Free Full Length CLEP College Mathematics Practice Test

The correct answer is $7$
Adding $6$ to each side of the inequality $4\ n\ -\ 3\ ≥\ 1$ yields the inequality $4\ n\ +\ 3\ ≥\ 7$.
Therefore, the least possible value of $4\ n\ +\ 3$ is $7$.

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

The correct answer is $38\%$
The population is increased by $15\%$ and $20\%$. $15\%$ increase changes the population to $115\%$ of original population.
For the second increase, multiply the result by $120\%$. $(1.15)\ ×\ (1.20)=1.38=138\%$
$38$ percent of the population is increased after two years.

Solve for $x$. $-\ 2\ ≤\ 2\ x\ -\ 4\ <\ 8\ ⇒$ (add $4$ all sides) $-\ 2\ +\ 4\ ≤\ 2\ x\ -\ 4\ + \ 4\ <\ 8\ +\ 4 ⇒$
$2\ ≤\ 2\ x\ <\ 12 ⇒$ (divide all sides by $2$) $1\ ≤\ x\ <\ 6$
$x$ is between $1$ and $6$. Choice $D$ represent this inequality.

The correct answer is $120$ cm$^3$
Volume of a box $=$ length $×$ width $×$ height $=4\ ×\ 5\ ×\ 6=120$

The correct answer is $60$
To find the number of possible outfit combinations, multiply number of options for each factor: $3\ ×\ 5\ ×\ 4=60$

The correct answer is $12,324$
In the stadium the ratio of home fans to visiting fans in a crowd is $5:7$. Therefore,
total number of fans must be divisible by $12: 5\ +\ 7 = 12$.
Let’s review the choices:
A. $12,324:$          $12,324\ ÷\ 12=1,027$
B. $42,326$           $42,326\ ÷\ 12=3,527.166$
C. $44,566$           $44,566\ ÷\ 12=3,713.833$
D. $66,812$           $66,812\ ÷\ 12=5,567.666$
Only choice $A$ when divided by $12$ results a whole number.

The correct answer is $60,000$
Three times of $24,000$ is $72,000$. One sixth of them cancelled their tickets.
One sixth of $72,000$ equals $12,000\ (\frac{1}{6}\ ×\ 72000 = 12000)$.
$60,000\ (72,000\ –\ 12,000=60,000)$ fans are attending this week

The correct answer is $97.6$
The area of the square is $595.36$. Therefore, the side of the square is square root of the area. $\sqrt{595.36}=24.4$
Four times the side of the square is the perimeter: $4\ ×\ 24.4=97.6$

The correct answer is $(-\ 2,\ 3)$
$x\ +\ 2\ y=4$. Plug in the values of $x$ and $y$ from choices provided. Then:
A.             $(-\ 2,\ 3)$ $x\ +\ 2\ y=4\ →\ -\ 2\ +\ 2\ (3)=4\ →\ -\ 2\ +\ 6=4$                This is true!
B.             $(1,\ 2)$ $x\ +\ 2\ y=4\ →\ 1\ +\ 2\ (2)=4\ →\ 1\ +\ 4 =4$ This is                   NOT true!
C.             $(-\ 1,\ 3)$ $x\ +\ 2\ y = 4\ →\ -\ 1\ +\ 2\ (3)=4\ →\ -\ 1\ +\ 6=4$               This is NOT true!
D.             $(-\ 3,\ 4)$ $x\ +\ 2\ y=4\ →\ -\ 3\ +\ 2\ (4)=4\ →\ -\ 3\ +\ 8=4$                 This is NOT true!

The correct answer is $10$ meters
The width of the rectangle is twice its length. Let $x$ be the length. Then, width$=2\ x$
Perimeter of the rectangle is $2$ (width $+$ length) $= 2\ (2\ x\ +\ x)=60\ ⇒\ 6\ x=60\ ⇒\ x=10$
Length of the rectangle is $10$ meters.

The correct answer is $\frac{2}{3} ,\ 67\%,\ 0.68,\ \frac{4}{5}$
Change the numbers to decimal and then compare. $\frac{2}{3}=0.666$… $67\%=0.67$
$\frac{4}{5}=0.80$ Then: $\frac{2}{3}\ <\ 67\%\ <\ 0.68\ <\ \frac{4}{5}$

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.
$4400\ –\ 25=4375$, mean$= \frac{sum\ of\ terms}{number\ of\ terms}\ ⇒$ mean$=\frac{4375}{50}=87.5$

The correct answer is $\frac{1}{4}$
To get a sum of $6$ for two dice, we can get $5$ different options: $(5,\ 1),\ (4,\ 2),\ (3,\ 3),\ (2,\ 4),\ (1,\ 5)$
To get a sum of $9$ for two dice, we can get $4$ different options: $(6,\ 3),\ (5,\ 4),\ (4,\ 5),\ (3,\ 6)$
Therefore, there are $9$ options to get the sum of 6 or $9$. Since, we have $6\ ×\ 6 = 36$ total options,
the probability of getting a sum of $6$ and $9$ is $9$ out of $36$ or $\frac{1}{4}$.

The correct answer is $8$
Use formula of rectangle prism volume. $V=$(length) (width) (height) $⇒ 2000=(25)\ (10)$ (height) $⇒$ height$=2000\ ÷\ 250=8$

The correct answer is $32$
The diagonal of the square is $8$. Let $x$ be the side.
Use Pythagorean Theorem: $a^2\ +\ b^2=c^2$
$x^2\ +\ x^2=8^2\ ⇒\ 2\ x^2 = 82\ ⇒\ 2\ x^2 = 64\ ⇒\ x^2 = 32\ ⇒\ x= \sqrt{32}$
The area of the square is: $\sqrt{32}\ ×\ \sqrt{32}=32$

The correct answer is $\frac{1}{4}$
Probability$=\frac{number\ of\ desired\ outcomes}{number\ of\ total\ outcomes} = \frac{18}{12\ +\ 18\ +\ 18\ +\ 24} = \frac{18}{72} =\frac{1}{4}$

The correct answer is $16$
average $=\frac{sum\ of\ terms}{number\ of\ terms}\ ⇒$ (average of $6$ numbers) $12 = \frac{sum\ of\ numbers}{6}\ ⇒$ sum of $6$ numbers is $12\ ×\ 6 = 72$
(average of $4$ numbers) $10 = \frac{sum\ of\ numbers}{4}\ ⇒$ sum of $4$ numbers is $10\ ×\ 4 = 40$
sum of $6$ numbers $–$ sum of $4$ numbers $=$ sum of $2$ numbers $72\ –\ 40 = 32$
average of $2$ numbers $= \frac{32}{2}=16$

The correct answer is $-\ 2$
Solving systems of Equations by Elimination
Multiply the first equation by ($-\ 2$), then add it to the second equation
$\underline{-\ 2\ (2\ x\ +\ 5\ y=11)\\ 4\ x\ -\ 2\ y =-\ 14}$ $\Rightarrow$ $- \ 4\ x\ -\ 10\ y= -\ 22\\ 4\ x\ -\ 2\ y= -\ 14$ $\Rightarrow$ $-\ 12\ y=-\ 36 \Rightarrow \ y=3$
Plug in the value of $y$ into one of the equations and solve for $x$.
$2\ x\ +\ 5\ (3)= 11\ ⇒ \ 2\ x\ +\ 15= 11\ ⇒\ 2\ x= -\ 4\ ⇒\ x= -\ 2$

The correct answer is $70$ cm$^2$
The perimeter of the trapezoid is $36$ cm. Therefore, the missing side (height) is $= 36\ –\ 8\ –\ 12\ –\ 6=10$.
Area of a trapezoid: $A=\frac{1}{2}\ h\ (b_1\ +\ b_2)= \frac{1}{2}\ (10)\ (6\ +\ 8)=70$

The correct answer is $45$
First, find the number. Let $x$ be the number. Write the equation and solve for $x$.
$150\%$ of a number is $75$, then: $1.5\ ×\ x=75\ ⇒\ x=75\ ÷\ 1.5=50$
$90\%$ of $50$ is: $0.9\ ×\ 50=45$

The correct answer is $\frac{3\ x\ -\ 1}{x^2\ -\ x}$
$(\frac{f}{g})\ (x) = \frac{f\ (x)}{g\ (x)} = \frac{3\ x\ –\ 1}{x^2\ -\ x}$

The correct answer is $\frac{1}{4}$
The probability of choosing a Hearts is $\frac{13}{52}=\frac{1}{4}$

The correct answer is $27$
First, find the sum of five numbers.
average $=\frac{sum\ of\ terms}{number\ of\ terms}\ ⇒\ 24 = \frac{sum\ of\ 5\ numbers}{5}\ ⇒$ sum of $5$ numbers $= 24\ ×\ 5 = 120$
The sum of $5$ numbers is $120$. If a sixth number that is greater than $42$ is added to these numbers,
then the sum of $6$ numbers must be greater than $162$. $120\ +\ 42 = 162$
If the number was $42$, then the average of the numbers is:
average $=\frac{sum\ of\ terms}{number\ of\ terms}=\frac{162}{6}=27$
Since the number is bigger than $42$. Then, the average of six numbers must be greater than $27$. Choice $D$ is greater than $27$.

The correct answer is $45$
Let $L$ be the length of the rectangular and $W$ be the with of the rectangular. Then, $L=4\ W\ +\ 3$
The perimeter of the rectangle is $36$ meters. Therefore: $2\ L\ +\ 2\ W=36$ $L\ +\ W=18$
Replace the value of $L$ from the first equation into the second equation and solve for $W$:
$(4\ W\ +\ 3)\ +\ W=18\ →\ 5\ W\ +\ 3=18\ →\ 5\ W=15\ →\ W=3$
The width of the rectangle is $3$ meters and its length is: $L=4\ W\ +\ 3=4\ (3)\ +\ 3=15$
The area of the rectangle is: length $×$ width $= 3\ ×\ 15 = 45$

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

The correct answer is $\frac{1}{22}$
$2,500$ out of $55,000$ equals to $\frac{2500}{55000}=\frac{25}{550}=\frac{1}{22}$

The correct answer is $60$
Jason needs an $75\%$ average to pass for five exams. Therefore, the sum of $5$ exams must be at lease $5\ ×\ 75 = 375$.
The sum of $4$ exams is: $68\ +\ 72\ +\ 85\ +\ 90=315$
The minimum score Jason can earn on his fifth and final test to pass is: $375\ –\ 315=60$

The correct answer is $\frac{1}{4}$
Isolate and solve for $x$. $\frac{2}{3}\ x\ +\ \frac{1}{6}= \frac{1}{3}\ ⇒\ \frac{2}{3}\ x= \frac{1}{3}\ -\ \frac{1}{6} = \frac{1}{6}\ ⇒\ \frac{2}{3}\ x= \frac{1}{6}$
Multiply both sides by the reciprocal of the coefficient of $x$. $(\frac{3}{2})\ \frac{2}{3}\ x=\frac{1}{6}$ $(\frac{3}{2})\ ⇒\ x= \frac{3}{12}=\frac{1}{4}$

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 $= 2\ π\ (3)\ (3\ +\ 8) = 66\ π$

The correct answer is $\frac{125}{512}$
The square of a number is $\frac{25}{64}$, then the number is the square root of $\frac{25}{64}$ $\sqrt{\frac{25}{64}}= \frac{5}{8}$
The cube of the number is: $(\frac{5}{8})^3 = \frac{125}{512}$

The correct answer is $28$
Write the numbers in order: $2,\ 19,\ 27,\ 28,\ 35,\ 44,\ 67$
Median is the number in the middle. So, the median is $28$.

The correct answer is $10$
Use Pythagorean Theorem: $a^2\ +\ b^2= c^2$                $6^2\ +\ 8^2= c^2\ ⇒\ 100= c^2 \ ⇒\ c=10$

The correct answer is $40$
Plug in $104$ for $F$ and then solve for $C$. $C= \frac{5}{9}\ (F\ –\ 32)\ ⇒\ C= \frac{5}{9}\ (104\ –\ 32)\ ⇒ C= \frac{5}{9}\ (72)=40$

The correct answer is $10$
Let $x$ be the number. Write the equation and solve for $x$.
$40\%$ of $x=4\ ⇒\ 0.40\ x=4 \ ⇒\ x=4\ ÷\ 0.40=10$

The correct answer is $$810$
Let $x$ be all expenses, then $\frac{22}{100}\ x=$660\ →\ x=\frac{100\ ×\ $660}{22}=$3,000$
He spent for his rent: $\frac{27}{100}\ ×\ $3,000=$810$

The correct answer is $6$ hours
The distance between Jason and Joe is $9$ miles.
Jason running at $5.5$ miles per hour and Joe is running at the speed of $7$ miles per hour.
Therefore, every hour the distance is $1.5$ miles less. $9\ ÷\ 1.5=6$

The correct answer is $80\%$
The failing rate is $11$ out of $55 = \frac{11}{55}$. Change the fraction to percent: $\frac{11}{55}\ ×\ 100\%=20\%$
$20$ percent of students failed. Therefore, $80$ percent of students passed the exam.

The correct answer is $$840$
Use simple interest formula: $I=$ prt ($I =$ interest, $p =$ principal, $r =$ rate, $t =$ time)
$I=(12000)\ (0.035)\ (2)=840$

The correct answer is $48\ x^8\ y^6$
Simplify. $6\ x^2\ y^3\ (2\ x^2\ y)^3= 6\ x^2\ y^3\ (8\ x^6\ y^3 ) = 48\ x^8\ y^6$

The correct answer is $g(x)=-\ 2\ x\ +\ 1$
Plugin 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.

The correct answer is $\frac{3\ x}{4}$
$\sqrt{\frac{x^2}{2}\ +\ \frac{x^2}{16}}=\sqrt{\frac{8\ x^2}{16}\ +\ \frac{x^2}{16}}=\sqrt{\frac{9\ x^2}{16}}=\sqrt{\frac{9}{16}\ x^2} =\sqrt{\frac{9}{16}}\ ×\ \sqrt{x^2}=\frac{3}{4}\ ×\ x=\frac{3\ x}{4}$

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 $-\ 4$.

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 $6\ a=6\ (5)=30$

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.

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:
$\frac{5\ x\ +\ 4}{x^3}\ +\ \frac{7\ x\ -\ 3}{x^3} =\frac{(5\ x\ +\ 4)\ +\ (7\ x\ -\ 3)}{x^3} =\frac{12\ x\ +\ 1}{x^3}$

The correct answer is $y=4\ (x\ -\ 3)^2\ -\ 3$
Let’s find the vertex of each choice provided:
A.                    $y=3\ x^2\ -\ 3$                    The vertex is: $(0,\ -\ 3)$
B.                    $y=-\ 3\ x^2\ +\ 3$                The vertex is: $(0,\ 3)$
C.                    $y=x^2\ +\ 3\ x\ -\ 3$
The value of $x$ of the vertex in the equation of a quadratic in standard form is: $x=\frac{-\ b}{2\ a}=\frac{-\ 3}{2}$
(The standard equation of a quadratic is: $a\ x^2\ +\ b\ x\ +\ c=0)$
The value of $x$ in the vertex is $3$ not $\frac{-\ 3}{2}$.
D.                     $y=4\ (x\ -\ 3)^2\ -\ 3$
Vertex form of a parabola equation is in form of $y=a\ (x\ -\ h)^2\ +\ k$, where $(h,\ k)$ is the vertex. Then $h=3$ and $k=-\ 3$. (This is the answer)

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: $\frac{(4\ x\ +\ 2)\ +\ (-\ 6\ x\ -\ 5)\ +\ (8\ x\ +\ 2)}{3}=\frac{6\ x\ -\ 1}{3}=2\ x\ -\frac{1}{3}$

The correct answer is $-\frac{1}{2}$
The equation of a line in slope intercept form is: $y=m\ x\ +\ b$.
Solve for $y$. $4\ x\ -\ 2\ y=12\ ⇒\ -\ 2\ y=12\ -\ 4\ x\ ⇒\ y=(12\ -\ 4\ x)\ ÷\ (-\ 2)\ ⇒\ y=2\ x\ -\ 6$. The slope is $2$.
The slope of the line perpendicular to this line is: $m\ 1\ ×\ m\ 2 = -\ 1\ ⇒\ 2\ ×\ m\ 2 = -\ 1\ ⇒\ m\ 2=-\ \frac{1}{2}$

The correct answer is $(1,\ 6),\ (2,\ 5),\ (−\ 5,\ 8)$
Since the triangle $A\ B\ C$ is reflected over the $y\ -$ axis,
then all values of y’s of the points don’t change and the sign of all x’s change.
(remember that when a point is reflected over the $y\ -$axis,
the value of $y$ does not change and when a point is reflected over the $x\ -$axis, the value of $x$ does not change).
Therefore: $(−\ 1,\ 6)$ changes to $(1,\ 6)$. $(−\ 2,\ 5)$ changes to $(2,\ 5)$. $(5,\ 8)$ changes to $(−\ 5,\ 8)$

The correct answer is $39$
The area of rectangle is:$9\ ×\ 4=36$ cm$^2$.
The area of circle is: $π\ r^2=π\ ×(\frac{10}{2})^2=3\ ×\ 25=75$ cm$^2$. Difference of areas is: $75\ -\ 36=39$

The correct answer is $\frac{2}{x^3}\ +\ 2$
$f(g(x) )=2\ ×(\frac{1}{x})^3\ +\ 2=\frac{2}{x^3}\ +\ 2$

The correct answer is $4$
$1269=6^4$ $→\ 6^x=6^4\ →\ x=4$

The correct answer is $170$ miles
Use the information provided in the question to draw the shape.
Use Pythagorean Theorem: $a^2\ +\ b^2 = c^2$
$80^2\ +\ 150^2 = c^2\ ⇒\ 6400\ +\ 22500 = c^2\ ⇒\ 28900 = c^2\ ⇒ c = 170$

The correct answer is $45$ m$^2$
Let $L$ be the length of the rectangular and $W$ be the with of the rectangular. Then, $L=4\ W\ +\ 3$
The perimeter of the rectangle is $36$ meters. Therefore: $2\ L\ +\ 2\ W=36$          $L\ +\ W=18$
Replace the value of $L$ from the first equation into the second equation and solve for $W$:
$(4\ W\ +\ 3)\ +\ W=18\ →\ 5\ W\ +\ 3=18\ →\ 5\ W=15\ →\ W=3$
The width of the rectangle is $3$ meters and its length is: $L=4\ W\ +\ 3=4\ (3)\ +\ 3=15$
The area of the rectangle is: length $×$ width $= 3\ ×\ 15 = 45$

The correct answer is $5$
Let $x$ be the number of adult tickets and $y$ be the number of student tickets. Then:
$x\ +\ y=12$,                  $12.50\ x\ +\ 7.50\ y=125$
Use elimination method to solve this system of equation. Multiply the first equation by $-\ 7$.$5$ and add it to the second equation.                                   $-\ 7.5\ (x\ +\ y=12)$,                        $-\ 7.5\ x\ -\ 7.5\ y=-\ 90$,                       $12.50\ x\ +\ 7.50\ y=125$.                         $5\ x=35$, $x=7$
There are $7$ adult tickets and $5$ student tickets.

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 $b$. $\frac{a}{b}=\frac{\frac{b}{25}}{b}\ →\ \frac{b}{25}\ ÷\ b=\frac{b}{25}\ ×\ \frac{1}{b}=\frac{b}{25\ b}=\frac{1}{25}$

The correct answer is $30$
Plug in the value of $x$ in the equation and solve for $y$.
$2\ y=\frac{ 2\ x^2}{3}\ +\ 6\ →\ 2\ y = \frac{2\ (9)^2}{3}\ +\ 6\ →\ 2\ y=\frac{2\ (81)}{3}\ +\ 6\ →\ 2\ y= 54\ +\ 6=60$
$2\ y = 60\ →\ y=30$

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

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: $y(x)=8\ x$