Did you take the DAT Quantitative Reasoning Practice Test? If so, then it’s time to review your results to see where you went wrong and what areas you need to improve.

## DAT Quantitative Reasoning Practice Test Answers and Explanations

**1- Choice A is correct**Solve the system of equations by elimination method.

\(\begin{cases}2x+y=7\\4x-2y=18\end{cases}\)

\(\begin{cases}2(2x+y=7)\\4x-2y=18\end{cases}⇒\begin{cases}4x+2y= 14\\4x-2y=18\end{cases}\)⇒ add the equations

\(8x=32⇒x=4,2x+y=7⇒y=7-2(4)=7-8=-1\)

**2- Choice C is correct.**100 times the number is 6.45. Let \(x\) be the number, then: \(100x=6.45\), \(x=\frac{6.45}{100}=0.0645\)

**3- Choice C is correct**The area of the floor is: 6 cm \(×\) 32 cm = 192 cm, The number is tiles needed =\( 192 ÷ 12 = 16\)

**4- Choice E is correct**The diagonal of the square is 12. Let \(x\) be the side.

Use Pythagorean Theorem: \(a^2+b^2=c^2\)

\(x^2 + x^2 = 12^2 ⇒ 2x^2=12^2 ⇒ 2x^2 = 144 ⇒ x^2 = 72 ⇒x= \sqrt{72}\)

The area of the square is: \(\sqrt{72} ×\sqrt{72}=72\)

**5- Choice E is correct**\(x=26+136=162\)

**6- Choice B is correct.**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

**7- Choice E is correct**To find the discount, multiply the number by (\(100\%\) – rate of discount).

Therefore, for the first discount we get: \((D) (100\% – 15\%) = (D) (0.85) = 0.85 D\)

For increase of \(22\%\): \((0.85 D) (100\% + 22\%) = (0.85 D) (1.22) = 1.037 D = 103.7\%\) of \(D\)

**8- Choice D is correct**If 14 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 15. Therefore, the probability of not choosing a red ball is 14 out of 15 and the probability of having not a red ball after removing 14 balls is the same.

**9- Choice D is correct**\(x\) and \(2z\) are colinear. \(y\) and \(3x\) are colinear. Therefore,

\(x+2z=y+3x\), subtract \(x\) from both sides,then, \(2z=y-2x\)

**10- Choice C is correct**\(45\%\) of \(30\) equals to: \(0.45×30=13.5\), \(15\%\) of \(420\) equals to: \(0.15×420=63\)

\(45\%\) of \(30\) is added to \(15\%\) of \(420\): \(13.5+63=76.5\)

**11- Choice C is correct**\(12\%\) of the volume of the solution is alcohol. Let \(x\) be the volume of the solution.

Then: \(12\%\) of \(x=48\) ml ⇒ \(0.12 x=48 ⇒ x=48÷0.12=400\)

**12- Choice D is correct**Plug in each pair of number in the equation:

A. \((-3,2): 3(-3)-2(2)=-13\) Nope!

B. \((-1,2): 3(–1)-2(2)=-7\) Nope!

C. \((1,2): 3(1)-2(2)=-1\) Nope!

D. \((2,-1): 3(2)-2(-1)=8\) Bingo!

E. \((3,0): 3(3)-2(0)=9\) Nope!

**13- Choice B is correct**The area of ∆BED is 21, then: \(\frac{6×AB}{2}=21→6×AB=42→AB=7\)

The area of ∆BDF is 32, then: \(\frac{4×BC}{2} =32→4×BC=64→BC=16\)

The perimeter of the rectangle is = \(2×(7+16)=46\)

**14- Choice B is correct**\(|x+8|≤2→-2≤x+8≤2→-2-8≤x+8-8≤2-8→-10≤x≤-6\)

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**15- Choice D is correct**Use distance formula: Distance = Rate \(×\) time ⇒ 564 \(= 60 × \)T, divide both sides by 60. \(\frac{564}{60}\) = T ⇒ T = 9.4 hours. Change hours to minutes for the decimal part. 0.4 hours \(= 0.4 × 60 = 24\) minutes.

**16- Choice B is correct**Write a proportion and solve for \(x⇒ \frac{3}{4}=\frac{x}{36} ⇒ 4x=3 ×36 ⇒ x=27\) ft

**17- Choice D is correct**\(tanθ=\frac{opposite}{adjacent}\)

\(tanθ=\frac{8}{15}\)⇒ we have the following right triangle. Then

\(c=\sqrt{8^2+15^2}=\sqrt{64+225}=\sqrt{289}=17\)

\(cosθ=\frac{adjacent}{hypotenuse}=\frac{15}{17}\)

**18- Choice C is correct**The percent of girls playing tennis is: \(65\%×20\%=0.65×0.20=0.13=13\%\)

**19- Choice B is correct**Add the first 4 numbers. \(55+62+44+63=224\), To find the distance traveled in the next 4 hours, multiply the average by number of hours. Distance = Average × Rate \(=65×4=260\), Add both numbers. \(224+260=484\)

**20- Choice D is correct**The question is this: 1.82 is what percent of 1.3? Use percent formula: \(part=\frac{percent}{100}×whole\)

\(1.82 = \frac{percent}{100} × 1.3 ⇒ 1.82=\frac{percent ×1.3}{100} ⇒182=percent×1.3 ⇒ percent=\frac{182}{1.3}= 140\)

**21- Choice E is correct.**Solve for \(x ⇒ \frac{5x}{18}=\frac{x-2}{3}\)⇒ Multiply the second fraction by \(6⇒ \frac{5x}{18}=\frac{6(x-2)}{6×3}\)

Tow denominators are equal. Therefore, the numerators must be equal.

\(5x=6(x-2)=6x-12, x=12\)

**22- Choice E is correct**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 \(30^\circ\) angle is half of the hypotenuse.

Draw the shape of this question:

The latter is the hypotenuse. Therefore, the latter is 40 ft.

**23- Choice D is correct**The weight of 14 meters of this rope is: \(14×732 \space g=10,248 \space g\)

\(1 \space kg = 1000 \space g\), therefore, \(10,248 \space g÷1000=10.248 \space kg\)

**24- Choice E is correct**The slop of line A is: \(m=\frac{y_2-y_1}{x_2-x_1}=\frac{2-(-6)}{4-2}=\frac{8}{2}=4\)

Parallel lines have the same slope and only choice E \((y=4x)\) has slope of 4.

**25- Choice E is correct**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 (4, 5).

**26- Choice C is correct**The area of trapezoid is: \(\frac{(9+12)}{2}×x =42→21x=84→x=4\), \(y=\sqrt{3^2+4^2}=5\)

Perimeter is: \(4+9+5+3+9=30\)

**27- Choice C is correct**Set of number that are not composite between 1 and 18: \(A = {1, 2, 3, 5, 7, 11, 13, 17}\)

Probability \(= \frac{number \space of \space desired \space outcomes}{number \space of \space total \space outcomes}=\frac{8}{18}=\frac{4}{9}\)

**28- Choice B is correct**Check each option provided:

A. \(2 → \frac{4+6+8+12}{4}=\frac{30}{4}=7.5\)

B. \(4 → \frac{2+6+8+12}{4}=\frac{28}{4}=7\)

C. \(6 → \frac{2+4+8+12}{4}=\frac{26}{4}=6.5\)

D. \(8 → \frac{2+4+6+12}{4}=\frac{24}{4}=6\)

E. \(12 → \frac{2+4+6+8}{4}=\frac{20}{4}=5\)

**29- Choice B is correct**\(0.5x=(0.2)×30→x=12→(x-4)^2=(12-4)^2=8^2=64\)

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*DAT Quantitative Reasoning*

**30- Choice D is correct.**Let the number be \(x\). Then: \(12=4\%×x→12=0.04x\), Solve for \(x→x=\frac{12}{0.04}=300\)

**31- Choice C is correct**\(f(x)=2x^2-x,g(x)=x^2-6 , (f-g)(x)=f(x)–g(x)=(2x^2-x)-(x^2-6 )=2x^2-x-x^2+6=x^2-x+6\)

**32- Choice A is correct**\(f(x)=-x^3-4x^2+6x+7,g(x)=-1\), then: \(f(g(x))= f(-1)=-(-1)^3-4(-1)^2+6(-1)+7=-(-1)-4(1)-6+7=1-4-6+7=-2\)

**33- Choice A is correct**Plug in \(\frac{z}{5}\) for \(z\) and simplify.

\(x=\frac{8y+\frac{r}{r+1}}{\frac{6}{\frac{z}{5}}}=\frac{8y+\frac{r}{r+1}}{\frac{5×6}{z}}=\frac{8y+\frac{r}{r+1}}{5×\frac{6}{z}}=\frac{1}{5}×\frac{8y+\frac{r}{r+1}}{\frac{6}{z}}=\frac{x}{5}\)

**34- Choice B is correct**Use the information provided in the question to draw the shape.

Use Pythagorean Theorem: \(a^2+b^2=c^2\)

\(12^2+35^2=c^2 ⇒ 144+1225= c^2⇒1369=c^2⇒ c=37\)

**35- Choice D is correct**\(y =ab^2-3a^2 b ,a=2,b=-2\), Plug in the values of a and b in the equation: \(y=2(-2)^2-3(2)^2 (-2)=2(4)-3(4)(-2)=8+24=32\)

**36- Choice A is correct**I. \(|a|<1→-1<a<1\)

Multiply all sides by \(b\). Since, \(b>0\) →\(-b<ba<b\) (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 \(4\), then: \(-4<4a<4\)

Subtract \(5\) from all sides. Then: \(-4-5<4a-5<4-5→-9<4a-5<-1\) (It is false!)

**37- Choice C is correct**\(One \space liter = 1000 \space cm^3→ 8 \space liters = 8000 \space cm^3⇒ 8000=16×5×h→h=\frac{8000}{80}=100 \space cm\)

**38- Choice E is correct**\(\frac{3}{4}×120=90\)

**39- Choice C is correct**\(tangent \space β= \frac{1}{cotangent \space β}=\frac{1}{1}=1\)

**40- Choice E is correct**Surface Area of a cylinder \(= 2πr (r+h)\), The radius of the cylinder is \(4(8÷2)\) inches and its height is \(12\) inches. Therefore, Surface Area of a cylinder \(=2π(4)(4+12)=128π\)