# Full-Length ACT Math Practice Test-Answers and Explanations

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 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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