## Full Length ACT Mathematics Practice Test

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ACT Mathematics
Practice Test 3
60 questions Total time for this section: 60 Minutes   You may use a scientific calculator on this test.

1- What is the median of these numbers? $$3, 10, 13, 8, 15, 19, 5$$
(A) $$8$$
(B) $$10$$
(C) $$5$$
(D) $$13$$
(E) $$19$$
2- What is the area of a square whose diagonal is $$10$$ cm?
(A) $$25$$ cm$$^2$$
(B) $$30$$ cm$$^2$$
(C) $$125$$ cm$$^2$$
(D) $$100$$ cm$$^2$$
(E) $$50$$ cm$$^2$$
3- David’s current age is $$42$$ years, and Ava’s current age is $$6$$ years. In how many years David’s age will be $$4$$ times Ava’s age?
(A) $$4$$
(B) $$6$$
(C) $$8$$
(D) $$10$$
(E) $$14$$
4- How long does a $$360–$$miles trip take moving at $$50$$ miles per hour (mph)?
(A) $$8$$ hours and $$10$$ minutes
(B) $$8$$ hours
(C) $$7$$ hours and $$24$$ minutes
(D) $$7$$ hours and $$12$$ minutes
(E) $$9$$ hours and $$12$$ minutes
5- The marked price of a computer is D dollar. Its price decreased by $$30\%$$ in January and later increased by $$10\%$$ in February. What is the final price of the computer in D dollar?
(A) $$0.77$$ D
(B) $$0.80$$ D
(C) $$0.88$$ D
(D) $$0.70$$ D
(E) $$0.90$$ D
6- Which is the correct statement?
(A) $$\frac{3}{4} \ > \ 0.8$$
(B) $$10\% = \frac{2}{5}$$
(C) $$3 \ < \ \frac{5}{2}$$
(D) $$\frac{5}{6} \ > \ 0.8$$
(E) None of them above
7- What is the solution of the following inequality?
$$|x \ - \ 12| \ ≤ \ 4$$
(A) $$7 \ ≤ \ x \ ≤ \ 13$$
(B) $$10 \ ≤ \ x \ ≤ \ 12$$
(C) $$8 \ ≤ \ x \ ≤ \ 16$$
(D) $$8 \ ≤ \ x$$
(E) $$x \ ≤ \ 16$$
8- The number $$42.5$$ is $$2,000$$ times greater than which of the following numbers?
(A) $$0.02678$$
(B) $$0.04498$$
(C) $$0.02125$$
(D) $$0.03864$$
(E) $$0.02018$$
9- How many tiles of $$6$$ cm$$^2$$ is needed to cover a floor of dimension $$9$$ cm by $$36$$ cm?
(A) $$58$$
(B) $$44$$
(C) $$62$$
(D) $$54$$
(E) $$60$$
10- Right triangle ABC is shown below. Which of the following is true for all possible values of angle and ?
(A) cos $$A =$$ cos $$B$$
(B) cos $$A =$$ tan $$B$$
(C) cos $$A =$$ sin $$B$$
(D) tan $$A =1$$
(E) cot $$A =$$ sin $$B$$
11- If $$40\%$$ of a class are girls, and $$30\%$$ of girls play tennis, what percent of the class play tennis?
(A) $$18\%$$
(B) $$15\%$$
(C) $$11\%$$
(D) $$12\%$$
(E) $$26\%$$
12- $$5^{\frac{7}{3}} \ × \ 5^{\frac{2}{3}} =$$
(A) $$5^{\frac{5}{2}}$$
(B) $$5^5$$
(C) $$5^2$$
(D) $$5^{\frac{2}{5}}$$
(E) $$5^{\frac{1}{5}}$$
13- A company pays its employer $$6500$$ plus $$4\%$$ of all sales profit. If $$x$$ is the number of all sales profit, which of the following represents the employer’s revenue?
(A) $$0.04 \ x \ - \ 6500$$
(B) $$0.96 \ x \ - \ 6500$$
(C) $$0.96 \ x \ + \ 6500$$
(D) $$0.04 \ x$$
(E) $$0.04 \ x \ + \ 6500$$
14- $$6$$ liters of water are poured into an aquarium that's $$20$$ cm long, $$5$$ cm wide, and $$90$$ cm high. How many cm will the water level in the aquarium rise due to this added water? ($$1$$ liter of water $$= 1000$$ cm$$^3$$)
(A) $$120$$ cm
(B) $$90$$ cm
(C) $$80$$ cm
(D) $$60$$ cm
(E) $$30$$ cm
15- Five years ago, Amy was four times as old as Mike was. If Mike is $$12$$ years old now, how old is Amy?
(A) $$33$$
(B) $$25$$
(C) $$22$$
(D) $$38$$
(E) $$41$$
16- What is the value of $$x$$ in the following figure?
(A) $$145^\circ$$
(B) $$152^\circ$$
(C) $$122^\circ$$
(D) $$136^\circ$$
(E) $$158^\circ$$

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17- If $$|a| \ < \ 1$$, then which of the following is true? $$(b \ > \ 0)$$?
I. $$– \ b \ < \ b \ a \ < \ b$$
II. $$- \ a \ < \ a^2 \ < \ a$$ if $$a \ < \ 0$$
III. $$- \ 5 \ <2 \ a \ - \ 3 \ < \ - \ 1$$
(A) I only
(B) II only
(C) I and III only
(D) III only
(E) I, II and III
18- If $$x$$ is a real number, and if $$x^3 \ + \ 18=130$$, then $$x$$ lies between which two consecutive integers?
(A) $$1$$ and $$2$$
(B) $$2$$ and $$3$$
(C) $$3$$ and $$4$$
(D) $$4$$ and $$5$$
(E) $$5$$ and $$6$$
19- From the figure, which of the following must be true? (figure not drawn to scale)
(A) $$y \ + \ 2 \ x=π§$$
(B) $$y \ + \ 4 \ x=π§$$
(C) $$y=π§$$
(D) $$x=π§$$
(E) $$2 \ x=π§$$
20- What is the value of $$y$$ in the following system of equation?
$$2 \ x \ - \ y= \ - \ 48$$
$$- \ x \ + \ 2 \ y= 12$$
(A) $$- \ 5$$
(B) $$- \ 10$$
(C) $$- \ 8$$
(D) $$5$$
(E) $$8$$
21- When $$30\%$$ of $$60$$ is added to $$15\%$$ of $$420,$$ the resulting number is:
(A) $$92$$
(B) $$81$$
(C) $$120$$
(D) $$143$$
(E) $$75$$
22- If a box contains red and blue balls in ratio of $$5 : 3$$, how many red balls are there if $$90$$ blue balls are in the box?
(A) $$150$$
(B) $$120$$
(C) $$100$$
(D) $$80$$
(E) $$200$$
23- Simplify $$(– \ 4 \ + \ 5 \ i) \ (2 \ + \ 7\ i)$$
(A) $$- \ 49 \ - \ 18 \ i$$
(B) $$39 \ - \ 18 \ i$$
(C) $$40 \ + \ 22\ i$$
(D) $$25 \ i$$
(E) $$25 \ i \ + \ 49$$
24- Which of the following has the same period and two times the amplitude of graph?
$$y =$$ cos $$x$$?
(A) $$y=$$ cos $$(x \ + 2)$$
(B) $$y=4\ +$$ cos $$x$$
(C) $$y=$$ cos $$2 \ x$$
(D) $$y=4$$ cos $$2 \ x$$
(E) $$y=2 \ + 2$$ cos $$x$$
25- A ladder leans against a wall forming a $$60^α΅$$ angle between the ground and the ladder. If the bottom of the ladder is $$30$$ feet away from the wall, how long is the ladder?
(A) $$50$$ feet
(B) $$25$$ feet
(C) $$40$$ feet
(D) $$100$$ feet
(E) $$125$$ feet
26- What is the surface area of the cylinder below?
(A) $$44 \ π$$ in$$^2$$
(B) $$55 \ π$$ in$$^2$$
(C) $$66 \ π$$ in$$^2$$
(D) $$77\ π$$ in$$^2$$
(E) $$88 \ π$$ in$$^2$$
27- Simplify:
$$5 \ x^2 \ + \ 2 \ y^5 \ - \ x^2 \ - \ 6 \ z^3 \ + \ 3 \ y^2 \ + \ 4 \ x^3 \ - \ 7 \ y^5 \ + \ 8 \ z^3$$
(A) $$2 \ x^2 \ + \ 4 \ x^3 \ + \ 4 \ y^2 \ - \ 8 \ y^5 \ + \ 2 \ z^3$$
(B) $$6 \ x^2 \ + \ 3 \ x^3 \ + \ 4 \ y^2 \ - \ 8 \ y^5$$
(C) $$4 \ x^2 \ + \ 4 \ x^3 \ + \ 3 \ y^2 \ - \ 5 \ y^5 \ + \ 2 \ z^3$$
(D) $$x^3 \ + \ 3 \ y^2 \ - \ 8 \ y^5 \ + \ 3 \ z^3$$
(E) $$4 \ x^2 \ + \ 5 \ y^5 \ + \ 2 \ z^3$$
28- A bag contains $$18$$ balls: two green, five black, eight blue, a brown, a red and one white. If $$17$$ balls are removed from the bag at random, what is the probability that a brown ball has been removed?
(A) $$\frac{17}{18}$$
(B) $$\frac{1}{18}$$
(C) $$\frac{1}{17}$$
(D) $$\frac{18}{17}$$
(E) $$17$$
29- Which of the following shows the numbers in increasing order?
(A) $$\frac{3}{4}, \frac{5}{7}, \frac{8}{13}, \frac{2}{5}$$
(B) $$\frac{3}{4}, \frac{8}{13}, \frac{5}{7}, \frac{2}{5}$$
(C) $$\frac{8}{13}, \frac{3}{4}, \frac{5}{7}, \frac{2}{5}$$
(D) $$\frac{2}{5}, \frac{5}{7}, \frac{8}{13}, \frac{3}{4}$$
(E) $$\frac{5}{7}, \frac{8}{13}, \frac{2}{5}, \frac{3}{4}$$
30- If a tree casts a $$24–$$foot shadow at the same time that a $$6$$ feet yardstick casts a $$3–$$foot shadow, what is the height of the tree?
(A) $$36$$ ft.
(B) $$52$$ ft.
(C) $$24$$ ft.
(D) $$48$$ ft.
(E) $$38$$ ft.

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31- From last year, the price of gasoline has increased from $$1.20$$ per gallon to $$1.80$$ per gallon. The new price is what percent of the original price?
(A) $$180\%$$
(B) $$200\%$$
(C) $$150\%$$
(D) $$120\%$$
(E) $$100\%$$
32- If $$(x \ - \ 25)^3=125$$ which of the following could be the value of $$(x \ - \ 21) \ (x \ - \ 10)$$?
(A) $$180$$
(B) $$112$$
(C) $$320$$
(D) $$220$$
(E) $$150$$
33- What are the values of mode and median in the following set of numbers?
$$1,2,2,5,4,4,3,3,3,1,1$$
(A) $$1, \ 3$$ Median: $$3$$
(B) $$1, \ 2$$ Median: $$2$$
(C) $$2, \ 3$$ Median: $$2$$
(D) $$1, \ 3$$ Median: $$2.5$$
(E) $$3$$ Median: $$3$$
34- If $$x \ \begin{bmatrix}2 & 0 \\0 & 3 \end{bmatrix} = \begin{bmatrix}x \ + \ 2 \ y \ - \ 4 & 0 \\0 & 2 \ y \ + \ 12 \end{bmatrix}$$, what is the product of $$x$$ and $$y$$?
(A) $$24$$
(B) $$36$$
(C) $$48$$
(D) $$52$$
(E) $$61$$
35- Removing which of the following numbers will change the average of the numbers to $$6$$?
$$2, 4, 6, 8, 10, 12$$
(A) $$2$$
(B) $$4$$
(C) $$6$$
(D) $$10$$
(E) $$12$$
36- If $$f(x) = 4 \ + \ x$$ and $$g(x) = \ – \ x^2 \ + \ 3 \ – \ 2 \ x$$, then find $$(g \ – \ f)(x)$$?
(A) $$– \ x^2 \ + \ 2 \ x \ – \ 1$$
(B) $$– \ x^2 \ - \ 3 \ x$$
(C) $$x^2 \ + \ 3 \ x \ – \ 2$$
(D) $$2 \ x^2 \ + \ 4 \ x \ – \ 2$$
(E) $$– \ x^2 \ – \ 3 \ x \ – \ 1$$
37- If $$50\%$$ of $$x$$ equal to $$40\%$$ of $$20$$, then what is the value of $$(x \ + \ 1)^2$$?
(A) $$169$$
(B) $$225$$
(C) $$25$$
(D) $$289$$
(E) $$324$$
38- If $$\frac{2 \ x}{18}=\frac{x \ - \ 2}{6}, x=$$
(A) $$4$$
(B) $$6$$
(C) $$8$$
(D) $$\frac{1}{6}$$
(E) $$\frac{1}{4}$$
39- In $$1999$$, the average worker's income increased $$2,500$$ per year starting from $$25,000$$ annual salary.  Which equation represents income greater than average? ($$I =$$ income, $$x =$$ number of years after $$1999$$)
(A) $$I \ > \ 2500 \ x \ + \ 50000$$
(B) $$I \ < \ 5000 \ x \ + \ 50000$$
(C) $$I \ > \ 2500 \ x \ - \ 25000$$
(D) $$I \ < \ 2500 \ x \ + \ 25000$$
(E) $$I \ > \ 2500 \ x \ + \ 25000$$
40- In five successive hours, a car travels $$32$$ km, $$40$$ km, $$51$$ km, $$39$$ km and $$53$$ km. In the next five hours, it travels with an average speed of $$50$$ km per hour. Find the total distance the car traveled in $$10$$ hours.
(A) $$425$$ km
(B) $$386$$ km
(C) $$412$$ km
(D) $$465$$ km
(E) $$491$$ km
41- A chemical solution contains $$3\%$$ alcohol. If there is $$24$$ ml of alcohol, what is the volume of the solution?
(A) $$600$$ ml
(B) $$500$$ ml
(C) $$200$$ ml
(D) $$250$$ ml
(E) $$800$$ ml
42- If cot $$\theta = \frac{12}{5}$$ and sin $$\theta \ > \ 0$$, then cos $$\theta =$$?
(A) $$- \ \frac{12}{13}$$
(B) $$- \ \frac{13}{12}$$
(C) $$- \ \frac{5}{12}$$
(D) $$\frac{12}{13}$$
(E) $$\frac{13}{12}$$
43- In the $$x \ y-$$plane, the point $$(5, 2)$$ and $$(6, 4)$$ are on line A. Which of the following equations of lines is parallel to line A?
(A) $$y= x$$
(B) $$y= 2$$
(C) $$y=2 \ x$$
(D) $$y=- \ 2 \ x$$
(E) $$y=\frac{1}{2} \ x$$
44- If $$f(x)=3^x$$ and $$g(x)=\log_{3}{x}$$, which of the following expressions is equal to $$f(3g(p))$$?
(A) $$P^9$$
(B) $$P^6$$
(C) $$3 \ P^2$$
(D) $$\frac{P}{3}$$
(E) $$P^3$$
45- The result of a research shows the number of men and women in four cities of a country.
What's the maximum ratio of the number of women to number of men in each city?
(A) $$0.98$$
(B) $$0.97$$
(C) $$0.96$$
(D) $$0.95$$
(E) $$0.94$$
46- The result of a research shows the number of men and women in four cities of a country.
What's the ratio of the percentage of men in city A to percentage of women in city C?
(A) $$\frac{19}{20}$$
(B) $$\frac{19}{2}$$
(C) $$1$$
(D) $$\frac{20}{19}$$
(E) $$\frac{2}{19}$$
(F) $$\frac{2}{19}$$
47- The result of a research shows the number of men and women in four cities of a country.
How many women should be added to city D to change the ratio of women to men to $$1.2$$?
(A) $$135$$
(B) $$134$$
(C) $$133$$
(D) $$132$$
(E) $$131$$
48- $$120$$ is equal to:
(A) $$20 \ − \ (4 \ × \ 10) \ + \ (6 \ × \ 30)$$
(B) $$(\frac{11}{8} \ × \ 72) \ + \ (\frac{125}{5})$$
(C) $$((\frac{30}{4} \ + \ \frac{13}{2}) \ × \ 7) \ − \ \frac{11}{2} \ + \ \frac{110}{4}$$
(D) $$(2 \ × \ 10) \ + \ (50 \ × \ 1.5) \ + \ 15$$
(E) $$\frac{481}{6} \ + \ \frac{121}{3}$$
49- Which of the following points lies on the line $$2 \ x \ + \ 4 \ y = 12$$?
(A) $$(2, 2)$$
(B) $$(– \ 1, 3)$$
(C) $$(– \ 2, 2)$$
(D) $$(2, 3)$$
(E) $$(2, 8)$$
50- In the following equation when $$z$$ is divided by $$2$$, what is the effect on $$x$$?
$$x=\frac{5 \ y \ + \ \frac{r}{r \ + \ 2}}{\frac{4}{z}}$$
(A) $$x$$ is divided by $$3$$
(B) $$x$$ is multiplied by $$3$$
(C) $$x$$ is multiplied by $$2$$
(D) $$x$$ is divided by $$2$$
(E) $$x$$ does not change
51- If $$y = 3 \ a \ b \ + \ 4 \ b^3$$, what is $$y$$ when $$a = 3$$ and $$b = 3$$?
(A) $$102$$
(B) $$78$$
(C) $$108$$
(D) $$24$$
(E) $$36$$
52- If tangent of an angel $$β$$ is $$2$$, then the cotangent of angle $$β$$ is:
(A) $$1$$
(B) $$\frac{1}{2}$$
(C) $$2$$
(D) $$0$$
(E) $$\frac{1}{4}$$
53- When point A $$(5, 3)$$ is reflected over the $$y-$$axis to get the point B, what are the coordinates of point B?
(A) $$(- \ 5, 3)$$
(B) $$(- \ 5, - \ 3)$$
(C) $$(5, - \ 3)$$
(D) $$(5, 3)$$
(E) $$(0, 3)$$
54- If $$f(x)=2 \ x^3 \ - \ 3 \ x^2 \ + \ 4 \ x$$ and $$g(x)= \ - \ 2$$, what is the value of $$f(g(x))$$?
(A) $$- \ 36$$
(B) $$- \ 22$$
(C) $$0$$
(D) $$32$$
(E) $$18$$
55- A boat sails $$40$$ miles south and then $$30$$ miles east. How far is the boat from its start point?
(A) $$50$$ miles
(B) $$80$$ miles
(C) $$40$$ miles
(D) $$70$$ miles
(E) $$60$$ miles
56- If the area of trapezoid is $$100$$, what is the perimeter of the trapezoid?
(A) $$24$$
(B) $$32$$
(C) $$18$$
(D) $$29$$
(E) $$36$$
57- In the following figure, ABCD is a rectangle, and E and F are points on AD and DC, respectively. The area of $$\triangle$$BED is $$18$$, and the area of $$\triangle$$BDF is $$24$$. What is the perimeter of the rectangle?
(A) $$70$$
(B) $$45$$
(C) $$50$$
(D) $$63$$
(E) $$54$$
58- A rope weighs $$500$$ grams per meter of length. What is the weight in kilograms of $$13.2$$ meters of this rope? ($$1$$ kilograms $$= 1000$$ grams)
(A) $$5.4$$ kg
(B) $$6.6$$ kg
(C) $$7.2$$ kg
(D) $$4.9$$ kg
(E) $$6.1$$ kg
59- $$z$$ is $$x\%$$ of what number?
(A) $$\frac{100 \ z}{x}$$
(B) $$\frac{1 \ z}{100 \ x}$$
(C) $$\frac{z}{\frac{1}{100} \ x}$$
(D) $$\frac{100 \ x}{z}$$
(E) $$\frac{x \ z}{100}$$
60- A number is chosen at random from $$1$$ to $$25$$. Find the probability of not selecting a composite number.
(A) $$1$$
(B) $$0$$
(C) $$\frac{1}{5}$$
(D) $$\frac{5}{2}$$
(E) $$\frac{2}{5}$$
 1- Choice B is correct 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 $$10$$. 2- Choice E is correct 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: $$\sqrt{50} \ × \ \sqrt{50} = 50$$ 3- Choice B is correct 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 $$20$$. 4- Choice D is correct The correct answer is $$7$$ hours and $$12$$ minutesUse 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. 5- Choice A is correct The correct answer is $$0.77$$ DTo find the discount, multiply the number by ($$100\% \ –$$ rate of discount).Therefore, for the first discount we get: (D) $$(100\% \ – \ 30\%) =$$ (D) $$(0.70) = 0.70$$ DFor increase of $$10\%$$: ($$0.70$$ D) $$(100\% \ + \ 10\%) = (0.70$$ D$$) (1.10) = 0.77$$ D $$= 77\%$$ of D 6- Choice D is correct 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! 7- Choice C is correct The correct answer is $$8 \ ≤ \ x \ ≤ \ 16$$$$|x \ - \ 12| \ ≤ \ 4→$$$$- \ 4 \ ≤ \ x \ - \ 12 \ ≤ \ 4→$$$$- \ 4 \ + \ 12 \ ≤ \ x \ - \ 12 \ + \ 12 \ ≤ \ 4 \ + \ 12 →$$$$8 \ ≤ \ x \ ≤ \ 16$$ 8- Choice C is correct The correct answer is $$0.02125$$$$2000$$ times the number is $$42.5$$.Let $$x$$ be the number, then:$$2000 \ x=42.5, \ x=\frac{42.5}{2000}=0.02125$$ 9- Choice D is correct The correct answer is $$54$$The area of the floor is: $$9$$ cm $$× \ 36$$ cm $$= 324$$ cmThe number is tiles needed $$= 324 \ ÷ \ 6 = 54$$ 10- Choice C is correct 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 $$B$$ 11- Choice D is correct The correct answer is $$12\%$$The percent of girls playing tennis is: $$40\% \ × \ 30\% = 0.40 \ × \ 0.30= 0.12 = 12\%$$ 12- Choice A is correct The correct answer is $$5^{\frac{5}{2}}$$$$5^{\frac{7}{4}} \ × \ 5^{\frac{3}{4}} = 5^{\frac{7}{4} \ + \ \frac{3}{4}} = 5^{\frac{10}{4}}=5^{\frac{5}{2}}$$ 13- Choice E is correct The correct answer is $$0.04 \ x \ + \ 6500$$Employer’s revenue: $$0.04 \ x \ + \ 6500$$ 14- Choice D is correct The correct answer is $$60$$ cmOne liter $$=1,000$$ cm$$^3→ 6$$ liters $$=6000$$ cm$$^3$$$$6000=20 \ × \ 5 \ × \ h→h=\frac{6000}{100}=60$$ cm 15- Choice A is correct 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: $$28 \ + \ 5 = 33$$ 16- Choice B is correct The correct answer is $$152^\circ$$$$x=25 \ + \ 127=152$$ 17- Choice C is correct The correct answer is I and III onlyI. $$|a| \ < \ 1→ \ - \ 1 \ < \ a\ < \ 1$$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!) 18- Choice D is correct 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. 19- Choice A is correct The correct answer is $$y \ + \ 2 \ x=π§$$ $$x$$ and $$z$$ are colinear.$$y$$ and $$5 \ x$$ are colinear.Therefore:$$x \ + \ z=y \ + \ 3 \ x$$,subtract $$x$$ from both sides,then,$$z=y \ + \ 2 \ x$$ 20- Choice C is correct 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}} 21- Choice B is correct 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 $$420: \ 18 \ + \ 63=81$$ 22- Choice A is correct The correct answer is $$150$$$$\frac{5}{3} \ × \ 90=150$$ 23- Choice A is correct The correct answer is $$- \ 49 \ - \ 18 \ i$$We know that: $$i=\sqrt{- \ 1}⇒i^2= \ - \ 1$$ $$(- \ 4 \ + \ 5 \ i) \ (2 \ + \ 7 \ i)= \ - \ 14 \ - \ 28 \ i \ + \ 10 \ i \ + \ 35 \ i^2= \ - \ 14 \ - \ 18 \ i \ + \ 35 \ i^2=- \ 49 \ - \ 18 \ i$$ 24- Choice E is correct 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 $$2 \ π$$. 25- Choice A is correct The correct answer is $$50$$ feetThe 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. 26- Choice C is correct 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$$^2$$ 27- Choice C is correct The correct answer is $$4 \ x^2 \ + \ 4 \ x^3 \ + \ 3 \ y^2 \ - \ 5 \ y^5 \ + \ 2 \ z^3$$$$5 \ x^2 \ + \ 2 \ y^5 \ - \ x^2 \ - \ 6 \ z^3 \ + \ 3 \ y^2 \ + \ 4 \ x^3 \ - \ 7 \ y^5 \ + \ 8 \ z^3$$$$5 \ x^2 \ - \ x^2 \ + \ 4 \ x^3 \ + \ 3 \ y^2 \ + \ 2 \ y^5 \ - \ 7 \ y^5 \ + \ 8 \ z^3 \ - \ 6 \ z^3=$$$$4 \ x^2 \ + \ 4 \ x^3 \ + \ 3 \ y^2 \ - \ 5 \ y^5 \ + \ 2 \ z^3$$ 28- Choice A is correct 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. 29- Choice D is correct The correct answer is $$\frac{2}{5}, \frac{5}{7}, \frac{8}{13}, \frac{3}{4}$$$$\frac{2}{5}=0.4 \ \ \ \ \frac{5}{7}≅0.71 \ \ \ \ \frac{8}{13}≅0.61 \ \ \ \ \frac{3}{4}=0.75$$ 30- Choice D is correct 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. 31- Choice C is correct 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 $$=\frac{ 180}{1.20} = 150$$ 32- Choice A is correct The correct answer is $$180$$$$(x \ - \ 25)^3=125→x \ - \ 25=5→x=30$$$$→(x \ - \ 21)\ (x \ - \ 10)=(30 \ - \ 21) \ (30 \ - \ 10)=(9) \ (20)=180$$ 33- Choice A is correct 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: $$3$$ 34- Choice C is correct 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.$$4 \ y=24→y=6→x=8→ x \ × \ y=48$$ 35- Choice E is correct 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. $$12 \ \ \frac{2 \ + \ 4 \ + \ 6 \ + \ 8 \ + \ 10}{5}=\frac{30}{5}=6$$ 36- Choice E is correct The correct answer is $$– \ x^2 \ – \ 3 \ x \ – \ 1$$$$(g \ – \ f)(x) = g(x) \ – \ f(x) = (– \ x^2 \ + \ 3 \ – \ 2 \ x) \ – \ (4 \ + \ x)$$$$– \ x^2 \ + \ 3 \ – \ 2 \ x \ – \ 4 \ – \ x = \ – \ x^2 \ – \ 3 \ x \ – \ 1$$ 37- Choice D is correct The correct answer is $$289$$$$0.5 \ x=(0.4) \ × \ 20→x=16→$$$$(16 \ + \ 1)^2=(17)^2=289$$ 38- Choice B is correct 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.$$2 \ x=3 \ x \ - \ 6$$, $$0=x \ - \ 6$$$$6=x$$ 39- Choice E is correct 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:$$I \ > \ 2500 \ x \ + \ 25000$$ 40- Choice D is correct The correct answer is $$465$$ kmAdd 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. $$250 \ + \ 215 = 465$$ 41- Choice E is correct 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 $$⇒ 0.03 \ x = 24 ⇒ x = 24 \ ÷ \ 0.03 = 800$$ 42- Choice D is correct 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 $$θ=\frac{adjacent}{hypotenuse}=\frac{12}{13}$$ 43- Choice C is correct 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 $$2$$. 44- Choice E is correct The correct answer is $$π^{ \ 3}$$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:$$f(\log_{3}{p}^{ \ 3})=3^{\log_{3}{p}^{ \ 3}}=p^{ \ 3}$$ 45- Choice B is correct 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: $$\frac{528}{550}=0.96$$ 46- Choice D is correct 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.$$\frac{\frac{600}{1170}}{\frac{665}{1365}}=\frac{20}{19}$$ 47- Choice D is correct The correct answer is $$132$$$$\frac{528 \ + \ x}{550}=1.2→528 \ + \ x=660→x=132$$ 48- Choice C is correct 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. $$\frac{481}{6} \ + \ \frac{121}{3}=\frac{481 \ + \ 242}{6}=120.5$$ 49- Choice A is correct 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! 50- Choice D is correct 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 $$2$$. 51- Choice C is correct 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 $$b = 3$$$$y = 3 \ (3) \ (3) \ + \ 4 \ (3)^3 = 27 \ + \ 3 \ (27) = 27 \ + \ 81 = 108$$ 52- Choice B is correct The correct answer is $$\frac{1}{2}$$cotangent $$β= \frac{1}{tangent \ β}=\frac{1}{2}$$ 53- Choice A is correct 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 $$(- \ 5, 3)$$. 54- Choice A is correct The correct answer is $$- \ 36$$$$g(x)= \ - \ 2$$, then:$$f(g(x))= f(- \ 2)=2 \ (- \ 2)^3 \ - \ 3 \ (- \ 2)^2 \ + \ 4 \ (- \ 2)=$$$$- \ 16 \ - \ 12 \ - \ 8=- \ 36$$ 55- Choice A is correct The correct answer is $$50$$ milesUse the information provided in the question to draw the shape.Use Pythagorean Theorem: $$a^2 \ + \ b^2 = c^2$$$$40^2 \ + \ 30^2 = c^2 ⇒$$$$1600 \ + \ 900 = c^2 ⇒$$$$2500 = c^2 ⇒ c = 50$$ 56- Choice D is correct 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: $$10 \ + \ 6 \ + \ 8 \ + \ 5=29$$ 57- Choice C is correct 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 $$= 2 \ × \ (9 \ + \ 16)=50$$ 58- Choice B is correct The correct answer is $$6.6$$ kgThe 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 59- Choice A is correct 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 $$\frac{100}{x}$$:$$z \ × \ \frac{100}{x}=\frac{x}{100} \ × \ \frac{100}{x} \ × \ A$$$$A=\frac{100 \ z}{x}$$ 60- Choice E is correct 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 $$=\frac{ number \ of \ desired \ outcomes}{number \ of \ total \ outcomes}=\frac{10}{25}=\frac{2}{5}$$

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