Full-Length ACT Math Practice Test-Answers and Explanations

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ACT Math Practice Test Answers and Explanations

1- Choice B is correct
Write the numbers in order: 10, 12, 14, 19, 23, 30, 32
Since we have 7 numbers (7 is odd), then the median is the number in the middle, which is 19.

2- Choice D is correct.
1,000 times the number is 5.08. Let \(x\) be the number, then: \(1,000x=5.08\), \(x=\frac{5.08}{1,000}=0.00508\)

3- Choice D is correct.
Let’s review the options provided.
4. In 4 years, David will be 29 and Ava will be 12. 29 is not 2 times 12.
6. In 6 years, David will be 31 and Ava will be 14. 31 is not 2 times 14!
8. In 7 years, David will be 32 and Ava will be 15. 32 is not 2 times 15.
10. In 9 years, David will be 34 and Ava will be 17. 34 is 2 times 17.
14. In 11 years, David will be 36 and Ava will be 19. 36 is not 2 times 19.

4- Choice C is correct
The area of the floor is: 7 cm \(×\) 20 cm = 140 cm, The number is tiles needed \(= 140 ÷ 8.75 = 16\)

5- Choice C is correct
To find the discount, multiply the number by (\(100\%\) – rate of discount).
Therefore, for the first discount we get: \((D)(100\% – 18\%) = (D) (0.82) = 0.82 D\)
For increase of \(12\%\): \((0.82 D) (100\% + 12\%) = (0.82 D) (1.12) = 0.9184 D = 91.84\%\) of \(D\)

6- Choice D is correct
Solve the system of equations by elimination method.
\(\begin{cases}5x-4y= -2\\2x+2y=10\end{cases}\)
Multiply the second equation by \(2\), then add it to the first equation.
\(\begin{cases}5x-4y= -2\\2(2x+2y=10)\end{cases} ⇒\begin{cases}5x-4y= -2\\4x+4y=20\end{cases}\)
⇒ add the equations \(9x=18⇒x=2\).Replace \(x\) to one of equations. \(2x+2y=10→2(2)+2y=10→2y=6→y=3\)

7- Choice E is correct
The diagonal of the square is 15. Let \(x\) be the side.
Use Pythagorean Theorem: \(a^2 + b^2 = c^2\)
\(x^2 + x^2 = 152 ⇒ 2x^2 = 152 ⇒ 2x^2 = 225 ⇒x^2 = 112.5 ⇒x= \sqrt{112.5}\)
The area of the square is: \(\sqrt{112.5} × \sqrt{112.5} = 112.5\)

8- Choice D is correct
\(x=36+112=148\)

9- Choice E is correct.
By definition, the sine of an acute angle is equal to the cosine of its complement.
Since angle A and B are complementary angles, therefore: sin A = cos B

10- Choice A is correct
Employer’s revenue: \(0.04x+11000\)

11- Choice E is correct
\(|x+8|≤5→-5≤x+8≤5→-13≤x≤-3\)

12- Choice B is correct
According to picture \(x+48=112→x=112-48=64\)

13- Choice B is correct
Check each option.
A. \(\frac{1}{6}> 0.2→ \frac{1}{6}=0.16\) and it is less than \(0.2\). Not true!
B. \(60\% = \frac{3}{5} → 60\% = \frac{3}{5}=0.6\). True!
C. \(2.5 > \frac{10}{3} → \frac{10}{3} =3.33\) and it is greater than \(2.5\). Not True!
D. \(\frac{5}{6}< 0.8 → \frac{5}{6}=0.8333…\) and it is greater than \(0.8\). Not True!
E. None of them above → Not True!

14- Choice A is correct
\(20\%\) of 120 equals to: \(0.20×120=24\), \(15\%\) of 300 equals to: \(0.15×300=45\)
\(20\%\) of 120 is added to \(15\%\) of 300: \(24+45=69\)

15- Choice D is correct
Use distance formula: Distance = Rate × time ⇒ 600 = 48 \(×\) T, divide both sides by 48. \(\frac{600}{48}\) = T ⇒ T = 12.5 hours. Change hours to minutes for the decimal part. 0.5 hours \(= 0.5 × 60 = 30\) minutes.

16- Choice A is correct
\(8^{\frac{9}{2}} × 8^{\frac{5}{2}} = 8^{\frac{9}{2} + \frac{5}{2}} = 8^{\frac{14}{2}} = 8^7\)

17- Choice D is correct
Write a proportion and solve for \(x\). ⇒ \(\frac{4}{3}=\frac{x}{30} ⇒ 3x=4 ×30 ⇒ x=40 \space ft\)

18- Choice B is correct
The relationship among all sides of the special right triangle
\(30^\circ -60^\circ – 90^\circ\) is provided in this triangle:

In this triangle, the opposite side of \(30^\circ\) angle is half of the hypotenuse.
Draw the shape of this question:

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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21- Choice D is correct
\(\frac{3}{7}×42=18\)

22- Choice A is correct
Add the first 5 numbers. \(24 + 33 + 52 + 45 + 46 = 200\)
To find the distance traveled in the next 5 hours, multiply the average by number of hours.
Distance = Average \(×\) Rate \(= 45 × 5 = 225\), Add both numbers. \(200 + 225 = 425\)

23- Choice B is correct
The question is this: 1.5 is what percent of 1.2? Use percent formula: \(part = \frac{percent}{100}× whole\)
\(1.5= \frac{percent}{100} × 1.2 ⇒ 1.5 =\frac{percent ×1.2}{100} ⇒150 = percent ×1.2 ⇒ percent = \frac{150}{1.2} = 125\)

24- Choice C is correct
\(One \space liter=1,000 \space cm^3→ 10.24 \space liters=10,240 \space cm^3\)⇒ \(10,240=32×10×h→h=\frac{10240}{320}=32 \space cm\)

25- Choice E is correct
\(3x^2+5y^2-4y^3-7z^2-3x^2+3x-5y^3+7z^2=3x^2-3x^2+3x+5y^2-4y^3-5y^3-7z^2+7z^2=3x+5y^2-9y^3\)

26- Choice A is correct
Solve for \(x\)→\(2x^3-40=210→x^3=125→x=5\)

27- Choice A is correct
Surface Area of a cylinder \(= 2πr (r + h)\), 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:
\(\begin{cases}5x=2x+y-2\\2x=2y+4)\end{cases}→\begin{cases}-3x+y=2\\2x-2y=4\end{cases}\)
⇒ Multiply first equation by 2⇒\(\begin{cases}-6x+2y=4\\2x-2y=4\end{cases}\)→ 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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