Full-Length ACT Math Practice Test-Answers and Explanations

Full-Length ACT Math Practice Test-Answers and Explanations: work through realistic questions in the same format and difficulty as the real test, then check every answer against a full step-by-step solution so you can see where you lost marks and what to review next.

The ladder is the hypotenuse. Therefore, the ladder is 40 ft.

19- Choice E is correct
The percent of girls playing tennis is: (45 % × 30 % = 0.45 × 0.30 = 0.135 = 13.5 %)

20- Choice C is correct
Probability (= frac{Number space of space favorable space outcomes}{Number space of space possible space outcomes}=frac{20}{45}=frac{4}{9})

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linder (= 2πr (r + h)), The radius of the cylinder is (5 (10 ÷ 2)) inches and its height is 14 inches. Therefore, Surface Area of a cylinder (= 2π (5) (5 + 14) = 190 π)

28- Choice C is correct
Five years ago, Amy was four times as old as Mike. Mike is 12 years now. Therefore, 5 years ago Mike was 7 years. Five years ago, Amy was: A(=4×7=28), Now Amy is 33 years old:( 28 + 5 = 33)

29- Choice D is correct
(7%) of the volume of the solution is alcohol. Let (x) be the volume of the solution.
Then: (7%) of (x =35) ml ⇒ (0.07 x = 35 ⇒ x = 35 ÷ 0.07 = 500)

30- Choice C is correct
I. (|a|<0.5→-0.5<a<0.5), Multiply all sides by b. Since, (b>0→-0.5b<ba<0.5b) (it is true!)
II. Since, (-0.5<a<0.5,) and (a<0 →-a(0.5)>a^2>a(0.5)) (plug in (-frac{1}{3}), and check!) (It’s NOT true)
III. (-0.5<a<0.5), multiply all sides by (3), then: (-1.5<3a<1.5)
Subtract 2 from all sides. Then: (-1.5-2<3a-2<1.5-2→-3.5<3a-2<-0.5) (It is true!)

31- Choice E is correct
The amplitude in the graph of the equation (y=acosbx) is (a). ((a) and (b) are constant)
In the equation (y=cosx), the amplitude is 3 and the period of the graph is (2π).
The only option that has three times the amplitude of graph (y = cos x) is (y=5+3 space cos x)
They both have the amplitude of 3 and period of (2π).

32- Choice C is correct
((x-4)^3=64→x-4=4→x=8, →(x+2)(x-8)=(8+2)(8-8)=0)

33- Choice A is correct
We know that: (i=sqrt{-1}⇒i^2=-1)
(frac{-3+4i}{1-3i}=frac{(-3+4i)(1+3i)}{1-9i^2 }
=frac{-3-9i+4i+12i^2}{10}=frac{-15}{10}-frac{5}{10} i=-frac{3}{2}-frac{1}{2} i)

34- Choice E is correct
(tan=frac{opposite}{adjacent}),
(tanθ=frac{4}{9})⇒ we have the following right triangle.

Then: (c=sqrt{4^2+9^2}=sqrt{16+81}=sqrt{97})
(cosθ=frac{adjacent}{hypotenuse}=frac{9}{sqrt{97}})

35- Choice B is correct.
Solve for (x).
(frac{10x}{12}=frac{2x-1}{3}). Multiply the second fraction by 4.
(frac{10x}{12}=frac{4(2x-1)}{4×3})
Two denominators are equal. Therefore, the numerators must be equal.
(10x=8x-4,2x=-4, -2=x)

36- Choice D is correct
(frac{4}{3}≅1.33), (frac{5}{8}≅0.625), (frac{3}{7}≅0.43), (frac{9}{15}=0.6)

37- Choice A is correct
ratio of A: (frac{400}{430}=0.93)
ratio of B: (frac{680}{720}=0.94)
ratio of C: (frac{600}{650}=0.93)
ratio of D: (frac{740}{800}=0.925)

38- Choice E is correct
First find percentage of men in city B and percentage of women in city D.
Percentage of men in city B (=frac{720}{1,400}) and percentage of women in city D (=frac{740}{1,540})
Find the ratio and simplify. (frac{frac{720}{1,400}}{frac{740}{1,540}}=frac{198}{185}=1.07)

39- Choice A is correct
(frac{600+x}{650}=1.3→600+x=845→x=245)

40- Choice B is correct
Use the information provided in the question to draw the shape. Use Pythagorean Theorem:

(a^2 + b^2 = c^2⇒ 50 + 120 = c^2 ⇒ 2500 + 14400 = c^2 ⇒ 16900 = c^2 ⇒ c = 130 space km)

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41- Choice A is correct
We write the numbers in the order: 12, 14, 14, 18, 22, 36, 44, 52
The mode of numbers is: 14 median is: 20

42- Choice D is correct
(0.2x=(0.05)×60→x=15→(x-3)^2=(12)^2=144)

43- Choice B is correct
The slop of line A is: (m=frac{y_2-y_1}{x_2-x_1}=frac{2-(-1)}{2-5}=-1)
Parallel lines have the same slope and only choice E ((y=-x)) has slope of (-1).

44- Choice D is correct
Replace (z) by (frac{z}{10}) and simplify.
(x_1=frac{2y+frac{2r}{r+1}}{frac{5}{frac{z}{10}}}=
frac{2y+frac{2r}{r+1}}{frac{10×5}{z}}
=frac{2y+frac{2r}{r+1}}{10×frac{5}{z}}=frac{1}{10}×frac{2y+frac{2r}{r+1}}{frac{5}{z}}=frac{x}{10})
When (z) is divided by 10, (x) is also divided by 10.

45- Choice C is correct
Let (x) be the number of years. Therefore, ($3,000) per year equals (3,000x).
starting from ($15,000) annual salary means you should add that amount to (3,000x).
Income more than that is: (I > 3,000 x + 15,000)

46- Choice D is correct
The weight of 25 meters of this rope is: 25 × 400 g = 10,000 g
1 kg = 1,000 g, therefore, 10,000 g ÷ 1,000 = 10 kg

47- Choice D is correct.
To solve for (f(4g(P))), first, find (4g(p))
(g(x)=log_5 x, g(p)=log_5 p, 4g(p)=4log_5 p=log_5 p^4)
Now, find (f(4g(p))): (f(x)=5^x⇒f(log_5 p^4 )=5^{log_5 p^4 })
Logarithms and exponential with the same base cancel each other. This is true because logarithms and exponential are inverse operations. Then: (f(log_5 p^4 )=5^{log_5 p^4 }=p^4)

48- Choice A is correct
Set of number that are not composite between 5 and 25: A= { 5, 7, 11, 13, 17, 19, 23}
Probability (= frac{number space of space desired space outcomes}{number space of space total space outcomes}=frac{7}{20})

49- Choice E is correct
Check each choice provided:
A. (4→ frac{6+9+7+9+10}{5}=frac{41}{5}=8.2)
B. (6→ frac{4+9+7+9+10}{5}=frac{37}{5}=7.8)
C. (7→ frac{4+6+9+9+10}{5}=frac{36}{5}=7.6)
D. (9→ frac{4+6+9+7+10}{5}=frac{30}{5}=7.2)
E. (10→ frac{4+6+9+7+9}{5}=frac{35}{5}=7)

50- Choice A is correct
Based on corresponding members from two matrices, we get:
(begin{cases}5x=2x+y-22x=2y+4)end{cases}→begin{cases}-3x+y=22x-2y=4end{cases})
⇒ Multiply first equation by 2⇒(begin{cases}-6x+2y=42x-2y=4end{cases})→ add two equations.
(-4x=8→x=-2). Replace (x) in second equation. (2(-2)-2y=4→-4-2y=4→y=-4.xy=(-2)(-4)=8)

51- Choice C is correct
(tangent space β= frac{1}{cotangent space β}=frac{1}{2})

52- Choice C is correct
((g + f)(x)= g(x)+f(x)=x^2-2x+6+4x-3=x^2+2x+3)

53- Choice D is correct.
Let the number be. Then: (10x=y% ×A)⇒ Solve for (A⇒ 10x=frac{y}{100}×A)
Multiply both sides by (frac{100}{y}: 10x×frac{100}{y}=frac{y}{100}×frac{100}{y}×A⇒ A=frac{1,000x}{y})

54- Choice B is correct
Simplify each choice provided.
A. (20-(4×10)+(6×30)=20-40+180=160)
B. (((frac{25}{2}+frac{30}{4})×(frac{32}{4}))-frac{8}{5}+frac{46}{10}=(12.5+7.5)×8+(frac{-16+46}{10})=160-3=157) (this is the answer)
C. ((frac{11}{8}×72)+(frac{125}{5})=99+25=124)
D. ((2×10)+(50×1.5)+15=20+75+15=110)
E. (frac{481}{6}+frac{121}{3}=frac{481+242}{6}=120.5)

55- Choice A is correct
(y = 8ab-5b^3)⇒ Plug in the values of a and b in the equation: (a=6) and (b=-2)
(y = 8ab-5b^3=8(6)(-2)-5(-2)^3=-96+40=-56)

56- Choice B is correct
The area of trapezoid is: ((frac{10+15}{2})×x=150→12.5x=150→x=12)
(y=sqrt{12^2+5^2}=13), Perimeter is: (12+10+13+5+10=50)

57- Choice E is correct
The area of ∆BED is 20, then: (frac{5×AB}{2}=20→5×AB=40→AB=8)
The area of ∆BDF is 32, then: (frac{4×BC}{2} =32→4×BC=64→BC=16)
The perimeter of the rectangle is (= 2×(8+16)=48)

58- Choice A is correct
When points are reflected over (y)-axis, the value of (y) in the coordinates doesn’t change and the sign of (x) changes. Therefore, the coordinates of point B is ((6,-12)).

59- Choice C is correct
Plug in each pair of number in the equation:
A. ((6, 1): 5(6)+3(1)=33≠6) Nope!
B. ((, 3, 3): 5(-3)+3(3)=-6≠6) Nope!
C. ((3, -3): 5(3)+3(-3)=6=6) Bingo!
D. ((2, 2): 5(2)+3(2)=16≠6) Nope!
E. ((2, 8): 5(2)+3(8)=34≠6) Nope!

60- Choice B is correct
(f(x)=2x^3+4x-18x^{-2}=2x^3+4x-frac{18}{x^2})
(g(x)=-3), then (2(-3)^3+4(-3)-frac{18}{(-3)^2} =-54-12-2=-68)

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Official source: the structure of the ACT, the sections it covers and the time allowed for each are published by ACT, Inc., the organization that writes and administers the test, at act.org.