ACT Form 67C Math Answer Explanation



2009 June(67C) Math Question 1

Word Problems

- 2 Variable Addition

1. The weekly fee for staying at the Pleasant Lake Campground is $20 per vehicle and $10 per person.
Last year, weekly fees were paid for v vehicles and p persons. Which of the following expressions gives the total amount, in dollars, collected for weekly fees last year?

Choice A

$$\Rightarrow $20\times v+$10\times p$$


Choice F

$$(9+5-(-6))(5+(-6))$$ $$=(20)(-1)$$ $$=-20$$


Choice E

$$1st\ machine:60\times8=480$$ $$2nd\ machine:80\times6=480$$ $$\Rightarrow 480+480=960$$


2009 June(67C) Math Question 4

Statistics

- Mean, Median

4. Marlon is bowling in a tournament and has the highest average after 5 games, with scores of 210, 225, 254, 231,and 280. In order to maintain this exact average.
what must be Marlon's score for his 6th game?

Choice J

$$210+225+254+231+280=1200$$ $$1200\div5=240$$ $$(1200+240)\div6=240$$


2009 June(67C) Math Question 5

Word Problems

- 1 Variable Equations

5. Joelle earns her regular pay of $7.50 per hour for up
40 hours of work in a week. For each hour over
40 hours of work in a week, Joelle is paid I3 times her regular pay. How much does Joelle earn for a week in which she works 42 hours?

Choice C

$$40\times7.5+2\times7.5\times1.5$$ $$=300+22.5$$ $$=322.5$$


Choice K

67c

$$\Rightarrow x^{2}=39+10x$$


Choice E

$$9(x-9)=-11$$ $$\Rightarrow 9x-81=-11$$ $$\Rightarrow 9x=70$$ $$\Rightarrow x=\frac{70}{9}$$


2009 June(67C) Math Question 8

Word Problems

- Cost

8. Discount tickets to a basketball tournament sell for
$4.00 each. Enrico spent S60.00 on discount tickets S37.50 less than if he had bought the tickets at the regular price. What was the regular ticket price?

Choice H

$$60\div4=15$$ $$15\times x=60+37.5$$ $$15x=97.5$$ $$x=6.5$$


2009 June(67C) Math Question 9

Quadratic Equations

- FOIL

9. The expression(3x-4y)(3x+ 4y)is equivalent to

Choice A

$$=(3x)(3x+4y^{2})-(4y^{2})(3x+4y^{2})$$ $$=(9x^{2})+(12xy^{2})-(12xy^{2})-(16y^{4})$$ $$=9x^{2}-(16y^{4})$$


Choice J

67c

$$WL=32$$ $$2(W+L)=24$$ $$W+L=12$$ $$W=4,L=8$$


2009 June(67C) Math Question 11

Angles

- Collinear Points & Triangles

11.In △ABC, the sum of the measures of ∠A and ∠B is
47°. What is the measure of ∠C?

Choice D

$$180° − 47° = 133°$$


2009 June(67C) Math Question 12

Probability

- Combinations

12. In the school cafeteria. students choose their lunch from 3 sandwiches, 3 soups, 4 salads, and 2 drinks How many different lunches are possible for a student who chooses exactly I sandwich. I soup, I salad, and I drink?

Choice K

$$3\times3\times4\times2=72$$


2009 June(67C) Math Question 13

Sequences and Patterns

- General Sequences

13. For 2 consecutive integers, the result of adding the smaller integer and triple the larger integer is 79. what are the 2 integers?

Choice B

Consecutive integers can be represented as (n, n + 1).

$$n+ 3 n + 1 = 79$$ $$4n + 3 = 79$$ $$n = 19$$


Choice F

$$f(-3)=-8(-3)^{2}$$ $$=-8(9)$$ $$=-72$$


Choice C

$$3^{x}=54$$ $$3^{3}=27$$ $$3^{4}=81$$ $$\Rightarrow 3^{3}\lt 3^{x}\lt 3^{4}$$ $$\Rightarrow 3\lt {x}\lt {4}$$


Choice J

$$70=7\times10$$ $$60=6\times10$$ $$50=5\times10$$ $$7\times6\times5\times10=2100$$


2009 June(67C) Math Question 17

Perimeter, Area, Volume

- Other Shapes

17. Hot Shot Electronics is designing a packing box for its new line of Acoustical Odyssey speakers. The box is a rectangular prism of length 45 centimeters, width 30 centimeters, and volume 81.000 cubic centimeters What is the height, in centimeters, of the box?

Choice B

$$Volume\ of\ a\ box\ =81,000$$ $$81,000=45\times30\times h$$ $$h=60$$


2009 June(67C) Math Question 18

Sequences and Patterns

- General Sequences

18. Four points, A, B, C, and D, lie on a circle having a circumference of 15 units, B is 2 units counterclockwise from A. C is 5 units clockwise from A. D is 7 units clock wise from A and 8 units counterclockwise from A What is the order of the points, starting with A and going clockwise around the circle?

Choice J

67c

$$\Rightarrow A,C,D,B$$


Choice D

$$t=5$$ $$y=16(2)^{5}=512$$


2009 June(67C) Math Question 20

Perimeter, Area, Volume

- Mixed Shapes

20. The length of a rectangle is 3 times the length of a smaller rectangle. The 2 rectangles have the same width. The area of the smaller rectangle is A square units. Which of the following is the value of? square units. The area of the larger rectangle is kA

Choice J

67c

$$Smaller\ Rectangle=l\cdot w=A$$ $$Larger\ Rectangle=3l \cdot w =kA$$ $$k=3$$


Choice E

$$=(a-4a)+(2b-6b)+(3c+5c)$$ $$=-3a-4b+8c$$


2009 June(67C) Math Question 22

Trigonometry

- Pythagorean Theorem & sin cos tan

22. The dimensions of the right triangle shown below are given in feet. What is sin θ ?

Choice G

$$\sin\theta=\frac{Opp}{Hyp}$$ $$\sin\theta=\frac{a}{c}$$


2009 June(67C) Math Question 23

Sequences and Patterns

- General Sequences

23. In a basketball passing drill, 5 basketball players stand evenly spaced around a circle. The player with the ball
(the passer) passes it to another player (the receiver)。
The receiver cannot be the player to the passer's immediate right or left and cannot be the player who last passed the ball. A designated player begins the drill as the first passer. This player will be the receiver for the first time on which pass of the ball .

Choice B

67c

Let five players be A, …


Choice H

$$y=0.12x+3000$$ $$\Rightarrow m=0.12$$ $$0.12+0.1=0.22$$


Choice A

$$=-8(7)(x^{3+6})+(-8)(-3)(x^{3+5})$$ $$=-56x^{9}+24x^{8}$$


2009 June(67C) Math Question 26

Absolute Values

- Absolute Value Expressions

26.  - 3 | -6+8 |=?

Choice G

$$=-3\mid2\mid$$ $$=-3(2)$$ $$=-6$$


2009 June(67C) Math Question 27

Trigonometry

- Pythagorean Theorem & sin cos tan

27. In right triangle △ACE below, BD is parallel to AE and BD is perpendicular to EC at D. The length of AC is 20 feet. the length of BD is 3 feet, and the length of CD is 4 feet. What is the length, in feet, of AE?

Choice B

$$3-4-5\ triangle \Rightarrow \overline{BC}=5$$ $$\overline{AC}:\overline{BC}=20:5=4:1$$ …


Choice H

“Constant rate.” - This is …


2009 June(67C) Math Question 29

Inequalities

- Inequality Expressions

29. The inequality 6(x+ 2)>7(x-5) is equivalent to which of the following inequalities?

Choice E

$$\Rightarrow 6x+12 \gt 7x-35$$ $$\Rightarrow …


Choice K

67c

Only K is 3 units …


2009 June(67C) Math Question 31

Trigonometry

- Pythagorean Theorem & sin cos tan

31. For △FGH, shown below, which of the following is an expression for y in terms of x ?

Choice E

$$A^{2}+B^{2}=C^{2}$$ $$y^{2}=x^{2}+4^{2}$$ $$y=\sqrt{x^{2}+16}$$


2009 June(67C) Math Question 32

Probability

- One Event

32. A bag contains 12 red marbles, 5 yellow marbles, and
15 green marbles. How many additional red marbles must be added to the 32 marbles already in the bag so that the probability of randomly drawing a red marble is 3/5 ?

Choice G

$$\frac{12+x}{32+x}=\frac{3}{5}$$ $$5(12+x)=3(32+x)$$ $$60+5x=96+3x$$ $$2x=36$$ $$x=18$$


Choice D

67c

$$4x-2y=8$$ $$\Rightarrow 2x-y=4$$ $$(x,y)\Rightarrow (0,-4),(2,0)$$


Choice F

$$2a=-5(1)^{2}+9$$ $$2a=-5+9=4$$ $$a=2$$


Choice D

$$\frac{1}{2}S+\frac{1}{3}S+xS=S$$ $$\Rightarrow x=\frac{1}{6}$$ $$\frac{1}{2}:\frac{1}{3}:\frac{1}{6}=3:2:1$$


2009 June(67C) Math Question 36

Conics

- Circles

36. A particular circle in the standard (x, y) coordinate plane has an equation of (x-5)+y=38. What are the radius of the circle, in coordinate units, and the coordinates of the center of the circle?

Choice F

The equation of a circle …


2009 June(67C) Math Question 37

Perimeter, Area, Volume

- Mixed Shapes

37. The figure below consists of a square and
2 semicircles, with dimensions as shown. What is the outside perimeter, in centimeters, of the figure?

Choice B

2 semicircles make one circle, …


2009 June(67C) Math Question 38

Perimeter, Area, Volume

- Mixed Shapes

38. In the figure below, points E and F are the midpoints of sides AD and BC of rectangle ABCD, point G is the intersection of AF and BE, and point H is the intersection of CE and DF, The interior of ABCD except for the interior of EGFH is shaded. What is the ratio of the area of EGFH to the area of the shaded region?

Choice G

67c

$$\triangle EGF=\triangle EHF=\triangle BAG=\triangle CDH$$


2009 June(67C) Math Question 39

Equation of a Line

- Distance & Midpoint

39. The coordinates of the endpoints of CD, in the standard (x, y) coordinate plane, are (-4, -2)and(14,2) What is the x-coordinate of the midpoint of CD?

Choice C

$$The\ midpoint$$ $$=(\frac{(-4+14)}{2},\frac{,(-2+2)}{2})$$ $$=(5,0)$$


Choice G

$$Area\ of\ one\ face\ of\ …


Choice B

$$ay+bx=x$$ $$ay$$ $$\Rightarrow y=-\frac{b}{a}+\frac{c}{a},\ m=-\frac{b}{a}$$ …


2009 June(67C) Math Question 42

Trigonometry

- Pythagorean Theorem & sin cos tan

42. According to the measurements given in the figure below, which of the following expressions gives the distance, in miles, from the boat to the dock?

Choice F

67c

From the angle of 52, …


Choice D

Odd- in age: not in …


Choice H

67c

“Lines of symmetry” are lines …


Choice A

$$Diameter=2, so radius=1$$ $$Area=\pi r …


Choice J

Diameter=2, 75% longer

$$2\times1.75=3.5$$


2009 June(67C) Math Question 47

Angles

- Parallel Lines

47. In the figure below,  AB‖CD . AE bisects ∠BAC, and CE bisects ∠ACD. If the measure of ∠ BAC is 82°, what is the measure of ∠AEC?

Choice C

$$\angle BAC=82°$$ $$\angle ACD=180°-82°=98°$$ $$\angle …


2009 June(67C) Math Question 48

Angles

- Circles

48. In the circle shown below, chords TR and QS intersect at P. which is the center of the circle and the measure of ∠PST is 30°. What is the degree measure of minor arc RS?

Choice H

67c

Arc angle is twice the …


Choice B

Infinite solutions means “same line.” …


Choice F

67c


2009 June(67C) Math Question 51

Word Problems

- Cost

51. For every hour that Marcia spends making frames in the second week of December each year. she donates s3 from that week's profit to a local charity. This year.
Marcia made 4 large frames and 2 small frames in that week. Which of the following is closest to the percent of that week's profit Marcia donated to the charity?

Choice C

$$"Every\ hour"=$3$$ $$ Large=3hrs\times4L=12$$ $$ …


Choice J

The maximum profit is when …


2009 June(67C) Math Question 53

Matrices

- Determinants

53. The determinant of a matrix [ c d ] equals  ad - cb 
What must be the value of x for the matrix [    ] to
have a determinant of -16?

Choice E

$$ab-cb$$ $$x\cdot x-8x=-16$$ $$x^{2}-8x+16=0$$ $$(x-4)(x-4)or(x-4)^{2}=0$$ …


2009 June(67C) Math Question 54

Equation of a Line

- Interpret Graphs or Figures

54. A formula for finding the value, A dollars, of P dollars invested at i% interest compounded annually for n years is A= P(1+0.01i)^n。 Which of the following is an expression for P in terms of i, n, and A?

Choice K

$$A=P(1+0.01i)^{n}$$ $$\Rightarrow\frac{A}{(1+0.01i)^{n}}=\frac{P(1+0.01i)^{n}}{(1+0.01i)^{n}}$$ $$\Rightarrow P=\frac{A}{(1+0.01i)^{n}}$$


2009 June(67C) Math Question 55

Inequalities

- Inequality Expressions

55. If x and y are real numbers such that  x > 1  and  y < -1  , then which of the following inequalities must be true?

Choice C

(A) Never true: A positive …


2009 June(67C) Math Question 56

Perimeter, Area, Volume

- Mixed Shapes

56. Triangles △ABC and △POR are shown below. The given side lengths are in centimeters. The area of  △ABC is 30 square centimeters.  What is the area of  △PQR, in square centimeters?

Choice J

67c

Set x as the base …


2009 June(67C) Math Question 57

Trigonometry

- Advanced sin, cos, tan functions

57. Triangle △ABC is shown in the figure below. The measure of ∠A is 40°, AB=18cm, and AC=12cm.
Which of the following is the length, in centimeters, of BC ?
(Note: For a triangle with sides of length a, b, and c opposite  angles ∠A, ∠B, and ∠C, respectively, the law
of sines states sin ∠A / a = sin ∠B / b = sin ∠C/c and the law of cosines states 
c^2 = a^2 + b^2 - 2ab cos ∠C.)

Choice E

Here it is the Law …


2009 June(67C) Math Question 58

Sequences and Patterns

- Arithmetic Sequences

58. What is the sum of the first 4 terms of the arithmetic sequence in which the 6th term is 8 and the 10th term is13?

Choice G

67c

$$\frac{13-8}{4}=\frac{5}{4}$$ $$8=a_{1}+5(\frac{5}{4})$$ $$a_{1}=1.75$$ $$a_{2}=1.75+1.25$$ $$a_{3}=1.75+1.25+1.25$$ …


2009 June(67C) Math Question 59

Quadratic Equations

- Factoring

59. In the equation x² + mx + n=0,  m and n are integers The only possible value for x is -3 . What is the value of m ?

Choice C

$$(x+3)^{2}=0$$ $$\Rightarrow x^{2}+6x+9=0$$ $$\Rightarrow m=6$$


2009 June(67C) Math Question 60

Inequalities

- Inequality Expressions

60. The solution set of which of the following equations is the set of real numbers that are 5 units from -3 ?

Choice F

67c

$$\mid x-(-3)\mid=5$$ $$(x+3)=-5 $$ $$(x+3)=5$$ …