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!)