Full Length DAT Quantitative Reasoning Practice Test

The correct answer is $30$
The area of the floor is: $5$ cm $× \ 30$ cm $= 150$ cm
The number is tiles needed $= 150\ ÷ \ 5=30$

The correct answer is $- \ 1$
Solving Systems of Equations by Elimination method.
$\cfrac{\begin{align} 4 \ x \ + \ 6 \ y \ = \ 10 \\  x \ + \  y \ =  3 \end{align}}{}$
Multiply the second equation by $- \ 4$, then add it to the first equation.
$\cfrac{\begin{align} 4 \ x \ + \ 6 \ y \ = 10 \\ - \ 4 \ (    x \ + \ y \ =  3) \end{align}}{} ⇒$
$\cfrac{ \begin{align} 4 \ x \ + \ 6 \ y \ =  10 \\ - \ 4 \ x \ - \ 4 \ y \ = 12 \end{align} }{\begin{align} 2\ y \ = - \ 2  \\ ⇒ y \ = - \ 1  \end{align}}$

The correct answer is $0.69$ D
To find the discount, multiply the number by ($100\% \ –$ rate of discount).
Therefore, for the first discount we get: (D) $(100\% \ – \ 30\%) =$ (D) $(0.60) = 0.60$ D
For increase of $10\%: \ (0.60$ D) $(100\% \ + \ 15\%) = (0.60$ D)$(1.15) = 0.69$ D $= 69\%$ of D

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

The correct answer is $0.0555$
$1000$ times the number is $55.5$. 
Let $x$ be the number, then:
$1000 \ x=55.5$
$x=\frac{55.5}{1000}=0.0555$

The correct answer is $9$ hours and $36$ minutes
Use distance formula:
Distance $=$ Rate $×$ time $⇒ 384 = 40 \ ×$ T, divide both sides by $40$.
$\frac{384}{40} =$ T $⇒$ T $= 9.6$ hours.
Change hours to minutes for the decimal part. $0.6$ hours $= 0.6 \ × \ 60=36$ minutes.

The correct answer is $165$
$x=35 \ + \ 130=165$

The correct answer is $88$
$35\%$ of $80$ equals to: $0.35 \ × \ 80=28$
$12\%$ of $600$ equals to: $0.15\ × \ 400=60$
$35\%$ of $80$ is added to $15\%$ of $400: \ 28 \ + \ 60=88$

The correct answer is $36$
The area of $\triangle$BED is $25$, then: $\frac{5 \ × \ AB}{2}=25→5 \ ×$ AB $=50→$ AB $=10$
The area of $\triangle$BDF is $18$, then: $\frac{3 \ × \ BC}{2}=12→3 \ ×$ BC $=24→$ BC $=8$
The perimeter of the rectangle is $= 2 \ × \ (10 \ + \ 8)=36$

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

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

The correct answer is $1,000$ ml
$3\%$ of the volume of the solution is alcohol.
Let $x$ be the volume of the solution. 
Then: $3\%$ of $x=30$ ml $⇒ 0.03 \ x=30 ⇒ x=30 \ ÷ \ 0.03=1,000$

The correct answer is $\frac{26}{27}$
If $26$ balls are removed from the bag at random, there will be one ball in the bag.
The probability of choosing a red ball is $1$ out of $27$.
Therefore, the probability of not choosing a red ball is $26$ out of $27$ and the probability of having not a red ball after removing $26$ balls is the same.

The correct answer is $12\%$
The percent of girls playing tennis is: $60\% \ × \ 20\%=0.60 \ × \ 0.20=0.12=12\%$

The correct answer is $(– \ 2,2)$
Plug in each pair of number in the equation:
A. $(2,1)$:         $4 \ (2) \ + \ 6 \ (1)=15$ Nope!
B. $(– \ 1,3)$:     $4 \ (– \ 1) \ + \ 6 \ (3)=14$ Nope!
C. $(– \ 2,2)$:     $4 \ (– \ 2) \ + \ 6 \ (2)=4$ Bingo!
D. $(2,2)$:        $4 \ (2) \ + \ 6 \ (2)=20$ Nope!
E. $(2,8)$:        $4 \ (2) \ + \ 6 \ (8)=56$ Nope!

The correct answer is $40$ ft
Write a proportion and solve for $x$.
$\frac{4}{3}=\frac{x}{30} ⇒ 3 \ x=4 \ × \ 30 ⇒ x=40$ ft

The correct answer is $584$ km
Add the first $5$ numbers.
$30 \ + \ 62 \ + \ 47 \ + \ 70 \ + \ 50=259$
To find the distance traveled in the next $5$ hours, multiply the average by number of hours.
Distance $=$ Average $×$ Rate $=65 \ × \ 5=325$
Add both numbers. $259 \ + \ 325=584$

The correct answer is $- \ 2$
Solve for $x$.
$\frac{2 \ x}{12}=\frac{x \ + \ 1}{3}$
Multiply the second fraction by $4$.
$\frac{2 \ x}{12}=\frac{4 \ (x \ +\ 1)}{3 \ × \ 4}$
Tow denominators are equal. Therefore, the numerators must be equal.
$2  \ x=4 \ x \ + \ 4$
$-  \ 2 \ x =  4$
$x= - \ 2$

The correct answer is $- \ 9 \ ≤ \ x \ ≤ \ 5$
$|x \ + \ 7 | \ ≤ \ 2→$
$- \ 2 \ ≤ \ x \ + \ 7 \ ≤ \ 2→$
$- \ 2 \ - \ 7 \ ≤ \ x \ + \ 7 \ - \ 7 \ ≤ \ 2 \ - \ 7→$
$-9 \ ≤ \ x \ ≤ \ 5$

The correct answer is $90\%$
The question is this: $1.50$ is what percent of $1.35$?
Use percent formula:
part $=\frac{percent}{100} \ ×$ whole 
$1.50 = \frac{percent}{100} \ × \ 1.35 ⇒$
$1.50=\frac{percent \ × \ 1.35}{100} ⇒$
$150=$ percent $× \ 1.35 ⇒$ percent $=\frac{150}{1.35}= 90$

The correct answer is $169$
$0.4 \ x=(0.25) \ × \ 32→x=20→(x \ - \ 7)^2=(13)^2=169$

The correct answer is $(12,5)$
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 $(12,5)$.

The correct answer is $\frac{8}{10}$
tan $θ=\frac{opposite}{adjacent}$
tan $θ=\frac{8}{6}⇒$ we have the following right triangle.
Then:
$c=\sqrt{8^2 \ + \ 6^2 }=\sqrt{64 \ + \ 36}=\sqrt{100}=10$
cos $θ=\frac{adjacent}{hypotenuse}=\frac{8}{10}$

The correct answer is $\frac{1}{3}$
Set of number that are not composite between $1$ and $45$:
A $= \left\{1, 2, 3, 5, 7, 11, 13, 17, 19, 23 , 29 , 31 ,37 ,41 ,43 \right\}$
Probability $= \frac{number \ of \ desired \ outomes}{number \ of \ total \ outcomes} =\frac{15}{45}=\frac{1}{3}$

The correct answer is $y= x$
The slop of line A is: $m=\frac{y_{2} \ - \ y_{1}}{x_{2} \ - \ x_{1}}=\frac{7 \ - \ 6}{5 \ - \ 4}=1$
Parallel lines have the same slope and only choice  D $(y=x)$ has slope of $1$.

The correct answer is $316$
$y = 5 \ a \ b \ + \  b^4$
Plug in the values of $a$ and $b$ in the equation: $a=3$ and $b=4$
$y = 5 \ (3) \ (4) \ + \   (4)^4 = 60 \ + \  (256) = 316$

The correct answer is $69$
The area of trapezoid is: $(\frac{30 \ + \ 12}{2}) \ × \ x=192→24 \ x=192→x=8$
$y=\sqrt{12^2 \ + \ 5^2}=13$
Perimeter is: $18 \ + \ 30 \ + \ 8 \ + \ 13=69$

The correct answer is $3 \ x^2 \ – \ 7 \ x \ + \ 13$
$(g \ – \ f)(x)=g(x) \ – \ f(x)=(3 \ x^2 \ + \ 5 \ – \ 4  \ x) \ – \ (-  \ 8 \ + \ 3 \ x)$
$3 \ x^2 \ + \ 5 \ – \ 4 \ x \ + \ 8 \ – \ 3 \  x = \ 3 \ x^2 \ - \ 7 \ x \ + \ 13$

The correct answer is $x$ is multiplied by $3$
Plug in $z\times 3$ for $z$ and simplify.
$x_{1}=\frac{8 \ y \ + \ \frac{r}{r \ + \ 1}}{\frac{6}{z \times 3 }}=\frac{8 \ y \ + \ \frac{r}{r \ + \ 1}}{\frac{1}{3}\times \frac{ 6}{z }}=$
$\frac{8 \ y \ + \ \frac{r}{r \ + \ 1}}{\frac{6}{z}}={3} \ × \ \frac{8 \ y \ + \ \frac{r}{r \ + \ 1}}{\frac{6}{z}}=x \times 3$

The correct answer is $4.788$
The weight of $10.5$ meters of this rope is: $10.5 \ × \ 456$ g $=4788$ g
$1$ kg $=1000$ g, therefore, $4788$ g $÷ \ 1000=4.788$ kg

The correct answer is $282$
$g(x)=- \ 3$, then $f(g(x))= f(- \ 3)=3 \ (- \ 3)^4 \ - \ 2 \ (- \ 3)^3 \ + \ 5 \ (- \ 3)= 243  \ + \ 54 \ - \ 15=282$

The correct answer is $20$
Check each option provided:
A. $4 \ \ \ \ \frac{7\ + \ 13 \ + \ 15 \ + \ 16 \ + \ 20}{5}=\frac{71}{5}=14.2$
B. $13 \ \ \ \ \frac{4 \ + \ 7 \ + \ 15 \ + \ 16 \ + \ 20}{5}=\frac{62}{5}=12.4$
C. $20 \ \ \ \ \frac{4 \ + \ 7 \ + \ 13 \ + \ 15 \ + \ 16}{5}=\frac{55}{5}=11$
D. $16\  \ \frac{4 \ + \ 7 \ + \ 13 \ + \ 15 \ + \ 20}{5}=\frac{59}{5}=11.8$
E. $7 \ \  \frac{4 \ + \ 13 \ + \ 15 \ + \ 16 \ + \ 20}{5}=\frac{68}{5}=13.6$

The correct answer is $130$ miles
Use the information provided in the question to draw the shape.
Use Pythagorean Theorem: $a^2 \ + \ b^2=c^2$
$50^2 \ + \ 120^2=c^2 ⇒$
$2500 \ + \ 14400= c^2⇒$
$16,900=c^2⇒ c=130$

The correct answer is $\frac{3}{2}$
tangent $\beta= \frac{1}{cotangent \ \beta}=\frac{3}{2}$

The correct answer is $72$
$\frac{3}{5} \ × \ 120=72$

The correct answer is $\frac{200 \ x}{y}$
Let the number be A. Then: $2 \ x=y\% \ ×$ A
Solve for A. $2 \ x=\frac{y}{100} \ ×$ A
Multiply both sides by $\frac{100}{y}$:
$2 \ x \ × \ \frac{100}{y}=\frac{y}{100} \ × \ \frac{100}{y} \ ×$ A
A $=\frac{200 \ x}{y}$

The correct answer is $125$ cm
One liter $= 1000$ cm$^3→ 5$ liters $= 5000$ cm$^3$
$5000=10 \ × \ 4 \ × \ h→h=\frac{5000}{40}=125$ cm

The correct answer is $110 \ π$ in$^2$
Surface Area of a cylinder $= 2 \ π \ r \ (r \ + \ h)$,
The radius of the cylinder is $5 \ (10 \ ÷ \ 2)$ inches and its height is $8$ inches. Therefore, 
Surface Area of a cylinder $=2 \ π \ (5) \ (5 \ + \ 6)=110 \ π$  in$^2$

The correct answer is $80$ feet
The relationship among all sides of special right triangle 
$30^\circ \ - \ 60^\circ \ - \ 90^\circ$ is provided in this triangle: 
In this triangle, the opposite side of $60^\circ$ angle is half of the hypotenuse. 
Draw the shape of this question: 
The latter is the hypotenuse. Therefore, the latter is $80$ ft.

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