Did you take the ACT Math Practice Test? If so, then it’s time to review your results to see where you went wrong and what areas you need to improve.

## 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 any 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 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 \(2(-3)^3+4(-3)-\frac{18}{(-3)^2} =-54-12-2=-68\)