Free Full Length THEA Mathematics Practice Test

Full Length THEA Mathematics Practice Test

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THEA Mathematics
Practice Test 2

(Calculator) 50 questions
Total time for this section: (Students have 5 hours to complete all three sections of the THEA Test)
You may use a non-programmable calculator for this test.

If $\frac{x \ - \ 3}{5}=N$ and $N=6$ , what is the value of $x$?

(A)

$25$

(B)

$28$

(C)

$30$

(D)

$33$

Which of the following is equal to $b^ \frac{3}{5}$?

(A)

$\sqrt{b^\frac{5}{3}}$

(B)

$b^\frac{5}{3}$

(C)

$\sqrt[5]{b^3}$

(D)

$\sqrt[3]{b^5}$

On Saturday, Sara read N pages of a book each hour for $3$ hours, and Mary read M pages of a book each hour for $4$ hours. Which of the following represents the total number of pages of book read by Sara and Mary on Saturday?

(A)

$12$ M N

(B)

$3$ N $ \ +\ 4$ M

(C)

$7$ M N

(D)

$4$ N $ + \ 3$ M

For $i=\sqrt{- \ 1}$ , which of the following is equivalent of $\frac{2 \ + \ 3 \ i}{5 \ - \ 2 \ i}$?

(A)

$\frac{3 \ + \ 2 \ i}{5}$

(B)

$5 \ + \ 3 \ i$

(C)

$\frac{4 \ + \ 19 \ i}{29}$

(D)

$\frac{4 \ + \ 19 \ i}{20}$

If function is defined as $f(x)=b \ x^2 \ + \ 15 $, and $b$ is a constant and $f(2)=35$ . What is the value of $f(3)$?

(A)

$25$

(B)

$35$

(C)

$60$

(D)

$65$

Find the solution $(x \ , \ y)$ to the following system of equations?
$- \ 3 \ x \ - \ y=6$
$6 \ x \ + \ 4 \ y=10$

(A)

$(14 \ ,\ 5) $

(B)

($6 \ ,\ 8 $)

(C)

($11 \ ,\ 17 $)

(D)

$(-\frac{17}{3} \ , \ 11)$

Calculate $f(4)$ for the function$f(x)=3\ x^2 \ - \ 4$.

(A)

$44$

(B)

$40$

(C)

$38$

(D)

$30$

What is the sum of all values of n that satisfies $2 \ n^2 \ + \ 16 \ n \ + \ 24=0$ ?

(A)

$8$

(B)

$4$

(C)

$- \ 4$

(D)

$- \ 8$

The equation above represents a parabola in the $x \ y-$plane. Which of the following equivalent forms of the equation displays the $x-$intercepts of the parabola as constants or coefficients?
$y=x^2 \ - \ 7 \ x \ + \ 12$

(A)

$y=x \ + \ 3$

(B)

$y=x \ (x \ - \ 7)$

(C)

$y=(x \ + \ 3) \ (x \ + \ 4)$

(D)

$y=(x \ - \ 3) \ (x \ - \ 4)$

10-

The function $g(x)$ is defined by a polynomial. Some values of $x$ and $g(x)$ are shown in the table below. Which of the following must be a factor of $g(x)$?

$x$	$g(x)$
$0$	$5$
$1$	$4$
$2$	$0$

(A)

$x $

(B)

$x \ - \ 1$

(C)

$x \ - \ 2$

(D)

$x \ + 1 $

11-

What is the value of $\frac{4 \ b}{c}$ when $\frac{c}{b}=2$ ?

(A)

$8$

(B)

$4$

(C)

$2$

(D)

$1$

12-

Which of the following is equivalent to $\frac{x \ + \ (4 \ x)^2 \ + \ (3 \ x)^3}{x}$ ?

(A)

$16 \ x^2 \ + \ 27 \ x \ + \ 1$

(B)

$27 \ x^2 \ + \ 16 \ x \ + \ 1$

(C)

$16 \ x^2 \ + \ 27 \ x $

(D)

$27 \ x^3\ + \ 16 \ x^2 \ + \ 1$

13-

If $\frac{a \ - \ b}{b}=\frac{10}{11}$ , then which of the following must be true?

(A)

$\frac{a}{b}=\frac{11}{10}$

(B)

$\frac{a}{b}=\frac{21}{11}$

(C)

$\frac{a}{b}=\frac{11}{21}$

(D)

$\frac{a}{b}=\frac{21}{10}$

14-

Which of the following lines is parallel to: $6 \ y \ - \ 2 \ x=24$

(A)

$y= \frac{1}{3} \ x \ + \ 2$

(B)

$y= 3 \ x \ + \ 5$

(C)

$y= x \ - \ 2$

(D)

$y= 2 \ x \ - \ 1$

15-

A construction company is building a wall. The company can build $30$ cm of the wall per minute. After $40$ minutes $\frac{3}{4}$ of the wall is completed. How many meters is the wall?

(A)

$10$

(B)

$15$

(C)

$16$

(D)

$20$

16-

If the ratio of $5 \ a$ to $2 \ b$ is $\frac{1}{10}$ , what is the ratio of $a$ to $b$?

(A)

$120^\circ$

(B)

$87^\circ$

(C)

$45^\circ$

(D)

$144^\circ$

17-

If the interior angles of a quadrilateral are in the ratio $1 \ : \ 4 \ : \ 7 \ : \ 8$ , what is the measure of the largest angle?

(A)

$\frac{1}{25}$

(B)

$\frac{2}{5}$

(C)

$\frac{5}{2}$

(D)

$\frac{1}{9}$

18-

When $5$ times the number $x$ is added to $10$, the result is $35$. What is the result when $3$ times $x$ is added to $6$?

(A)

$10$

(B)

$15$

(C)

$21$

(D)

$25$

19-

If $3 \ h \ + \ g=8 \ h \ + \ 4$ , what is g in terms of $h$?

(A)

$h=5 \ g \ - \ 4$

(B)

$g=5 \ h \ + \ 4$

(C)

$h=5 \ g$

(D)

$g=h \ + \ 1$

20-

If $8 \ + \ 2 \ x$ is $16$ more than $20$ , what is the value of $6 \ x$?

(A)

$40$

(B)

$55$

(C)

$62$

(D)

$84$

21-

If $a\ - \ b \ > \ 10$ and $ a \ + \ b \ < \ 14$ , which of the following pairs could not be the values of $a$ and $b$ ?

(A)

($11\ , \ 0$)

(B)

($13\ , \ 2$)

(C)

($13\ , \ 0$)

(D)

($12\ , \ 1$)

22-

What is the value of $x$ in the following system of equations?
$5 \ x \ + \ 2 \ y=3$
$y=x$

(A)

$x= \frac{3}{7}$

(B)

$x= \frac{1}{3}$

(C)

$x= \frac{2}{3}$

(D)

$x= \frac{4}{3}$

23-

In a hotel, there are $5$ floors and $x$ rooms on each floor. If each room has exactly $y$ chairs, which of the following gives the total number of chairs in the hotel?

(A)

$5 \ x \ y$

(B)

$2 \ x \ y$

(C)

$ x \ + \ y$

(D)

$ x \ + \ 5 \ y$

24-

If $α=2\ β$ and $β=3 \ γ$ , how many $α$ are equal to $36 \ γ$?

(A)

$ 12$

(B)

$ 2$

(C)

$ 6$

(D)

$ 4$

25-

Which of the following is the sum of the two polynomials shown above?
$4 \ x^2 \ + \ 6 \ x \ - \ 3$ , $3 \ x^2 \ - \ 5 \ x \ + \ 8$

(A)

$5 \ x^2 \ + \ 3 \ x \ + \ 4$

(B)

$4 \ x^2 \ - \ 6 \ x \ + \ 3$

(C)

$ 7 \ x^2 \ + \ x \ + \ 5 $

(D)

$7 \ x^2 \ + \ 5 \ x \ + \ 1$

26-

The table below shows some values of linear function $g(x)$ . Which of the following defines $g(x)$?

$x$	$1$	$2$	$3$
$g(x)$	$- \ 1$	$- \ 3$	$- \ 5$

(A)

$g(x)=2 \ x \ + \ 1$

(B)

$g(x)=2 \ x \ - \ 1$

(C)

$g(x)=- \ 2 \ x \ + \ 1$

(D)

$g(x)= x \ + \ 2 $

27-

In a certain set of numbers, the ratio of integers to non-integers is $1:4$. What percent of the numbers in the set are integers?

(A)

$10\%$

(B)

$20\%$

(C)

$25\%$

(D)

$30\%$

28-

What is the $y-$intercept of the line with the equation $ x \ - \ 3 \ y=12$ ?

(A)

$ 1$

(B)

$ - \ 2 $

(C)

$ 3$

(D)

$ - \ 4$

29-

If $4 \ a \ - \ 3=14$ what is the value of $6 \ a$?

(A)

$5$

(B)

$15$

(C)

$30$

(D)

$45$

30-

If $x≠0$ and $x=x^{- \ 6}$ , what is the value of $x$?

(A)

$- \ 2$

(B)

$ 1$

(C)

$ 2$

(D)

$ 3$

31-

Which of the following is equal to expression
$\frac{5}{x^2} \ + \frac {7 \ x \ - \ 3}{x^3}$ ?

(A)

$\frac{6 \ x \ + \ 1}{x^3} $

(B)

$\frac{10 \ x \ + \ 6}{x^3} $

(C)

$\frac{12\ x \ + \ 1}{x^3} $

(D)

$\frac{13\ x \ + \ 2}{x^3} $

32-

The results of a research show the number of men and women in four cities of a country.
What's the maximum ratio of number of women to number of men in the four cities?

THEA Math

(A)

$0.98$

(B)

$0.97$

(C)

$0.96$

(D)

$0.95$

33-

What's the ratio of percentage of men in city A to percentage of women in city C?

THEA Math1

(A)

$0.9$

(B)

$0.95$

(C)

$1$

(D)

$1.05$

34-

How many women should be added to city D until the ratio of the number of women to number of men will be $1.2$ ?

THEA Math2

(A)

$120$

(B)

$128$

(C)

$132$

(D)

$160$

35-

The system of equations below has solution $(x \ , \ y)$. What is the value of $x$?
$\frac{3}{2} \ y=5$
$x \ - \frac{3}{2} \ y=3$

(A)

$3$

(B)

$5$

(C)

$8$

(D)

$10$

36-

What is the average of $4 \ x \ + \ 2$ , $- \ 6 \ x \ - \ 5$ and $ 8 \ x \ + \ 2$ ?

(A)

$3 \ x \ + \ 2$

(B)

$3 \ x \ - \ 2$

(C)

$2 \ x \ + 1$

(D)

$2 \ x \ - \frac {1}{3}$

37-

Tickets for a talent show cost $$3$ for children and $$4$ for adults. If John spends at least $$10$ but no more than $$15$ on $x$ children tickets and $2$ adult ticket, what are two possible values of $x$ ?

(A)

$0 \ , \ 1$

(B)

$1\ , \ 2$

(C)

$1\ , \ 4$

(D)

$1\ , \ 5$

38-

Sara orders a box of pen for $$3$ per box. A tax of $8.5 \%$ is added to the cost of the pens before a flat shipping fee of $$6$ closest out the transaction. Which of the following represents total cost of $ p$ boxes of pens in dollars?

(A)

$1.085 \ (3 \ p) \ + \ 6$

(B)

$6 \ p \ + \ 3$

(C)

$1.085 \ (6 \ p) \ + \ 3$

(D)

$3 \ p \ + \ 6$

39-

A plant grows at a linear rate. After five weeks, the plant is $40$ cm tall. Which of the following functions represents the relationship between the height $(y)$ of the plant and number of weeks of growth $(x)$?

(A)

$y(x)=40 \ x \ + \ 8$

(B)

$y(x)=8 \ x \ + \ 40$

(C)

$y(x)=40\ x $

(D)

$y(x)=8\ x $

40-

If $b^\frac{- \ 3}{2}=x$ , where $b \ > \ 0$ , $x \ > \ 0$. Which of the following equations gives $b$ in term of $x$?

(A)

$\sqrt{x}$

(B)

$^3\sqrt\frac{1}{x^2}$

(C)

$\frac{1}{x^2} $

(D)

$\sqrt{x^2}$

41-

In five successive hours, a car travels $40$ km , $45$ km , $50$ km , $35$ km and $55$ 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)

$450$ km

(C)

$475$ km

(D)

$500$ km

42-

In the $x \ y \ -$plane , the point $(4 \ , \ 4)$ lies on the graph of the function $g(x)=4 \ x^2 + \ c \ x \ - \ 10$ . What is the value of $c$ ?

$x$	$g(x)$
$a$	$b$
$7$	$18$
$8$	$20$

(A)

$50$

(B)

$12.25$

(C)

$- \ 2.5$

(D)

$- \ 12.5$

43-

For what value of $x$ is $|x \ - \ 3| \ + \ 3$ equal to $0$?

(A)

$1$

(B)

$2$

(C)

no value of $x$

(D)

$- \ 3$

44-

The table below shows some values for the function $g(x)$ . If $g(x)$ is a linear function , what is the value of $a$ in terms of $b$?

(A)

$4 \ a \ - \ b$

(B)

$b=4 \ a \ + 2$

(C)

$b=2 \ a \ + 4$

(D)

$b=a \ + 1$

45-

A restaurant sells salads for $$5$ each and drinks for $$2$ each. The restaurant’s revenue from selling a total of $300$ salads and drinks in two days was $$960$. How many drinks were sold in total?

(A)

$600$

(B)

$420$

(C)

$350$

(D)

$180$

46-

The mean of $50$ test scores was calculated as $88$ . But, it turned out that one of the scores was misread as $94$ but it was $69$. What is the mean?

(A)

$87.5$

(B)

$67.5$

(C)

$75.6$

(D)

$85.7$

47-

Simplify the expression.
$(6 \ x^3 \ - \ 8 \ x^2 \ + \ 2 \ x^4 ) \ - \ (4 \ x^2 \ - \ 2 \ x^4 \ + \ 2 \ x^3 )$

(A)

$4 \ x^4 \ + \ 4 \ x^3 \ - \ 12 \ x^2$

(B)

$4 \ x^3 \ - \ 12 \ x^2$

(C)

$4 \ x^4 \ - \ 4 \ x^3 \ + \ 12 \ x^2$

(D)

$8 \ x^3 \ - \ 12 \ x^2$

48-

In the following sequence $5$ is the first term. What is the value of the $50^{th}$ term of the sequence?
$5 \ , \ 9 \ , \ 13 \ , \ 17 \ , \ …$

(A)

$105$

(B)

$145$

(C)

$191$

(D)

$201$

49-

Two cars are $240$ miles apart. They both drive in a straight line toward each other. If Car A drives at $56$ mph and Car B drives at $64$ mph , then how many miles apart will they be exactly $40$ minutes before they meet?

(A)

$120$

(B)

$80$

(C)

$40$

(D)

$20$

50-

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)

$\frac{1}{9}$

(B)

$\frac{1}{6}$

(C)

$\frac{16}{17}$

(D)

$\frac{17}{18}$

THEA Mathematics Practice Test Answers and Explanations

1- Choice D is correct

The correct answer is $33$
Since $N=6$ , substitute $6$ for N in the equation $\frac{x \ - \ 3}{5}=N$ , which gives $\frac{x \ - \ 3}5=6$.
Multiplying both sides of $\frac{x \ - \ 3}{5}=6$ by $5$ gives $x \ - \ 3=30$
and then adding $3$ to both sides of $x \ - \ 3=30$ then , $x=33$.

2- Choice C is correct

The correct answer is $\sqrt[5]{b^3}$
$b^\frac{m}{n}=\sqrt[n]{b^m}$ For any positive integers $ m$ and $n$.
Thus, $b^\frac{3}{5}=\sqrt[5]{b^3}$

3- Choice B is correct

The correct answer is $3$ N $ + \ 4 $ M.
The total number of pages read by Sara is $3$ (hours she spent reading) multiplied by her rate of reading: $\frac{N \ pages}{ hour}$ × $3$ hours $=3$ N
Similarly, the total number of pages read by Mary is $4$ (hours she spent reading) multiplied by her rate of reading: $\frac{M \ pages}{ hour} \ × \ 4 $ hours $=4$ M
the total number of pages read by Sara and Mary is the sum of the total number of pages read by Sara and the total number of pages read by Mary: $3$ N $ + \ 4 $ M.

4- Choice C is correct

The correct answer is $\frac{4 \ + \ 19 \ i}{29}$
To rewrite $\frac{2 \ + \ 3 \ i}{5 \ - \ 2 \ i}$ in the standard form$a \ + \ b \ i$ ,
multiply the numerator and denominator of $\frac{2 \ + \ 3 \ i}{5 \ - \ 2 \ i}$ by the conjugate, $5 \ + \ 2 \ i$.
This gives$\frac{(2 \ + \ 3 \ i)}{(5 \ - \ 2 \ i)}\frac{(5 \ + \ 2 \ i)}{(5 \ + \ 2 \ i)}=\frac{10 \ + \ 4 \ i \ + \ 15 \ i \ + \ 6 \ i^2}{5^2 \ - \ (2 \ i)^2 }$ .
Since $i^2=- \ 1$ , this last fraction can be rewritten as $\frac{10 \ + \ 4 \ i \ + \ 15 \ i \ + \ 6 \ (- \ 1)}{25 \ - \ 4(- \ 1)}=\frac{4 \ + \ 19 \ i}{29}$.

5- Choice C is correct

The correct answer is $60$
First find the value of $b$, and then find $f(3)$.
Since $f(2)=35$, substuting $2$ for $x$ and $35$ for $f(x)$ gives $35=b \ (2)^2 \ + \ 15=4 \ b \ + \ 15$ .
Solving this equation gives $b=5$. Thus
$f(x)=5 \ x^2 \ + \ 15$ , $f(3) = 5 \ (3)^2 \ + \ 15 →f(3) = 45 \ + \ 15$ , $f(3) = 60$

6- Choice D is correct

The correct answer is $(-\frac{17}{3} , 11)$
Multiplying each side of $- \ 3 \ x \ - \ y=6$ by $2$ gives $- \ 6 \ x - \ 2 \ y = 12$.
Adding each side of $- \ 6 \ x \ – \ 2 \ y = 12$ to the corresponding side of $6 \ x \ + \ 4 \ y=10$ gives $2 \ y=22$ or $y=11$.
Finally, substituting $11$ for $y$ in $6 \ x \ + \ 4 \ y=10$ gives $6 \ x \ + \ 4 \ ( \ 11 \ )=10$ or $x=-\frac{17}{3}$.

7- Choice A is correct

The correct answer is $44$
Identify the input value. Since the function is in the form $f(x)$ and the question asks to calculate $f(4)$ ,
the input value is four. $f(4)→x=4$ , Using the function, input the desired value.
Now substitute $4$ in for every $x$ in the function.
$f(x)=3 \ x^2 \ - \ 4$ , $f(4)=3 \ ( \ 4 \ )^2 \ - \ 4$, $f(4)=48 \ - \ 4$ , $f(4)=44$

8- Choice D is correct

The correct answer is $ - \ 8$
The problem asks for the sum of the roots of the quadratic equation $2 \ n^2 \ + \ 16 \ n \ + \ 24=0$.
Dividing each side of the equation by $2$ gives $n^2 \ + \ 8 \ n \ + \ 12=0$.
If the roots of
$n^2 \ + \ 8 \ n \ + \ 12=0$ are $n _1$ and $n_2$ ,
then the equation can be factored as
$n^2 \ + \ 8 \ n \ + \ 12=(n \ - \ 1) \ (n \ - \ n_2 )=0$.
Looking at the coefficient of$n$ on each side of $n^2 \ + \ 8 \ n \ + \ 12=(n \ + \ 6) \ (n \ + \ 2)$ gives $n=- \ 6$ or $n=- \ 2$ ,
then, $- \ 6 \ + \ (- \ 2)=- \ 8$

9- Choice D is correct

The correct answer is $y=(x \ - \ 3) \ (x \ - \ 4)$
The $x-$intercepts of the parabola represented by $y=x^2 \ - \ 7 \ x \ + \ 12$ in the $x \ y- $plane are the values of $x$ for which $y$ is equal to $0$.
The factored form of the equation, $y=(x \ - \ 3) \ (x \ - \ 4)$,
shows that yequals $0$ if and only if
$x=3$ or $x=4$.
Thus, the factored form $y=(x \ - \ 3) \ (x \ - \ 4)$,
displays the $x-$intercepts of the parabola as the constants $3$ and $4$.

10- Choice C is correct

The correct answer is $x \ - \ 2$
If $x \ - \ a$ is a factor of $g(x)$ , then $g(a)$ must equal $0$.
Based on the table $g(2)=0$.
Therefore, $x \ - \ 2$ must be a factor of $g(x)$.

11- Choice C is correct

The correct answer is $2$
To solve this problem first solve the equation for $c$ .
$\frac{c}{b}=2$
Multiply by $b$ on both sides.
Then: $b \ × \frac {c}{b}=2 \ × \ b → c=2 \ b$
Now to calculate $\frac{4 \ b}{c}$,
substitute the value for $c$ into the denominator and simplify.
$ \frac{4 \ b}{c}=\frac{4 \ b}{2 \ b}=\frac{4}{2}=\frac{2}{1}=2$

12- Choice B is correct

The correct answer is $27 \ x^2 \ + \ 16 \ x \ + \ 1$
Simplify the numerator.
$\frac{x \ + \ (4 \ x)^2 \ + \ (3 \ x)^3}{x}=\frac{x \ + \ 4^2 \ x^2 \ + \ 3^3 \ x^3}{x}=\frac{x \ + \ 16 \ x^2 \ + \ 27 \ x^3}{x}$
Pull an $x$ out of each term in the numerator.
$\frac{x \ (1 \ + \ 16 \ x \ + \ 27 \ x^2)}{x}$
The $x$ in the numerator and the $x$ in the denominator cancel:
$1 \ + \ 16 \ x \ + \ 27 \ x^2=27 \ x^2 \ + \ 16 \ x \ + \ 1$

13- Choice B is correct

The correct answer is $\frac{a}{b}=\frac{21}{11}$.
The equation $\frac{a \ - \ b}{b}=\frac{10}{11}$ can be rewritten as $\frac{a}{b} \ - \frac{b}{b}=\frac{10}{11}$,
from which it follows that $\frac{a}{b} \ - \ 1=\frac{10}{11}$ ,
or $\frac{a}{b}=\frac{10}{11} \ + \ 1=\frac{21}{11}$.

14- Choice A is correct

The correct answer is $y= \frac{1}{3} \ x \ + \ 2$
First write the equation in slope intercept form.
Add$2$ to both sides to get $6 \ y=2 \ + \ 24$.
Now divide both sides by $6$ to get $y=\frac{1}{3} \ x \ + \ 4$.
The slope of this line is $\frac{1}{3}$,
so any line that also has a slope of $\frac{1}{3}$ would be parallel to it.
Only choice A has a slope of $\frac{1}{3}$.

15- Choice C is correct

The correct answer is $16$
The rate of construction company$=\frac{30 \ cm}{1 \ min}=30$ cm/min
Height of the wall after $40$ minutes $=\frac{30 \ cm}{1 \ min} \ × 40$ min$=1200$ cm
Let $x$ be the height of wall,
then $\frac{3}{4} \ x=1200$ cm→$x =\frac{4 \ × \ 1200}{3}→x=1600$ cm$=16 \ m$

16- Choice D is correct

The correct answer is $144^\circ$
The sum of all angles in a quadrilateral is $360$ degrees.
Let $x$ be the smallest angle in the quadrilateral.
Then the angles are: $x \ , \ 4 \ x \ ,7 \ x \ , \ 8 \ x$
$x \ + \ 4 \ x \ + \ 7 \ x \ + \ 8 \ x=360→20 \ x=360→x=18$
The angles in the quadrilateral are: $18^\circ$ , $72^\circ$ , $126^\circ$ , and $144^\circ$

17- Choice A is correct

The correct answer is $\frac{1}{25}$
Write the ratio of $5 \ a$ to $2 \ b$.
$\frac{5 \ a}{2 \ b}=\frac{1}{10}$
Use cross multiplication and then simplify.
$5 \ a \ × \ 10=2 \ b \ × \ 1 → 50 \ a=2 \ b → a=\frac{2 \ b}{50}=\frac{b}{25}$
Now, find the ratio of $a$ to $b$.
$\frac{a}{b}=\frac{\frac{b}{25}}{b} → \frac{b}{25} \ ÷ \ b=\frac{b}{25} \ × \frac{1}{b}=\frac{b}{25 \ b}=\frac{1}{25}$

18- Choice C is correct

The correct answer is $21$
When $5$ times the number $x$ is added to $10$, the result is $10 \ + \ 5 \ x$.
Since this result is equal to $35$, the equation $10 \ + \ 5 \ x = 35$ is true.
Subtracting $10$ from each side of $10 \ + \ 5 \ x = 35$ gives $5 \ x=25$,
and then dividing both sides by $5$ gives $x=5$.
Therefore, $3$ times $x$ added to $6$, or $6 \ + \ 3 \ x$,
is equal to $6 \ + \ 3 \ (5)=21$.

19- Choice B is correct

The correct answer is $g=5 \ h \ + \ 4$
Fining $g$ in term of $h$ , simply means “solve the equation for $g$ ”.
To solve for $g$, isolate it on one side of the equation.
Since $g$ is on the left-hand side, just keep it there.
Subtract both sides by $3h$. $3 \ h \ + \ g \ - \ 3 \ h=8 \ h \ + \ 4 \ - \ 3 \ h$
And simplifying makes the equation $g=5 \ h \ + \ 4$ ,
which happens to be the answer.

20- Choice D is correct

The correct answer is $84$
The description $8 \ + \ 2 \ x \ i \ 16$ more than $20$ can be written as the equation $8 \ + \ 2 \ x=16 \ + \ 20$,
which is equivalent to $8 \ + \ 2 \ x=36$.
Subtracting $8$ from each side of $8 \ + \ 2 \ x=36$ gives $2 \ x=28$.
Since $6 \ x $ is $3$ times $2 \ x$ ,
multiplying both sides of $2 \ x=28$ by $3$ gives $6 \ x=84$
Method 2: To solve a quadratic equation, put it in the form of $a \ x^2 \ + \ b \ x \ + \ c=0$ ,
factor the left side (if it is factorable), and set each factor equal to $0$ separately to get the two solutions.
To solve $x^2=4 \ x \ - \ 3$,
first, rewrite it as $x^2 \ - \ 4 \ x \ + \ 3=0$.
Then factor the left side: $(x \ - \ 1) \ (x \ - \ 3)=0$ , $x=1$ Or $x=3$
The equation has $2$ distinct real solutions.

21- Choice B is correct

The correct answer is ($13 \ , \ 2$)
From the choices provided, plugin the values of $a$ and $b$ into both inequalities and check.
A: $(11 \ , \ 0)→a \ - \ b=11 \ - \ 0=11 \ > \ 10$ and $a \ + \ b=11 \ + \ 0=11 \ < \ 14$
B: $(13 \ , \ 2)→a \ - \ b=13 \ - \ 2=11 \ > \ 10$ and $a \ + \ b=13 \ + \ 2=15 \ > \ 14$
C: $(13 \ , \ 0)→a \ - \ b=13 \ - \ 0=13 \ > \ 10$ and $a \ + \ b=13 \ + \ 0=13 \ < \ 14$
D: $(12 \ , \ 1)→a \ - \ b=12 \ - \ 1=11 \ > \ 10$ and $a \ + \ b=12 \ + \ 1=13 \ < \ 14$
For choice B, $15$ is not less than $14$.
Therefore, choice B does not provide the correct values of $a$ and $b$.

22- Choice A is correct

The correct answer is $x= \frac{3}{7}$
Substituting $x$ for $y$ in first equation.
$5 \ x \ + \ 2 \ y=3$ , $5 \ x \ + \ 2 \ ( \ x \ )=3$ , $7 \ x=3$
Divide both side of $7 \ x=3$ by $3$ gives $x=\frac{3}{7}$

23- Choice A is correct

The correct answer is $5 \ x \ y$
There are $5$ floors, $x$ rooms in each floor, and $y$ chairs per room.
If you multiply $5$ floors by $x$, there are $5 \ x$ rooms in the hotel.
To get the number of chairs in the hotel, multiply $5 \ x$ by $y$.
$5 \ x \ y$ is the number of chairs in the hotel.

24- Choice C is correct

The correct answer is $6$
If$=3 \ γ$ , then multiplying both sides by $12$ gives $12 \ β=36 \ γ$.
$α=2 \ β$ , thus $α=6 \ γ$ .
Multiply both sides of the equation by $6$ gives $6 \ α=36 \ y$

25- Choice C is correct

The correct answer is $7 \ x^2 \ + \ x \ + \ 5$
The sum of the two polynomials is $(4 \ x^2 \ + \ 6 \ x \ - \ 3) \ + \ (3 \ x^2 \ - \ 5 \ x \ + \ 8)$
This can be rewritten by combining like terms:
$(4 \ x^2 \ + \ 6 \ x \ - \ 3) \ + \ (3 \ x^2 \ - \ 5 \ x \ + \ 8)=(4 \ x^2 \ + \ 3 \ x^2 ) \ + \ (6 \ x \ - \ 5 \ x) \ + \ (- \ 3 \ + \ 8)=
7 \ x^2 \ + \ x \ + \ 5$

26- Choice C is correct

The correct answer is $g(x)=- \ 2 \ x \ + \ 1$
Plug in the values of $x$ in the choices provided.
The points are $(1 \ , \ - \ 1) \ , \ (2 \ , \ - \ 3)$ , and $(3 \ , \ - \ 5)$
For $(1 \ , \ - \ 1)$ check the options provided:
A. $g(x)=2 \ x \ + \ 1 \ → \ - \ 1=2 \ (1) \ + \ 1 \ → \ - \ 1=3$ This is NOT true.
B. $g(x)=2 \ x \ - \ 1 → - \ 1=2 \ (1) \ - \ 1=1$ This is NOT true.
C. $g(x)=- \ 2 \ x \ + \ 1 → - \ 1=2(- \ 1) \ + \ 1 → - \ 1=- 1$ This is true.
D. $g(x)=x \ + \ 2 → - \ 1 =1 \ + \ 2 → - \ 1=3$ This is NOT true.
From the choices provided, only choice C is correct.

27- Choice B is correct

The correct answer is $20\%$
Since all the numbers in the set are either integers or non-integers and the ratio of integers to non-integers is $1 \ : \ 4$ ,
one out of five numbers is integer.
Therefore the ratio of integers to all number is $1 \ : \ 5$
that is equal $\frac{1}{5} \ × \ 100=20\%$.

28- Choice D is correct

The correct answer is $- \ 4$
To find the $y-$intercept of a line from its equation,
put the equation in slope-intercept form:
$x \ - \ 3 \ y=12→ \ - \ 3 \ y=- \ x \ + \ 12 → $
$3 \ y=x \ - \ 12→ $
$y=\frac{1}{3} x\ - \ 4 $
The $y-$intercept is what comes after the $x$.
Thus, the $y-$intercept of the line is $- \ 4$.

29- Choice C is correct

The correct answer is $30$
Adding both side of $4 \ a \ - \ 3=17$ by $3$ gives $4 \ a=20$
Divide both side of $4 \ a=20$ by $4$ gives $a=5$,
then $6 \ a=6 \ (5)=30$

30- Choice B is correct

The correct answer is $1$
The easiest way to solve this one is to plug the answers into the equation.
When you do this, you will see the only time $x=x^{- \ 6}$ is when $x=1$ or $x=0$.
Only $x=1$ is provided in the choices.

31- Choice C is correct

The correct answer is $\frac{12 \ x \ + \ 1}{x^3} $
First find a common denominator for both of the fractions in the expression $\frac{5}{x^2} + \frac{7 \ x \ - \ 3}{x^3}$ .
of $x^3$ , we can combine like terms into a single numerator over the denominator:
$\frac{5 \ x \ + \ 4}{x^3}+\frac{7 \ x \ - \ 3}{x^3} =\frac{(5 \ x \ + \ 4) \ + \ (7 \ x \ - \ 3)}{x^3}=\frac{12 \ x \ + \ 1}{x^3}$

32- Choice B is correct

The correct answer is $0.97$
First, review ratio of number of women to number of men in the four cities.
Ratio of women to men in city A: $\frac{570}{600}=0.95$
Ratio of women to men in city B: $\frac{291}{300}=0.97$
Ratio of women to men in city C: $\frac{665}{700}=0.95$
Ratio of women to men in city D: $\frac{528}{550}=0.96$
From four ratios, $0.97$ is the maximum.

33- Choice D is correct

The correct answer is $1.05$
Percentage of men in city A $= \frac{600}{1170} \ × \ 100=51.28 \%$
Percentage of men in city C $= \frac{665}{1365} \ × \ 100=48.72 \%$
Percentage of men in city A to percentage of women in C $= \frac{51.28}{48.72}=1.05$

34- Choice C is correct

The correct answer is $132$
Let $x$ be the number of women that should be added to city D until the ratio of the number of women to number of men will be $1.2.$
Then: $\frac{528 \ + \ x}{550}=1.2 → 528 \ + \ x=660 → x=132$

35- Choice C is correct

The correct answer is $8$
Adding the two equations side by side eliminates $y$ and yields $x = 8$.
$\frac{3}{2} \ y = 5 \ , \frac{x \ - \ 3}{2} \ y = 3$ , $→ x \ + \ 0 = 8 → x=8$

36- Choice D is correct

The correct answer is $2 \ x \ - \frac {1}{3}$
To find the average of three numbers even if they’re algebraic expressions,
add them up and divide by$3$.
Thus, the average equals:
$\frac{(4 \ x \ + \ 2) \ + \ (- \ 6 \ x \ - \ 5) \ + \ (8 \ x \ + \ 2)}{3}=\frac{6 \ x \ - \ 1}{3}=2 \ x \ - \frac {1}{3}$

37- Choice B is correct

The correct answer is $1 \ , \ 2$
Because each children ticket costs $$3$ and each adult ticket costs $$4$,
the total amount, in dollars, that John spends on $x$ student tickets and $2$ adult ticket is $3(x) \ + \ 4 \ (2)$.
Because John spends at least $$10$ but no more than $$15$ on the tickets,
you can write the compound inequality $3 \ x \ + \ 8 \ ≥ \ 10$ and $3 \ x \ + \ 8 \ ≤ \ 15$.
Subtracting $8$ from each side of both inequalities and then dividing each side of both inequalities by $3$ gives $x \ ≥ \ 0.66$ and $x \ ≤ \ 2.3$.
Thus, the value of must be an integer that is both greater than or equal to $0.66$ and less than or equal to $2.3$.
Therefore, $x = 1$ or $x =2$.
Either $1$ or $2$ may be gridded as the correct answer.

38- Choice A is correct

The correct answer is $1.085 \ (3 \ p) \ + \ 6$
Since a box of pen costs $$3$, then $3 \ p$ Represents the cost of $p$ boxes of pen.
Multiplying this number times $1.085$ will increase the cost by the $8.5 \%$ for tax.
Then add the $$6$ shipping fee for the total: $1.085 \ (3 \ p) \ + \ 6$

39- Choice D is correct

The correct answer is $y(x)=8\ x$
Rate of change (growth or $x$) is $8$ per week . $40 \ ÷ \ 5=8$
Since the plant grows at a linear rate,
then the relationship between the height $(y)$ of the plant and number of weeks of growth $(x)$ can be written as: $y \ (x)=8 \ x$

40- Choice B is correct

The correct answer is $\sqrt[3]\frac{1}{x^2}$
According to the properties of exponents,
you can rewrite the equation as $\frac{1}{\sqrt[2]{b^3}}=x$
Remember that $a^{-n}=\frac{1}{a^n}$ and ${1}\ ^\frac{1}{n}=\sqrt[n]{a}$
Proceed to solve for $b$ first by squaring both sides,
which gives $\frac{1}{b^3} =x^2$
And then multiplying both sides by $b^3$ to find $1=b^3 \ x^2$.
finally, dividing both sides by $x^2$ isolates the desired variable.
$\frac{1}{x^2} =b^3 → \sqrt[3]\frac{1}{x^2}=b$

41- Choice C is correct

The correct answer is $475$ km
Add the first $5$ numbers. $40 \ + \ 45 \ + \ 50 \ + \ 35 \ + \ 55 = 225$
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. $250 \ + \ 225 = 475$

42- Choice D is correct

The correct answer is $- \ 12.5$
It is given that the function $g()$ passes through the point $(4 \ , \ 4)$.
Thus, if $x=4$ , the value of $g(x)$ is $4$ (since the graph of $g$ in the $x \ y-$plane is the set of all points $(x \ , \ g(x))$.
Substituting $4$ for $x$ and $4$ for $g(x)$ in $g(x)=4\ x^2 \ + \ c \ x \ - \ 10$ gives $4=4 \ (4)^2 \ + \ c \ (4) \ - \ 10$
Solve this equation for $c$.
Then: $4=64 \ + \ 4 \ c \ - \ 10$
$- \ 50=4 \ c → c= -\ 12.5$

43- Choice C is correct

The correct answer is no value of $x$
If the value of $|x \ - \ 3| \ + \ 3$ is equal to $0$ , then $|x \ - \ 3| \ + \ 3=0$ .
Subtracting $3$ from both sides of this equation gives $|x \ - \ 3|=- \ 3$.
The expression $|x \ - \ 3|$ on the left side of the equation is the absolute value of $x \ - \ 3$ ,
and the absolute value can never be a negative number.
Thus $|x \ - \ 3|=- \ 3$ has no solution.
Therefore, there are no values for $x$ for which the value of $|x \ - \ 3| \ + \ 3$ is equal to $0$.

44- Choice C is correct

The correct answer is $b=2 \ a + 4$
Since function $g(x)$ is a linear function,
we can write it in the form of $g(x)=m \ x \ + \ d$
We start by finding the slope, $m$ .
The slope is given by $\frac{d_1 \ - \ d_2}{a_1 \ - \ a_2}$
Choosing two values $(7 \ , \ 18)$ and $(8 \ , \ 20)$ gives us $m=\frac{20 \ - \ 18}{8 \ - \ 7}=2$
Next we solve for $d$ in the equation $g(x)=2 \ x \ + \ d$.
Looking at the first set of numbers on the table,
we can see that when $x=7$ , $g(x)=18$
Substituting these values in the equation $g(x)=2 \ x \ + \ d$ gives us $18=(2) \ (7) \ +\ d$
And simplifying gives us $d=4$.
Therefore, the linear equation of the function is
$g(x)=2 \ x \ + \ 4$
But that’s not one of the choices!
The table shows that $g(x)=b$ , $x=a$ then we can merely substitute them.
Which gives us the equation $b=2 \ a \ + \ 4$

45- Choice D is correct

The correct answer is $180$
To determine the number of drinks sold,
write and solve a system of two equations.
Let $x$ be the number of salads sold and let $y$ be the number of drinks sold.
Then:
$x \ + \ y=300$
Since each salad cost $$5$ , each drink cost $$2$, and the total revenue was $$960$,
the equation
$5 \ x \ + \ 2 \ y=960$ is true.
The equation $x \ + \ y=150$ is equivalent to $2 \ x \ + \ 2 \ y=600$,
and subtracting each side of
$2 \ x \ + \ 2 \ y=600$ from the respective side of $5 \ x \ + \ 2 \ y=960$ gives $3 \ x=360$.
Therefore, the number of salads sold, $x$ was $x=120$ and the number of drinks sold was
$y=300 \ - \ 120=180$.

46- Choice A is correct

The correct answer is $87.5$
average (mean) $=\frac{sum \ of \ terms}{number \ of \ terms} ⇒ 88 = \frac{sum \ of \ terms}{50} ⇒ sum = 88 \ × \ 50 = 4400$
The difference of $94$ and $69$ is $25$.
Therefore, $25$ should be subtracted from the sum.
$4400 \ – \ 25= 375$
mean $=\frac{sum \ of \ terms}{number \ of \ terms} ⇒ mean =\frac{4375}{50}=87.5$

47- Choice A is correct

The correct answer is $4 \ x^4 \ + \ 4 \ x^3 \ - \ 12 \ x^2$
Simplify and combine like terms.
$(6 \ x^3 \ - \ 8 \ x^2 \ + \ 2 \ x^4 ) \ - \ (4 \ x^2 \ - \ 2 \ x^4 \ + \ 2 \ x^3)⇒
(6 \ x^3 \ - \ 8 \ x^2 \ + \ 2 \ x^4 ) \ - \ 4 \ x^2 \ + \ 2 \ x^4 \ - \ 2 \ x^3 ⇒
4 \ x^4 \ + \ 4 \ x^3 \ - \ 12 \ x^2$

48- Choice D is correct

The correct answer is $201$
The first term in the sequence is $5$ and each term in this sequence is found by adding $4$ to the preceding term.
This is an arithmetic sequence.
Given an arithmetic sequence with the first term $a_1$ and the common difference $d$ , the $n^{th}$(or general) term is given by:
$a_n=a_1 \ + \ (n \ - \ 1) \ d$ , Then: $a_n=5 \ + \ (50 \ - \ 1) \ 4=201$

49- Choice B is correct

The correct answer is $80$
The speed of car A is 56 mph and the speed of car B is $64$ mph.
When both cars drive in a straight line toward each other,
the distance between the cars decreases at the rate of $120$ miles per hour: $56 \ + \ 64=120$
$40$ minutes is two third of an hour.
Therefore, they will be $80$ miles apart $40$ minutes before they meet. $\frac{2}{3} \ × \ 120=80$

50- Choice D 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.