ACT Form 67C Math Answer Explanation


2009 June(67C) Math Question 1

Word Problems Practice

- 2 Variable Addition

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 Practice

- Mean, Median

Choice J

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

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 Practice

- Cost

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 Practice

- FOIL

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 Practice

- Collinear Points & Triangles

Choice D

$$180° − 47° = 133°$$

2009 June(67C) Math Question 12

Probability Practice

- Combinations

Choice K

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

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

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

Choice B

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

Choice J

67c

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

Choice D

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

Choice J

67c

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

Choice E

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

Choice G

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

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

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

Choice G

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

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 Practice

- Inequality Expressions

Choice E

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

Choice K

67c

Only K is 3 units …

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 Practice

- One Event

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 Practice

- Circles

Choice F

The equation of a circle …

Choice B

2 semicircles make one circle, …

Choice G

67c

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

Choice C

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

Choice G

$$Area\ of\ one\ face\ of\ …

Choice B

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

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 Practice

- Parallel Lines

Choice C

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

2009 June(67C) Math Question 48

Angles Practice

- Circles

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 Practice

- Cost

Choice C

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

Choice J

The maximum profit is when …

2009 June(67C) Math Question 53

Matrices Practice

- Determinants

Choice E

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

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 Practice

- Inequality Expressions

Choice C

(A) Never true: A positive …

Choice J

67c

Set x as the base …

Choice E

Here it is the Law …

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 Practice

- Factoring

Choice C

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

2009 June(67C) Math Question 60

Inequalities Practice

- Inequality Expressions

Choice F

67c

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