Free Full Length SHSAT Mathematical Practice Test

The correct answer is $− \ \frac{1}{8}$
The equation of a line in slope intercept form is: $y=m \ x \ + \ b$
Solve for $y$.
$8 \ x \ - \ y=2→ \ - \ y= \ - \ 8 \ x \ + \ 2$
Divide both sides by $(- \ 1)$.
Then: $- \ y= \ - \ 8 \ x \ + \ 2→y=8 \ x \ - \ 2$
The slope of this line is $8$.
The product of the slopes of two perpendicular lines is $- \ 1$. 
Therefore, the slope of a line that is perpendicular to this line is:
$m_{1} \ × \ m_{2} = \ - \ 1 ⇒ 8 \ × \ m_{2} = \ - \ 1 ⇒ m_{2} = \frac{- \ 1}{8}= \ - \ \frac{1}{8}$

The correct answer is $104$
Plug in the value of $x$ and $y$.
$4 \ (x \ - \ 5 \ y) \ + \ (6 \ - \ x)^2$ when $x=2$ and $y= - \ 4$
$x=2$ and $y= - \ 4$
$4 \ (x \ - \ 5 \ y) \ + \ (6 \ - \ x)^2=4 \ (2 \ - \ 5 \ (- \ 4)) \ + \ (6 \ - \ 2)^2=4 \ (2 \ + \ 20) \ + \ ( 4)^2 = 88\ + \ 16=104$

The correct answer is $67.5$
Use PEMDAS (order of operation):
$\left[ 5 \ × \ (18) \ - \ 57 \ \right] \ – \ (– \ 12) \ + \ \left[ \ 5 \ × \ 9 \ \right] \ ÷ \ 2=$
$\left[ 90 \ - \ 57 \ \right] \ + \ 12 \ + \ 45 \ ÷ \ 2=$
$33\ + \ 12 \ + \ 22.5  = \ 67.5$

The correct answer is $− \ 153$
To solve absolute values equations, write two equations. 
$2 \ x \ - \ 8$ can equal positive $26$, or negative $26$. Therefore, $2 \ x \ - \ 8= 26 ⇒ 2 \ x=34⇒ x=17$
$2 \ x \ - \ 8= - \ 26 ⇒ 2 \ x= - \ 26 \ + \ 8=  - \ 18⇒ x= - \ 9$
Find the product of solutions: $- \ 9 \ × \ 17=  - \ 153$

The correct answer is $5.73$ ft
Write a proportion and solve for the missing number.
$\frac{46}{22} = \frac{12}{x}→ 46\ x=12\ × \ 22=264$
$46  \ x=264 →x=\frac{246}{46}=5.73$

The correct answer is $5.73$ ft
Write a proportion and solve for the missing number.
$\frac{46}{22} = \frac{12}{x}→ 46\ x=12\ × \ 22=264$
$46  \ x=264 →x=\frac{246}{46}=5.73$

Solve for $x$.
$- \ 2 \ ≤ \ 7 \ x \ - \ 2 \ < \ 12 ⇒$ (add $4$ all sides)
$- \ 2 \ + \ 2 \ ≤ \  7 \ x \ - \ 2 \ + \ 2 \ < \ 12\ + \ 2 ⇒$ 
$0 \ ≤ \ 7 \ x \ < \ 14 ⇒$ (divide all sides by $7$) 
$0 \ ≤ \ x \ < \ 2$
$x$ is between $0$ and $2$.
Choice B represent this inequality.

The correct answer is $-\ 3 \ x^5 \ + \ 7 \ x^4 \ + \ 15 \ x^2\ + \ x^3$
Simplify and combine like terms.
$(6 \ x^2 \ - \ 7 \ x^4 \ + \ 6 \ x^3 ) \ - \ (3 \ x^5 \ - \ 9 \ x^2 \ + \ 5 \ x^3 ) ⇒$
$(6 \ x^2 \ - \ 7 \ x^4 \ + \ 7 \ x^3 ) \ - \ 3 \ x^5 \ + \ 9 \ x^2 \ - \ 5 \ x^3⇒$
$-\ 3 \ x^5 \ + \ 7 \ x^4 \ + \ 15 \ x^2\ + \ x^3$

The correct answer is $343$
$7^3 = 7 \ × \ 7 \ × \ 7  = 343$

The correct answer is $96$ cm$^3$
Volume of a box $=$ length $×$ width $×$ height $= 6 \ × \ 8 \ × \ 2 = 96$

The correct answer is $78.4$
The area of the square is $387.3$.
Therefore, the side of the square is square root of the area.
$\sqrt{387.3}=19.6$
Four times the side of the square is the perimeter:
$4 \ × \ 19.6 = 78.4$

The correct answer is $68\%$
the population is increased by $25\%$ and $35\%$.
$15\%$ increase changes the population to $125\%$ of original population.
For the second increase, multiply the result by $135\%$.
$(1.25) \ × \ (1.35) =168\%$
$68$ percent of the population is increased after two years.

The correct answer is $\frac{4}{3}, 55\%, 0.48, \frac{2}{5}$
Change the numbers to decimal and then compare.
$\frac{4}{3} = 1.333$…
$0.48$
$55\% = 0.55$
$\frac{2}{5} = 0.4$
Therefore
$\frac{4}{3} \ < \ 55\% \ < \ 0.48 \ < \ \frac{ 2}{5}$

The correct answer is $144,000$
Three times of $45,000$ is $180,000$.
One sixth of them cancelled their tickets.
One sixth of $180,000$ equals $36,000\ (\frac{1}{5} \ × \ 180000 = 36000)$. 
$144,000 \ (180000 \ – \ 36000 = 144000)$ fans are attending this week

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

The correct answer is $180$ cm$^2$
The perimeter of the trapezoid is $36$ cm.
Therefore, the missing side (height) is $= 54 \ – \ 6 \ – \ 10 \ – \ 4 = 36$
Area of a trapezoid: A $= \frac{1}{2} \ h \ (b_1 \ + \ b_2) = \frac{1}{2} \ (36) \ (4 \ + \ 6) = 180$

The correct answer is $67.8$
average (mean) $= \frac{sum \ of \ terms}{number \ of \ terms} ⇒$
$60= \frac{sum \ of \ terms}{68} ⇒$ sum $=  60\ × \ 68 = 4080$
The difference of $32$ and $23$ is $10$.
Therefore, $10$ should be subtracted from the sum.
$4080 \ – \ 10 = 4070$
mean $= \frac{sum \ of \ terms}{number \ of \ terms} ⇒$
mean $= \frac{4070}{60} = 67.8$

The correct answer is $33,000$ and $78,120$
(If you selected $3$ choices and $2$ of them are correct, then you get one point. 
If you answered $2$ or $3$ choices and one of them is correct, you receive one point. 
If you selected more than $3$ choices, you won’t get any point for this question.)
In the stadium the ratio of home fans to visiting fans in a crowd is $9:6$.
Therefore, total number of fans must be divisible by $15: \ 9 \ + \ 6 = 15$. 
Let’s review the choices: 
☐A. $12,324$:      $12,324 \ ÷ \ 15 = 821.6$
☐B. $78,120$:      $16,788 \ ÷ \ 15 = 5,208$
☐C. $42,326$:      $42,326 \ ÷ \ 15 =2,821.7$
☐D. $33,000$:      $33,000 \ ÷ \ 15 = 22,000$
☐E. $66,812$:      $66,812 \ ÷ \ 15 =4,454.1333$
Only choices D and B when divided by $15$ result a whole number.

The correct answer is $6$
Use formula of rectangle prism volume.
V $=$ (length) (width) (height) $⇒ 4000 = (20) \ (25)$ (height) $⇒$ height $= 3000 \ ÷ \ 500 = 6$

The correct answer is $18$
The diagonal of the square is $6$.
Let $x$ be the side. 
Use Pythagorean Theorem: $a^2 \ + \ b^2 = c^2$
$x^2 \ + \ x^2 = 6^2 ⇒$
$2 \ x^2 = 6^2 ⇒$
$2 \ x^2 = 36 ⇒$
$x^2 = 18 ⇒$
$x= \sqrt{18}$
The area of the square is:
$\sqrt{18} \ × \ \sqrt{18} =18$

The correct answer is $\frac{1}{6}$
Probability $= \frac{number \ of \ desired \ outcomes}{number \ \ of \ total \ outcomes} = \frac{8}{11 \ + \ 8 \ + \ 16 \ + \ 13} = \frac{8}{48} = \frac{1}{6}$

The correct answer is $42$
To find the number of possible outfit combinations, multiply number of options for each factor:
$7 \ × \ 3 \ × \ 2= 42$

The correct answer is $\frac{3}{4}$
The probability of choosing a Hearts is $\frac{14}{64}=\frac{3}{4}$

The correct answer is $17$
average $= \frac{sum \ of \ terms}{number \ of \ terms} ⇒$ 
(average of $4$ numbers) $31 = \frac{sum \ of \ numbers}{4} ⇒$
sum of $4$ numbers is $31 \ × \ 4 = 124$
(average of $3$ numbers) $23 = \frac{sum \ of \ numbers}{3} ⇒$ sum of $3$ numbers is 
$23 \ × \ 3 = 56$
sum of $4$ numbers $–$ sum of $3$ numbers $=$ sum of $4$ numbers
$124 \ – \ 56 = 68$, average of $4$ numbers $= \frac{68}{4 }= 17$

The correct answer is $(− \ 2,3)$ and $(0, 2)$
(If you selected $3$ choices and $2$ of them are correct, then you get one point.
If you answered $2$ or $3$ choices and one of them is correct, you receive one point.
If you selected more than $3$ choices, you won’t get any point for this question.)
$4\ x \ - \ y=8$. 
Plug in the values of $x$ and $y$ from choices provided. Then:
☐A. $(- \ 2,3)$:      $4 \ x \ -   \ y=8→ \ - \ 4 \ (2)\  - \ 3 =8→ \ - \ 8 \ - \ 3=11$ This is NOT  true!
☐B. $(1,2)$:          $4 \ x \ -\  y=8→4\ (1) \ - \ 2 =8→4 \ - \ 2=2$ This is NOT true!
☐C. $(- \ 1,3)$:      $4 \ x \ - \  y=8→ \  4\ (- \ 1) \ - \ 3=4→ \ - \ 4 \ -  \ 3=- \ 7$ This is NOT true!
☐D. $(- \ 3,4)$:     $4 \ x \ - \ y=8→ \ 4\ (- \ 3) \ -  \ 4=8→ \ - \ 12 \ - \ 4=8$ This is  true!
☐E. $(4,8)$:         $4 \ x \ - \  y=8→4 \ (4) \ -  \ 8 =8→16 \ - \ 8=8$ This is true!

The correct answer is $8$ meters
The width of the rectangle is twice its length.
Let $x$ be the length. Then, width $=3 \ x$
Perimeter of the rectangle is $2$ (width $+$ length) $= 2 \ (3 \ x \ + \ x)=56 ⇒ 7\ x=56 ⇒ x=8$
Length of the rectangle is $8$ meters.

The correct answer is $30$ and $25$
(If you selected $3$ choices and $2$ of them are correct, then you get one point.
If you answered $2$ or $3$ choices and one of them is correct, you receive one point.
If you selected more than $3$ choices, you won’t get any point for this question.)
First, find the sum of five numbers. 
average $= \frac{sum \ of \ terms}{number \ of \ terms} ⇒$
$12 = \frac{sum \ of \ 3\ numbers}{3 }⇒$
sum of $3$ numbers $= 12 \ × \ 3 = 36$
The sum of $3$ numbers is $36$.
If a fouth number that is greater than $52$ is added to these numbers, then the sum of $4$ numbers must be greater than $88$. 
$36 \ + \ 52 = 88$
If the number was $52$, then the average of the numbers is: 
average $= \frac{sum \ of \ terms}{number \ of \ terms} = \frac{88}{4} = 22$
Since the number is bigger than $52$.
Then, the average of six numbers must be greater than $22$. 
Choices B and E are greater than $22$.

The correct answer is $2$
Solving Systems of Equations by Elimination
Multiply the first equation by $(– \ 3)$, then add it to the second equation.
$\cfrac{\begin{align} \ 3 \ x \ + \ 6 \ y \ = \ 18 \\- \ 3 \ ( x \ - \ 4 \ y \ = \ - \ 6 )\end{align}}{\cfrac{\begin{align} 3 \ x \ + \ 6\ y \ = 18 \\ -\ 3 \ x \ + \ 12 \ y \ = 18 \end{align}}{{\begin{align} 18\ y \ =  36 \\ ⇒ y \ = \ 2 \end{align}} } } ⇒$
Plug in the value of $y$ into one of the equations and solve for $x$.
$3 \ x \ + \ 6 \ (2)= 18 ⇒$
$3 \ x \ + \ 12= 18 ⇒$
$3 \ x= 6 ⇒ x= 2$

The correct answer is $\frac{1}{10}$
$3,000$ out of $45,000$ equals to $\frac{3000}{45000 }= \frac{1500}{15000} =\frac{1}{10}$

The correct answer is $14$
Let $x$ be the width of the rectangle. 
Use Pythagorean Theorem: 
$a^2 \ + \ b^2 = c^2$
$x^2 \ + \ 3^2 = 5^2 ⇒$
$x^2 \ + \ 9 = 25 ⇒$
$x^2 = 25 \ – \ 9 = 16 ⇒ x = 4$
Perimeter of the rectangle $= 2$ (length $+$ width) $= 2 \ (3 \ + \ 4) = 2 \ (7) = 14$

The correct answer is $720 \ π$ in$^2$
Surface Area of a cylinder $= 2 \ π \ r \ (r \ + \ h)$,
The radius of the cylinder is $6 \ (12 \ ÷ \ 2)$ inches and its height is $10$ inches. Therefore, 
Surface Area of a cylinder $= 2 \ π \ (6) \ (6 \ + \ 10) = 720 \ π$ in$^2$

The correct answer is $8$
The ratio of boy to girls is $6:8$.
Therefore, there are $6$ boys out of $14$ students.
To find the answer, first divide the total number of students by $14$, then multiply the result by $4$. 
$56 \ ÷ \ 14 = 4 ⇒ 4 \ × \ 6 = 24$
There are $24$ boys and $32 \ (56 \ – \ 24)$ girls.
So, $8$ more boys should be enrolled to make the ratio $1:1$

The correct answer is $85$
Jason needs an $82\%$ average to pass for five exams.
Therefore, the sum of $5$ exams must be at least $5 \ × \ 82 =410$
The sum of $4$ exams is: $62\ + \ 95 \ + \ 78 \ + \ 90 = 325$.
The minimum score Jason can earn on his fifth and final test to pass is: $410\ – \ 325 = 85$

The correct answer is $80 \ x^{10} \ y^8$
Simplify: 
$5 \ x^4 \ y^6 \ (4 \ x^3 \ y)^2= 5 \ x^4 \ y^6 \ (16 \ x^6 \ y^2 ) = 80 \ x^{10} \ y^8$

The correct answer is $\frac{64}{343}$
The square of a number is $\frac{16}{49}$, then the number is the square root of $\frac{16}{49}$
$\sqrt{\frac{16}{49}}= \frac{4}{7}$
The cube of the number is: $(\frac{4}{7})^3 = \frac{64}{343}$

The correct answer is $\frac{2}{3}$
Isolate and solve for $x$. 
$\frac{4}{5} \ x \ + \ \frac{1}{3}= \frac{1}{5} ⇒$
$\frac{4}{5} \ x= \frac{1}{5} \ - \ \frac{1}{3} = \frac{8}{15} ⇒$
$\frac{4}{5} \ x= \frac{8}{15}$ 
Multiply both sides by the reciprocal of the coefficient of $x$.
$(\frac{5}{4}) \ \frac{4}{5} \ x= \frac{8}{15} \ (\frac{5}{4}) ⇒$
$x= \frac{8}{12}=\frac{2}{3}$

The correct answer is $30$
Write the numbers in order:
$3, 14, 27, 30, 46, 52, 85$
Median is the number in the middle.
So, the median is $30$.

The correct answer is $$625$
Use simple interest formula: $I=prt$
($I =$ interest, $p =$ principal, $r =$ rate, $t =$ time)
$I=(25000) \ (0.005) \ (5)=625$

The correct answer is $50$
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 $$336$
Let $x$ be all expenses, then:
$\frac{22}{100} \ x=$264 →x=\frac{100 \ × \ $264}{22}=$1,200$
He spent for his rent:
$\frac{28}{100} \ × \ $1,200=$336$

The correct answer is $16$
First, find the number.
Let $x$ be the number.
Write the equation and solve for $x$. 
$75\%$ of a number is $48$, then: $0.75\ × \ x=48 ⇒ x=48 \ ÷ \ 0.75=64$
$25\%$ of $64$ is: 
$0.25\ × \ 64 = 16$

The correct answer is $7$ hours
The distance between Jason and Joe is $14$ miles.
Jason running at $6$ miles per hour and Joe is running at the speed of $8$ miles per hour.
Therefore, every hour the distance is $2$ miles less. 
$14 \ ÷ \ 2 = 7$

The correct answer is $56$
Plug in $100$ for F and then solve for C.
C $= \frac{7}{8}$ (F $– \ 36) ⇒$ C $= \frac{7}{8} (100 \ – \ 36) ⇒$ C $= \frac{7}{8} \ (64) = 56$

The correct answer is $85\%$
The failing rate is $12$ out of $80 = \frac{12}{80}$
Change the fraction to percent: $\frac{12}{80} \ × \ 100\%=15%$
$15$ percent of students failed.
Therefore, $85$ percent of students passed the exam.

The correct answer is $56$
Let $x$ be the number.
Write the equation and solve for $x$. 
$25\%$ of $x=14⇒ 0.25 \ x=14 ⇒ x=14 \ ÷ \ 0.25=56$

The correct answer is $100 \ x \ + \ 11,000 \ ≤ \ 18,000$
Let $x$ be the number of new shoes the team can purchase.
Therefore, the team can purchase $100 \ x$.
The team had $$18,000$ and spent $$11,000$.
Now the team can spend on new shoes $$7,000$ at most. 
Now, write the inequality: $100 \ x \ + \ 11,000 \ ≤ \ 18,000$

The correct answer is $3$
Solve for $y$.
$3 \ x \ - \  y=9 ⇒  -\  y=9 \ - \ 3 \ x ⇒ y=3 \ x \ - \ 9$
The slope of the line is $3$.