ACT Math Practice Test

The correct answer is $10$
Write the numbers in order:
$3, 5, 8, 10, 13, 15, 19$
Since we have $7$ numbers ($7$ is odd), then the median is the number in the middle, which is $10$.

The correct answer is $50$ cm$^2$
The diagonal of the square is $10$. 
Let $x$ be the side. 
Use Pythagorean Theorem: $a^2 \ + \ b^2 = c^2$
$x^2 \ + \ x^2 = 10^2 ⇒$
$2 \ x^2 = 10^2 ⇒$
$2 \ x^2 = 100⇒$
$x^2 = 50 ⇒x= \sqrt{50}$
The area of the square is: $\sqrt{50} \ × \ \sqrt{50} = 50$

The correct answer is $6$
Let’s review the options provided.
A. $4$. In $4$ years, David will be $46$ and Ava will be $10$. $46$ is not $4$ times $10$.
B. $6$. In $6$ years, David will be $48$ and Ava will be $12$. $48$ is $4$ times $12$!
C. $8$. In $8$ years, David will be $80$ and Ava will be $14$. $50$ is not $4$ times $14$.
D. $10$. In $10$ years, David will be $52$ and Ava will be $16$. $52$ is not $4$ times $16$.
E. $14$. In $14$ years, David will be $56$ and Ava will be $20$. $56$ is not $4$ times $20$.

The correct answer is $7$ hours and $12$ minutes
Use distance formula:
Distance $=$ Rate $×$ time $⇒ 360 = 50 \ ×$ T, divide both sides by $50$.
$\frac{360 }{50} =$ T $⇒$ T $= 7.2$ hours.
Change hours to minutes for the decimal part.
$0.2$ hours $= 0.2 \ ×\ 60 = 12$ minutes.

The correct answer is $0.77$ D
To find the discount, multiply the number by ($100\% \ –$ rate of discount).
Therefore, for the first discount we get: (D) $(100\% \ – \ 30\%) =$ (D) $(0.70) = 0.70$ D
For increase of $10\%$: ($0.70$ D) $(100\% \ + \ 10\%) = (0.70$ D$) (1.10) = 0.77$ D $= 77\%$ of D

The correct answer is $\frac{5}{6} \ > \ 0.8$
Check each option.
A. $\frac{3}{4} \ > \ 0.8$      $\frac{3}{4}=0.75$ and it is less than $0.8$. Not true!
B. $10\% = \frac{2}{5}$      $10\% = \frac{1}{10} \ < \ \frac{2}{5}$. Not True!
C. $3 \ < \ \frac{5}{2}$         $\frac{5}{2}=2.5 \ < \ 3$. Not True!
D. $\frac{5}{6} \ > \ 0.8$      $\frac{5}{6}=0.8333$… and it is greater than $0.8$. Bingo!
E. None of them above Not True!

The correct answer is $8 \ ≤ \ x \ ≤ \ 16$
$|x \ - \ 12| \ ≤ \ 4→$
$- \ 4 \ ≤ \ x \ - \ 12 \ ≤ \ 4→$
$- \ 4 \ + \ 12 \ ≤ \ x \ - \ 12 \ + \ 12 \ ≤ \ 4 \ + \ 12 →$
$8 \ ≤ \ x \ ≤ \ 16$

The correct answer is $0.02125$
$2000$ times the number is $42.5$.
Let $x$ be the number, then:
$2000 \ x=42.5, \ x=\frac{42.5}{2000}=0.02125$

The correct answer is $54$
The area of the floor is: $9$ cm $× \ 36$ cm $= 324$ cm
The number is tiles needed $= 324 \ ÷ \ 6 = 54$

The correct answer is cos $A =$ sin $B$
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: cos $A =$ sin $B$

The correct answer is $12\%$
The percent of girls playing tennis is: $40\% \ × \ 30\% = 0.40 \ × \ 0.30= 0.12 = 12\%$

The correct answer is $5^{\frac{5}{2}}$
$5^{\frac{7}{4}} \ × \ 5^{\frac{3}{4}} = 5^{\frac{7}{4} \ + \ \frac{3}{4}} = 5^{\frac{10}{4}}=5^{\frac{5}{2}}$

The correct answer is $0.04 \ x \ + \ 6500$
Employer’s revenue: $0.04 \ x \ + \ 6500$

The correct answer is $60$ cm
One liter $=1,000$ cm$^3→ 6$ liters $=6000$ cm$^3$
$6000=20 \ × \ 5 \ × \ h→h=\frac{6000}{100}=60$ cm

The correct answer is $33$
Five years ago, Amy was three 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: $28 \ + \ 5 = 33$

The correct answer is $152^\circ$
$x=25 \ + \ 127=152$

The correct answer is I and III only
I. $|a| \ < \ 1→ \ - \ 1 \ < \ a\ < \ 1$
Multiply all sides by $b$.
Since, $b \ > \ 0→ \ - \ b \ < \ b \ a \ < \ b$ (it is true!)
II. Since, $- \ 1 \ < \ a \ < \ 1$, and $a \ < \ 0→ \ - \ a \ > \ a^2 \ > \ a$ (plug in $\frac{- \ 1}{2}$, and check!) (It’s false)
III. $- \ 1 \ < \ a \ < \ 1$, multiply all sides by $2$, then: $- \ 2 \ < \ 2 \ a \ < \ 2$ 
Subtract $3$ from all sides. Then:
$- \ 2 \ - \ 3 \ < \ 2 \ a \ - \ 3 \ < \ 2 \ - \ 3→ \ - \ 5 \ < \ 2 \ a \ - \ 3 \ < \ - \ 1$ (It is true!)

The correct answer is $4$ and $5$
Solve for $x$.
$x^3 \ + \ 18=130, \ x^3=112$
Let’s review the choices. 
A. $1$ and $2$.    $13 = 1$ and       $23 = 8, \ 112$ is not between these two numbers.
B. $2$ and $3$.    $23 = 8$ and       $33 = 27, \ 112$ is not between these two numbers.
C. $3$ and $4$.    $33 = 27$ and     $43 = 64, \ 112$ is not between these two numbers.
D. $4$ and $5$.    $43 = 64$ and     $53 = 125, \ 112$ is between these two numbers.
E. $5$ and $6$.    $53 = 125$ and   $63 = 216, \ 112$ is not between these two numbers.

The correct answer is $y \ + \ 2 \ x=𝑧$ 
$x$ and $z$ are colinear.
$y$ and $5 \ x$ are colinear.
Therefore:
$x \ + \ z=y \ + \ 3 \ x$,subtract $x$ from both sides,then,$z=y \ + \ 2 \ x$

The correct answer is $- \ 8$
Solving Systems of Equations by Elimination m8ethod.
$\cfrac{\begin{align} 2 \ x \ -  \ y \ = \ - \ 48 \\ - \ x \ + \ 2 \ y \ = \ 12 \end{align}}{}$
Multiply the second equation by $2$, then add it to the first equation.
$\cfrac{\begin{align} 2 \ x \ -  \ y \ = \ - \ 48 \\ 2 \ ( \ - \ x \ + \ 2 \ y \ = \ 12) \end{align}}{}$ 
$\cfrac{ \begin{align} 2 \ x \ -  \ y \ = \ - \ 48 \\ - \ 2 \ x \ + \ 4\ y \ = \ 24 \end{align} } ⇒$ add the equations
${\begin{align} 3\ y \ = - \ 24 \\ ⇒ y \ = -\ 8 \end{align}}$

The correct answer is $81$
$30\%$ of $60$ equals to: $0.30 \ × \ 60=18$
$15\%$ of $430$ equals to: $0.15 \ × \ 420=63$
$30\%$ of $60$ is added to $15\%$ of $420: \ 18 \ + \ 63=81$

The correct answer is $150$
$\frac{5}{3} \ × \ 90=150$

The correct answer is $- \ 49 \ - \ 18 \ i$
We know that: $i=\sqrt{- \ 1}⇒i^2= \ - \ 1$ 
$(- \ 4 \ + \ 5 \ i) \ (2 \ + \ 7 \ i)= \ - \ 14 \ - \ 28 \ i \ + \ 10 \ i \ + \ 35 \ i^2= \ - \ 14 \ - \ 18 \ i \ + \ 35 \ i^2=- \ 49 \ - \ 18 \ i$

The correct answer is $y=2 \ + 2$ cos $x$
The amplitude in the graph of the equation $y=a$ cos $b \ x$ is $a$. ($a$ and $b$ are constant)
In the equation $y=$ cos $x$, the amplitude is $2$ and the period of the graph is $2 \ π$.
The only option that has two times the amplitude of graph $y =$ cos $x$ is $y=2 \ + \ 2$ cos $x$
They both have the amplitude of $2$ and period of $2 \ π$.

The correct answer is $50$ feet
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 latter is the hypotenuse. Therefore, the latter is $50$ ft.

The correct answer is $66 \ π$ in$^2$
Surface Area of a cylinder $= 2 \ π \ r \ (r \ + \ h)$,
The radius of the cylinder is $3 \ (6 \ ÷ \ 2)$ inches and its height is $8$ inches. Therefore, 
Surface Area of a cylinder $= 2 \ π \ (3) \ (3 \ + \ 8) = 66 \ π$ in$^2$

The correct answer is $4 \ x^2 \ + \ 4 \ x^3 \ + \ 3 \ y^2 \ - \ 5 \ y^5 \ + \ 2 \ z^3$
$5 \ x^2 \ + \ 2 \ y^5 \ - \ x^2 \ - \ 6 \ z^3 \ + \ 3 \ y^2 \ + \ 4 \ x^3 \ - \ 7 \ y^5 \ + \ 8 \ z^3$
$5 \ x^2 \ - \ x^2 \ + \ 4 \ x^3 \ + \ 3 \ y^2 \ + \ 2 \ y^5 \ - \ 7 \ y^5 \ + \ 8 \ z^3 \ - \ 6 \ z^3=$
$4 \ x^2 \ + \ 4 \ x^3 \ + \ 3 \ y^2 \ - \ 5 \ y^5 \ + \ 2 \ z^3$

The correct answer is $\frac{17}{18}$
If $17$ balls are removed from the bag at random, there will be one ball in the bag.
The probability of choosing a brown ball is $1$ out of $18$.
Therefore, the probability of not choosing a brown ball is $17$ out of $18$ and the probability of having not a brown ball after removing $17$ balls is the same.

The correct answer is $\frac{2}{5}, \frac{5}{7}, \frac{8}{13}, \frac{3}{4}$
$\frac{2}{5}=0.4 \ \ \ \  \frac{5}{7}≅0.71 \ \ \ \  \frac{8}{13}≅0.61 \  \ \ \ \frac{3}{4}=0.75$

The correct answer is $48$ ft.
Write a proportion and solve for $x$.
$\frac{6}{3}=\frac{x}{24} ⇒ 3 \ x=6 \ ×\ 24 ⇒ x=48$ ft.

The correct answer is $150\%$
The question is this: $1.80$ is what percent of $1.20$?
Use percent formula: part $= \frac{percent}{100} \ ×$ whole 
$1.80 =\frac{ percent}{100} \ × \ 1.20 ⇒$
$1.80 = \frac{percent \ × \ 1.20}{100} ⇒$
$180 =$ percent $× \ 1.20⇒$
percent $=\frac{ 180}{1.20} = 150$

The correct answer is $180$
$(x \ - \ 25)^3=125→x \ - \ 25=5→x=30$
$→(x \ - \ 21)\ (x \ - \ 10)=(30 \ - \ 21) \ (30 \ - \ 10)=(9) \ (20)=180$

The correct answer is Mode: $1, \ 3$ Median: $3$
We write the numbers in the order: $1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5$
The mode of numbers is: $1$ and $3$ median is: $3$

The correct answer is $48$
Based on corresponding members from two matrices, we get: $\begin{cases}2 \ x = x \ + \ 2 \ y \ - 4\\3 \ x =2 \ y \ + \ 12\end{cases}→$
$\begin{cases} x \ - \ 2 \ y = \ - \ 4\\3 \ x \ - \ 2 \ y = 12\end{cases}$ Multiply first equation by $- \ 3$.
$\begin{cases}- \ 3 \ x \ + \ 6 \ y = 12\\3 \ x \ - \ 2 \ y = 12\end{cases}→$ add two equations.
$4 \ y=24→y=6→x=8→ x \ × \ y=48$

The correct answer is $12$
Check each choice provided:
A. $2 \ \ \ \ \frac{4 \ + \ 6 \ + \ 8 \ + \ 10 \ + \ 12}{5}=\frac{40}{5}=8$
B. $4 \ \ \ \ \frac{2 \ + \ 6 \ + \ 8 \ + \ 10 \ + \ 12}{5}=\frac{38}{5}=7.6$
C. $6 \ \ \ \ \frac{2 \ + \ 4 \ + \ 8 \ + \ 10 \ + \ 12}{5}=\frac{36}{5}=7.2$
D. $10 \ \ \frac{2 \ + \ 4 \ + \ 6 \ + \ 8 \ + \ 12}{5}=\frac{32}{5}=6.4$
E. $12 \ \ \frac{2 \ + \ 4 \ + \ 6 \ + \ 8 \ + \ 10}{5}=\frac{30}{5}=6$

The correct answer is $– \ x^2 \ – \ 3 \ x \ – \ 1$
$(g \ – \ f)(x) = g(x) \ – \ f(x) = (– \ x^2 \ + \ 3 \ – \ 2 \ x) \ – \ (4 \ + \ x)$
$– \ x^2 \ + \ 3 \ – \ 2 \ x \ – \ 4 \ – \ x = \ – \ x^2 \ – \ 3 \ x \ – \ 1$

The correct answer is $289$
$0.5 \ x=(0.4) \ × \ 20→x=16→$
$(16 \ + \ 1)^2=(17)^2=289$

The correct answer is $6$
Solve for $x$. 
$\frac{2 \ x}{18}=\frac{x \ - \ 2}{6}$.
Multiply the second fraction by $3$.
$\frac{2 \ x}{18}=\frac{3 \ (x \ - \ 2)}{6 \ × \ 3}$
Tow denominators are equal.
Therefore, the numerators must be equal.
$2 \ x=3 \ x \ - \ 6$, 
$0=x \ - \ 6$
$6=x$

The correct answer is $I \ > \ 2500 \ x \ + \ 25000$
Let $x$ be the number of years.
Therefore, $$2,500$ per year equals $2500 \ x$. 
starting from $$25,000$ annual salary means you should add that amount to $2500 \ x$. 
Income more than that is:
$I \ > \ 2500 \ x \ + \ 25000$

The correct answer is $465$ km
Add the first $5$ numbers.
$32 \ + \ 40 \ + \ 51 \ + \ 39 \ + \ 53 = 215$
To find the distance traveled in the next $5$ hours, multiply the average by number of hours.
Distance $=$ Average $×$ Rate $= 50 \ × \ 5 = 250$
Add both numbers. $250 \ + \ 215 = 465$

The correct answer is $800$ ml
$3\%$ of the volume of the solution is alcohol.
Let $x$ be the volume of the solution. 
Then: $3\%$ of $x = 24$ ml $⇒ 0.03 \ x = 24 ⇒ x = 24 \ ÷ \ 0.03 = 800$

The correct answer is $\frac{12}{13}$
cot $=\frac{adjacent}{opposite}$
cot $θ=\frac{12}{5}⇒$ we have the following right triangle.
Then:
$c=\sqrt{5^2 \ + \ 12^2 }=\sqrt{25 \ + \ 144}=\sqrt{169}=13$
cos $θ=\frac{adjacent}{hypotenuse}=\frac{12}{13}$

The correct answer is $y=2 \ x$
The slop of line A is: $m=\frac{y_{2} \ - \ y_{1}}{x_{2} \ - \ x_{1}}=\frac{4 \ - \ 2}{6 \ - \ 5}=2$
Parallel lines have the same slope and only choice C $(y=2 \ x)$ has slope of $2$.

The correct answer is $𝑃^{ \ 3}$
To solve for $f(3 \ g(P))$, first, find $3 \ g(p)$
$g(x)=\log_{3}{x}$
$g(p)=\log_{3}{p}$
$3 \ g(p)=3 \ \log_{3}{p}=\log_{3}{p}^3$
Now, find $f(3 \ g(p))$:
$f(x)=3^{ \ x}$
$f(\log_{3}{p}^{ \ 3} )=3^{\log_{3}{p}^{ \ 3}}$
Logarithms and exponentials with the same base cancel each other.
This is true because logarithms and exponentials are inverse operations. Then:
$f(\log_{3}{p}^{ \ 3})=3^{\log_{3}{p}^{ \ 3}}=p^{ \ 3}$

The correct answer is $0.97$
ratio of A: $\frac{570}{600}=0.95$ ratio of B: $\frac{291}{300}=0.97$ ratio of C: $\frac{665}{700}=0.95$ ratio of D: $\frac{528}{550}=0.96$

The correct answer is $\frac{20}{19}$
First find percentage of men in city A and percentage of women in city C.
Percentage of men in city A $=\frac{600}{1170}$ and percentage of women in city C $=\frac{665}{1365}$
Find the ratio and simplify.
$\frac{\frac{600}{1170}}{\frac{665}{1365}}=\frac{20}{19}$

The correct answer is $132$
$\frac{528 \ + \ x}{550}=1.2→528 \ + \ x=660→x=132$

The correct answer is $((\frac{30}{4} \ + \ \frac{13}{2}) \ × \ 7) \ − \ \frac{11}{2} \ + \ \frac{110}{4}$
Simplify each choice provided. 
A. $20 \ − \ (4 \ × \ 10) \ + \ (6 \ × \ 30)=20 \ - \ 40 \ + \ 180=160$
B. $(\frac{11}{8} \ × \ 72) \ + \ (\frac{125}{5})=99 \ + \ 25=124$
C. $((\frac{30}{4} \ + \ \frac{13}{2}) \ × \ 7) \ − \ \frac{11}{2} \ + \ \frac{110}{4}=((\frac{30 \ + \ 26}{4}) \ × \ 7) \ - \ \frac{11}{2} \ + \ \frac{55}{2}=((\frac{56}{4}) \ × \ 7) \ + \ \frac{55 \ - \ 11}{2}=$
$(14 \ × \ 7) \ + \ \frac{44}{2}=98 \ + \ 22=120$ (this is the answer)
D. $(2 \ × \ 10) \ + \ (50 \ × \ 1.5) \ + \ 15=20 \ + \ 75 \ + \ 15=110$
E. $\frac{481}{6} \ + \ \frac{121}{3}=\frac{481 \ + \ 242}{6}=120.5$

The correct answer is $(2, 2)$
Plug in each pair of number in the equation:
A. $(2, 2): \ \ \ \  \  \ 2 \ (2) \ + \ 4 \ (2) = 12$ Bingo!
B. $(– \ 1, 3): \ \ \ 2 \ (– \ 1) \ + \ 4 \ (3) = 10$ Nope!
C. $(– \ 2, 2): \ \ \ 2 \ (– \ 2) \ + \ 4 \ (2) = 4$ Nope!
D. $(2, 3): \ \ \ \ \  \  2 \ (2) \ + \ 4 \ (3) = 14$ Nope!
E. $(2, 8): \ \ \ \  \  \ 2 \ (2) \ + \ 4 \ (8) = 36$ Nope!

The correct answer is $x$ is divided by $2$
Replace $z$ by $\frac{z}{2}$ and simplify.
$x_{1}=\frac{5 \ y \ + \ \frac{r}{r \ + \ 2}}{\frac{4}{\frac{z}{2}}}= \frac{5 \ y \ + \ \frac{r}{r \ + \ 2}}{\frac{4 \ × \ 2}{z}}=$
$\frac{5 \ y \ + \ \frac{r}{r \ + \ 2}}{2 \ \times \frac{4}{z}} = \frac{1}{2} \ \times \ \frac{5 \ y \ + \ \frac{r}{r \ + \ 2}}{\frac{4}{z}} =\frac{x}{2}$
When $z$ is divided by $2, \ x$ is also divided by $2$.

The correct answer is $108$
$y = 3 \ a \ b \ + \ 4 \ b^3$
Plug in the values of a and b in the equation: $a = 3$ and $b = 3$
$y = 3 \ (3) \ (3) \ + \ 4 \ (3)^3 = 27 \ + \ 3 \ (27) = 27 \ + \ 81 = 108$

The correct answer is $\frac{1}{2}$
cotangent $β= \frac{1}{tangent \ β}=\frac{1}{2}$

The correct answer is $(- \ 5, 3)$
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 $(- \ 5, 3)$.

The correct answer is $- \ 36$
$g(x)= \ - \ 2$, then:
$f(g(x))= f(- \ 2)=2 \ (- \ 2)^3 \ - \ 3 \ (- \ 2)^2 \ + \ 4 \ (- \ 2)=$
$- \ 16 \ - \ 12 \ - \ 8=- \ 36$

The correct answer is $50$ miles
Use the information provided in the question to draw the shape.
Use Pythagorean Theorem: $a^2 \ + \ b^2 = c^2$
$40^2 \ + \ 30^2 = c^2 ⇒$
$1600 \ + \ 900 = c^2 ⇒$
$2500 = c^2 ⇒ c = 50$

The correct answer is $29$
The area of trapezoid is: $(\frac{6 \ + \ 10}{2}) \ × \ x=64→8 \ x=64→x=8$
$y=\sqrt{3^2 + \ 4^2}=5$
Perimeter is: $10 \ + \ 6 \ + \ 8 \ + \ 5=29$

The correct answer is $50$
The area of $\triangle$BED is $16$, then:
$\frac{4 \ × \ AB}{2}=18→4 \ ×$ AB $=36→$ AB $=9$
The area of $\triangle$BDF is $24$, then:
$\frac{3 \ × \ BC}{2}=24→3 \ ×$ BC $=48→$ BC $=16$
The perimeter of the rectangle is $= 2 \ × \ (9 \ + \ 16)=50$

The correct answer is $6.6$ kg
The weight of $12.2$ meters of this rope is: $13.2 \ × \ 500$ g $= 6600$ g
$1$ kg $= 1000$ g, therefore, $6600$ g $÷ \ 1000 = 6.6$ kg

The correct answer is $\frac{100 \ z}{x}$
Let the number be $A$. Then:
$z=x\% \ × \ A$
Solve for $A$.
$z=\frac{x}{100} \ × \ A$
Multiply both sides by $\frac{100}{x}$:
$z \ × \ \frac{100}{x}=\frac{x}{100} \ × \ \frac{100}{x} \ × \ A$
$A=\frac{100 \ z}{x}$

The correct answer is $\frac{2}{5}$
Set of number that are not composite between $1$ and $25$: A $= \left\{1, 2, 3, 5, 7, 11, 13, 17, 19, 23\right\}$
Probability $=\frac{ number \ of \ desired \ outcomes}{number \ of \ total \ outcomes}=\frac{10}{25}=\frac{2}{5}$