, and 1 side measures 14 inches. If it can be determined, what are the lengths. in inches, of the other 3 sides?

Choice H

It's the definition of a …


2010 April(68G) Math Question 31

Inequalities

- Inequality Expressions

31. Which of the following is a solution statement for the inequality 3x-7 < 5+9x?

Choice A

$$3x-7 \lt 5+9x $$ $$\Rightarrow …


Choice H

$$2\pi r$$ $$=2\times3.14\times4$$ $$=25.12$$ $$\approx …


2010 April(68G) Math Question 33

Absolute Values

- Absolute Value Expressions

33.-5  | -8 + 9 |=?

Choice B

$$-5\mid-8+9\mid$$ $$=-5(1)$$ $$=-5$$


2010 April(68G) Math Question 34

Sequences and Patterns

- Logical Expressions

34. In a large high school, some teachers teach only I subject, and some teachers teach more than 1 subject Using the information given in the table below about the math, science, and gym teachers in the school, how many teachers teach math only?

Choice H

68g

$$x=(at\ least\ 1\ class\ of\ …


Choice D

$$Marigolds:\ 60\times1+40\times4+?\times6$$ $$=60+160+6?$$ $$Petunias:\ 25\times4+100\times6$$ …


Choice K

$$1\times60\times\frac{1}{10}=6$$ $$6\times x=15$$ $$x=2.5$$


Choice B

$$1\times30=30$$ $$30=4\times7...2$$


2010 April(68G) Math Question 38

Graphing Properties

- Interpret Graphs or Figures

Use the following information to answer questions 38-40.
Parallelogram WXYZ is shown in the standard (x, y ) coordinate plane below. The coordinates for 3 of its vertices are W(0, 0),  X (a , b), and  Z (c , 0).
38. What are the coordinates of y?

Choice K

68g

$$Y=(c+a,b)$$


2010 April(68G) Math Question 39

Angles

- Other Shapes

39. The measure of ∠y is 50°.
 what is the measure of the angle between WX and the y-axis?

Choice B

68g

$$x=90°-50°=40°$$


Choice F

68g


2010 April(68G) Math Question 41

Perimeter, Area, Volume

- Other Shapes

41. A liter is 1,000 cubic centimeters. Which of the following is closest to the height, in centimeters, of a cylindrical container, shown below, with diameter
10 cm and capacity 1 liter?
(Note: The volume of a cylinder with radius r and height h is x r^2 h)

Choice D

$$r=\frac{10}{2}=5$$ $$V=\pi r^{2}h$$ $$1000= 3.14\times5^{2}\times …


Choice J

$$1-\frac{1}{12}\times3=\frac{9}{12}$$ $$\frac{9}{12}\times\frac{1}{5}=\frac{3}{20}$$


2010 April(68G) Math Question 43

Quadratic Equations

- Factoring

43. Which of the following equations shows a correct use of the quadratic formula to solve x-5x+3=0?

Choice A

$$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$ $$\Rightarrow x=\frac{5\pm\sqrt{25-4(1)(3)}}{2(1)}$$


2010 April(68G) Math Question 44

Equation of a Line

- Distance & Midpoint

44. In the standard (x, y) coordinate plane, the point (1, -6)
is the midpoint of the line segment with endpoints
( 9, -13)and (a, b).  What is (a , b)?

Choice F

$$\frac{(9,-13)+(a,b)}{2}=(1,-6)$$ $$9+a=1\times2$$ $$\Rightarrow a=-7$$ $$-13+b=-6\times2$$ …


2010 April(68G) Math Question 45

Trigonometry

- Pythagorean Theorem & sin cos tan

45. A straight 10-foot-tall ladder is leaning against a house at an angle of 75, as shown in the figure below.
Which of the following expressions gives the distance, in feet, the base of the ladder is from the house along the level ground?

Choice B

68g

$$\cos 75°=\frac{x}{10}$$ $$x=10\cos 75°$$


2010 April(68G) Math Question 46

Graphing Properties

- Interpret Graphs or Figures

46. One of the following equations is graphed in the standard(x,y) coordinate plane below. Which one?

Choice K

68g

The intercept with the Y …


2010 April(68G) Math Question 47

Perimeter, Area, Volume

- Other Shapes

47. The vertices of a trapezoid have the (x,y)coordinates indicated in the figure below. What is the area, in square coordinate units, of the trapezoid?

Choice D

68g

$$(18+10)\times3\times\frac{1}{2}=42$$


Choice G

$$y=mx+b$$ $$C=0.99\times H+0$$ $$m=0.99$$ $$0.99\ …


2010 April(68G) Math Question 49

Sequences and Patterns

- Geometric Sequences

49. The first 3 terms of a geometric sequence are
4,10, and 25. What is the next term in the sequence?

Choice D

$$\frac{10}{4}=2.5$$ $$\frac{25}{10}=2.5$$ $$25\times2.5=62.5$$


Choice J

$$64=4\times4\times4$$ $$4\times4\times6=96$$


2010 April(68G) Math Question 51

Angles

- Circles

51. In the figure below, a sector is shown shaded in a circle with radius 4 decimeters. The length of the are of the unshaded sector is 7r decimeters, What is the measure of the central angle of the shaded sector ?

Choice C

$$2\pi r=2\pi(4)=8\pi$$ $$8\pi-7\pi=\pi$$ $$\frac{\pi}{8\pi}\times360°=45°$$


2010 April(68G) Math Question 52

Conics

- Circles

52. In the standard (x, y) coordinate plane, the graph of which of the following equations is a circle with center(4, -2)and radius 3 coordinate units?

Choice K

The general equation for a …


2010 April(68G) Math Question 53

Trigonometry

- Advanced sin, cos, tan functions

53.In△ABC, the measure of ∠A is47°, the measure of ZB is 76, and the length of BC is 18 centimeters Which of the following is an expression for the length, in centimeters, of Ac Note: The law of sines states that for any triangle, the ratios of the lengths of the sides to the sines of the angles opposite those sides are equal .)

Choice D

68g

$$\frac{18}{\sin 47°}=\frac{x}{\sin 76°}$$ $$x=\frac{18\sin 76°}{\sin …


2010 April(68G) Math Question 54

Equation of a Line

- Distance & Midpoint


Choice F

68g

$$\sqrt{2}(l+x)-\sqrt{2}x$$ $$=\sqrt{2}l$$


2010 April(68G) Math Question 55

Absolute Values

- Absolute Value Expressions

55.If x ≤ 2,  then  | x-2 |=?

Choice E

$$If\ x\leq2...$$ $$Let\ x=1$$ $$\mid …


2010 April(68G) Math Question 56

Sequences and Patterns

- Logical Expressions

56. There are 25 buildings on Elm Street. Of these
25 buildings, 10 have fewer than 6 rooms, 10 have more than 7 rooms, and 4 have more than 8 rooms What is the total number of buildings on Elm Street that have 6, 7 or 8 rooms?

Choice H

68g

$$More\ or\ equal\ to\ 6=25-(fewer\ …


2010 April(68G) Math Question 57

Trigonometry

- Advanced sin, cos, tan functions

57. If cos x=-1/3 , what is the value of cos 2x?
(Note:(cos x²)= (1+cos2x) / 2 )

Choice B

$$\cos x=-\frac{1}{3}$$ $$(\cos x)^{2}=\frac{1+\cos 2x}{2}$$ …


Choice F

$$y=f(g(x))$$ $$\Rightarrow 6=f(g(4))$$ $$g(4)=10(4)+b=40+b$$ $$f(40+b)=\sqrt{40+b}=6=\sqrt{36}$$ …


Choice A

68g

$$\Rightarrow10\times10=100$$


2010 April(68G) Math Question 60

Inequalities

- Inequality Expressions

60. When a, b, and c are real numbers and a b^2 c^4 >0, which of the following must be greater than 0?

Choice F

$$a\gt 0$$ $$b^{2}\gt 0$$ $$c^{4}\gt …