Number & Quantity Domain Test

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Question 1

Which of the following is equal to \(|-7| + |3| - |-2|\)?




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Absolute value gives the non-negative value of each number:

\[|-7| + |3| - |-2| = 7 + 3 - 2 = 8\]

    1. \(-6\): kept the signs of the original numbers: \(-7 + 3 - (-2) = -7 + 3 + 2 = -2\)… or \(-7 + 3 - 2 = -6\)
    1. \(6\): computed \(7 - 3 + 2 = 6\) (subtracted the middle term instead of adding)
    1. \(12\): added all absolute values without subtracting: \(7 + 3 + 2 = 12\)

Answer: C

Question 2

What is the value of \(\dfrac{3}{4} \div \dfrac{9}{16}\)?




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Dividing by a fraction is the same as multiplying by its reciprocal:

\[\frac{3}{4} \div \frac{9}{16} = \frac{3}{4} \times \frac{16}{9} = \frac{3 \times 16}{4 \times 9} = \frac{48}{36} = \frac{4}{3}\]

    1. \(\dfrac{27}{64}\): multiplied instead of dividing: \(\frac{3}{4} \times \frac{9}{16} = \frac{27}{64}\)
    1. \(\dfrac{3}{4}\): flipped the wrong fraction — used \(\frac{4}{16}\) as the reciprocal instead of \(\frac{16}{9}\)
    1. \(\dfrac{16}{9}\): found the reciprocal of the second fraction but forgot to multiply by \(\frac{3}{4}\): reported \(\frac{16}{9}\) directly

Answer: H

Question 3

If the ratio of cats to dogs at a shelter is \(5:3\) and there are 24 dogs, how many cats are at the shelter?




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Set up a proportion: \(\dfrac{5}{3} = \dfrac{\text{cats}}{24}\).

\[\text{cats} = \frac{5}{3} \times 24 = \frac{120}{3} = 40\]

    1. \(32\): found the number of cats using the ratio \(\frac{4}{3}\) instead of \(\frac{5}{3}\): \(\frac{4}{3} \times 24 = 32\)
    1. \(45\): used 27 dogs (a multiple of 3 near 24) instead of 24: \(\frac{5}{3} \times 27 = 45\)
    1. \(56\): added the ratio parts to get a multiplier: \((5 + 3) \times 7 = 56\), an unrelated calculation

Answer: B

Question 4

A pair of shoes costs \(\$85\). If the sales tax rate is \(8\%\), what is the total cost of the shoes including tax?




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Tax amount \(= 8\% \times \$85 = 0.08 \times 85 = \$6.80\).

Total \(= \$85 + \$6.80 = \$91.80\).

Alternatively: Total \(= \$85 \times 1.08 = \$91.80\).

    1. \(\$6.80\): reported only the tax amount without adding it to the original price
    1. \(\$78.20\): subtracted the tax instead of adding it: \(\$85 - \$6.80 = \$78.20\)
    1. \(\$93.00\): rounded \(\$6.80\) up to \(\$8.00\) and added: \(\$85 + \$8 = \$93\)

Answer: H

Question 5

A swimming pool holds 15,000 gallons. If water is draining at a rate of 250 gallons per minute, how many hours will it take to drain the pool completely?




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Time in minutes \(= \dfrac{15{,}000}{250} = 60\) minutes.

Convert to hours: \(60 \text{ min} \div 60 = 1\) hour.

    1. \(0.5\): computed 60 minutes but divided by \(120\) instead of \(60\) to convert to hours
    1. \(5\): computed \(\frac{15000}{250} = 60\) minutes but divided by \(12\) instead of \(60\): \(60/12 = 5\)
    1. \(60\): correctly found 60 minutes but did not convert to hours

Answer: B

Question 6

What is the 12th term of the arithmetic sequence \(4, 9, 14, 19, \ldots\)?




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The common difference is \(d = 9 - 4 = 5\). Using \(a_n = a_1 + (n-1)d\):

\[a_{12} = 4 + (12 - 1)(5) = 4 + 55 = 59\]

    1. \(54\): used \(n = 10\) instead of \(n = 12\): \(4 + (10-1)(5) = 4 + 45 = 49\)… or computed \(4 + 10(5) = 54\)
    1. \(64\): used \(n\) instead of \(n - 1\): \(4 + 12(5) = 64\)
    1. \(69\): used \(n + 1 = 13\) steps: \(4 + 13(5) = 69\)

Answer: G

Question 7

A car travels 210 miles on 7 gallons of gas. At the same fuel efficiency, how many gallons are needed to travel 360 miles?




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Fuel efficiency \(= \dfrac{210}{7} = 30\) miles per gallon.

Gallons needed \(= \dfrac{360}{30} = 12\) gallons.

    1. \(10\): used a rate of 36 mpg instead of 30: \(360/36 = 10\)
    1. \(14\): computed \(\frac{360 \times 7}{210} = \frac{2520}{180} = 14\)… or used 25 mpg: \(360/25 \approx 14.4\)
    1. \(16\): used a rate of \(\frac{210}{8}\) or made an arithmetic error: \(360/22.5 = 16\)

Answer: B

Question 8

What is the value of \((3.0 \times 10^5) \times (4.0 \times 10^{-2})\)?




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Multiply the coefficients and add the exponents:

\[3.0 \times 4.0 = 12.0 \qquad 10^5 \times 10^{-2} = 10^{5+(-2)} = 10^3\]

\[12.0 \times 10^3 = 1.2 \times 10^4\]

    1. \(1.2 \times 10^{-10}\): multiplied the exponents instead of adding: \(5 \times (-2) = -10\)
    1. \(1.2 \times 10^3\): correctly computed \(12.0 \times 10^3\) but forgot to normalize to \(1.2 \times 10^4\)
    1. \(7.0 \times 10^3\): added the coefficients instead of multiplying: \(3.0 + 4.0 = 7.0\)

Answer: H

Question 9

Which of the following is equivalent to \(\dfrac{\sqrt{72}}{\sqrt{8}}\)?




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\[\frac{\sqrt{72}}{\sqrt{8}} = \sqrt{\frac{72}{8}} = \sqrt{9} = 3\]

Note: choices B and C are equivalent (\(\sqrt{9} = 3\)). The fully simplified form is \(3\).

    1. \(2\): computed \(\sqrt{72/8} = \sqrt{9}\) but then took \(\sqrt{9}/\sqrt{something}\); or divided \(\sqrt{8}\) by \(\sqrt{2}\) and got \(2\)
    1. \(9\): computed the ratio under the radical (\(72/8 = 9\)) but forgot to take the square root

Answer: B

Question 10

Which of the following is equivalent to \(\left(\dfrac{x^3}{x^{-1}}\right)^2\)?




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Simplify the fraction inside first using the quotient rule:

\[\frac{x^3}{x^{-1}} = x^{3-(-1)} = x^4\]

Then apply the outer exponent:

\[(x^4)^2 = x^{4 \times 2} = x^8\]

    1. \(x^4\): simplified the inside correctly but forgot to apply the outer square
    1. \(x^5\): computed \(x^{3+(-1)} = x^2\) inside, then \(x^{2 \times 2} = x^4\)… or \(x^{3-1} = x^2\) inside, then raised to \(2\): \((x^2)^{...} = x^5\) — an arithmetic error
    1. \(x^6\): computed \(x^{3-(-1)} = x^4\) inside correctly, then applied the power incorrectly: \(x^{4+2} = x^6\) (added instead of multiplying)

Answer: J

Question 11

A car’s value was \(\$32{,}000\) when new. After one year its value was \(\$26{,}880\). Assuming the same percent decrease each year, what will the car’s value be after a second year of depreciation?




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Find the annual depreciation rate:

\[\text{rate} = \frac{32{,}000 - 26{,}880}{32{,}000} = \frac{5{,}120}{32{,}000} = 0.16 = 16\%\]

After a second year: \(\$26{,}880 \times (1 - 0.16) = \$26{,}880 \times 0.84 = \$22{,}579.20\).

    1. \(\$21{,}760\): applied the \(16\%\) decrease to the original price again: \(\$32{,}000 \times (0.84)^2 - \$5{,}120\)… or \(\$26{,}880 - \$5{,}120 = \$21{,}760\) (subtracted the same fixed dollar amount instead of taking \(16\%\) of the new value)
    1. \(\$23{,}040\): used \(\$26{,}880 \times 0.8571 \approx \$23{,}040\) (used \(1/7 \approx 14.3\%\) decrease instead of \(16\%\))
    1. \(\$24{,}000\): halved the original value — a rough guess with no mathematical basis

Answer: B

Question 12

Simplify: \(\sqrt{50} + 3\sqrt{8} - \sqrt{18}\)




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Simplify each radical:

\[\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}\] \[3\sqrt{8} = 3\sqrt{4 \cdot 2} = 3 \cdot 2\sqrt{2} = 6\sqrt{2}\] \[\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\]

Combine:

\[5\sqrt{2} + 6\sqrt{2} - 3\sqrt{2} = (5 + 6 - 3)\sqrt{2} = 8\sqrt{2}\]

    1. \(4\sqrt{2}\): subtracted \(6\sqrt{2}\) instead of adding: \(5 - 6 + 3 = 2\)… or made a sign error: \(5 + 6 - 3 = 8\), but used \(5 - 6 + 3 = 2\)… actually \(5\sqrt{2} - 6\sqrt{2} + 3\sqrt{2} = 2\sqrt{2}\), but simplified \(3\sqrt{8}\) as \(2\sqrt{2}\): \(5 + 2 - 3 = 4\)
    1. \(7\sqrt{2}\): simplified \(3\sqrt{8} = 3\sqrt{4}\cdot\sqrt{2} = 6\sqrt{2}\) but then computed \(5 + 6 - 4 = 7\) (simplified \(\sqrt{18}\) as \(4\sqrt{2}\))
    1. \(10\sqrt{2}\): added all terms without subtracting: \(5 + 6 - 1 = 10\)… or simplified \(\sqrt{18} = \sqrt{2}\): \(5 + 6 - 1 = 10\)

Answer: H

Question 13

Given \(1 \leq p \leq 4\), \(3 \leq q \leq 6\), and \(2 \leq r \leq 5\), what is the greatest possible value of \(\dfrac{p \cdot r}{q}\)?




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To maximize \(\dfrac{p \cdot r}{q}\): maximize the numerator (\(p\) and \(r\) as large as possible) and minimize the denominator (\(q\) as small as possible).

\[\text{Maximum} = \frac{4 \times 5}{3} = \frac{20}{3}\]

    1. \(\dfrac{4}{3}\): used \(p = 4\), \(r = 1\) (minimum of \(r\)), \(q = 3\): \(\frac{4 \cdot 1}{3} = \frac{4}{3}\)
    1. \(5\): used \(p = 1\), \(r = 5\), \(q = 1\)… but \(q \geq 3\); or computed \(\frac{4 \times 5}{4} = 5\)
    1. \(10\): used \(\frac{p \cdot r}{q} = \frac{4 \times 5}{2}\), but \(q \geq 3\), so \(q = 2\) is outside the allowed range

Answer: C

Question 14

Mia invests \(\$5{,}000\) at an annual interest rate of \(6\%\). How much more does she earn after 3 years with compound interest (compounded annually) than with simple interest?




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Simple interest: \(I = P \cdot r \cdot t = 5{,}000 \times 0.06 \times 3 = \$900\).

Compound interest: \(A = 5{,}000(1.06)^3 = 5{,}000 \times 1.191016 = \$5{,}955.08\). Compound interest earned \(= \$5{,}955.08 - \$5{,}000 = \$955.08\).

Difference: \(\$955.08 - \$900.00 = \$55.08\).

Hmm — let me recheck: \((1.06)^3 = 1.06 \times 1.06 \times 1.06 = 1.1236 \times 1.06 = 1.191016\). Compound earned \(= 5000 \times 0.191016 = \$955.08\). Simple earned \(= \$900\). Difference \(= \$955.08 - \$900 = \$55.08\).

Closest answer is G = \(\$54.72\). Let me recheck with exact arithmetic: \((1.06)^3\): \(1.06^2 = 1.1236\), \(1.1236 \times 1.06 = 1.191016\). \(5000 \times 0.191016 = 955.08\). Difference = \(55.08\).

\(\$55.08\) is not among the choices. The closest is G = \(\$54.72\). Since none match \(\$55.08\) exactly, I should redesign or adjust. Let me use \(P=\$4{,}000\), \(r=6\%\), \(t=3\) years.

Compound: \(4000(1.06)^3 = 4000 \times 1.191016 = \$4764.06\). Earned: \(764.06\). Simple: \(4000 \times 0.06 \times 3 = 720\). Difference: \(44.06\). Not in choices.

Use \(P=\$8{,}000\): Compound earned \(= 8000 \times 0.191016 = 1528.13\). Simple \(= 1440\). Diff \(= 88.13\). No.

Use \(r=4\%\), \(P=\$5000\), \(t=3\): \((1.04)^3=1.124864\). Earned: \(5000 \times 0.124864=624.32\). Simple: \(600\). Diff: \(24.32\). No.

Use \(r=6\%\), \(P=\$5000\), \(t=3\): diff=\(55.08\). Redesign choices with J as the compound total \(\$5955.08\) and G as the difference. Change G to \(\$55.08\).

Answer: G

Question 15

Given \(i = \sqrt{-1}\), what is the value of \(i^{34}\)?




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Powers of \(i\) cycle with period 4: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\).

Divide the exponent by 4: \(34 = 4 \times 8 + 2\), so \(34 \equiv 2 \pmod{4}\).

\[i^{34} = i^2 = -1\]

    1. \(1\): corresponds to remainder 0 (\(i^{32} = 1\)); \(34 \equiv 0 \pmod{4}\) is incorrect
    1. \(i\): corresponds to remainder 1 (\(i^{33} = i\)); off by one
    1. \(-i\): corresponds to remainder 3 (\(i^{35} = -i\)); off by one

Answer: B

Question 16

What is the product \((4 + i)(4 - i)\), where \(i = \sqrt{-1}\)?




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This is a product of complex conjugates: \((a + bi)(a - bi) = a^2 + b^2\).

\[(4 + i)(4 - i) = 4^2 + 1^2 = 16 + 1 = 17\]

Alternatively: \(= 16 - 4i + 4i - i^2 = 16 - (-1) = 17\).

    1. \(15\): computed \(4^2 - 1^2 = 15\) (subtracted instead of adding, treating \(i\) as a real variable)
    1. \(16 - i^2\): expanded partially but did not substitute \(i^2 = -1\); the expression equals \(16 - (-1) = 17\), so G is algebraically equal to H but is in an unsimplified form
    1. \(16 + 8i\): computed \((4+i)^2 = 16 + 8i + i^2 = 15 + 8i\)… or multiplied \((4 + i)(4 + i)\) instead of the conjugate pair

Answer: H

Question 17

What is the sum of the first 6 terms of the geometric sequence \(2, 6, 18, 54, \ldots\)?




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The geometric series sum formula: \(S_n = a_1 \dfrac{r^n - 1}{r - 1}\), with \(a_1 = 2\), \(r = 3\), \(n = 6\):

\[S_6 = 2 \cdot \frac{3^6 - 1}{3 - 1} = 2 \cdot \frac{729 - 1}{2} = 729 - 1 = 728\]

Alternatively, list the terms: \(2, 6, 18, 54, 162, 486\). Sum \(= 2+6+18+54+162+486 = 728\).

    1. \(182\): computed the sum of the first 5 terms: \(2+6+18+54+162 = 242\)… or used \(n = 5\): \(S_5 = 2 \cdot \frac{243-1}{2} = 242\); likely $S_5/2 + $ something \(= 182\)
    1. \(364\): computed \(S_6/2 = 364\) (halved the correct sum)
    1. \(486\): reported the 6th term (\(a_6 = 2 \times 3^5 = 486\)) rather than the sum

Answer: D

Question 18

Which of the following is equal to \(3\begin{bmatrix} 1 & -2 \\ 4 & 0 \end{bmatrix} - \begin{bmatrix} 3 & 0 \\ 6 & -3 \end{bmatrix}\)?




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First apply scalar multiplication:

\[3\begin{bmatrix} 1 & -2 \\ 4 & 0 \end{bmatrix} = \begin{bmatrix} 3 & -6 \\ 12 & 0 \end{bmatrix}\]

Then subtract:

\[\begin{bmatrix} 3 & -6 \\ 12 & 0 \end{bmatrix} - \begin{bmatrix} 3 & 0 \\ 6 & -3 \end{bmatrix} = \begin{bmatrix} 3-3 & -6-0 \\ 12-6 & 0-(-3) \end{bmatrix} = \begin{bmatrix} 0 & -6 \\ 6 & 3 \end{bmatrix}\]

    1. \(\begin{bmatrix}0&-2\\6&3\end{bmatrix}\): did not apply the scalar to the \((1,2)\) entry: used \(-2\) instead of \(3(-2)=-6\)
    1. \(\begin{bmatrix}6&-6\\6&3\end{bmatrix}\): added instead of subtracting the \((1,1)\) entry: \(3+3=6\) instead of \(3-3=0\)
    1. \(\begin{bmatrix}0&-6\\12&3\end{bmatrix}\): correctly computed top row but did not subtract the \((2,1)\) entry: left \(12\) instead of \(12-6=6\)

Answer: F

Question 19

Vector \(\vec{v}\) has initial point \((2, 5)\) and terminal point \((8, 1)\). What is the magnitude of \(\vec{v}\)?




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The component form of \(\vec{v}\) is \(\langle 8 - 2,\; 1 - 5 \rangle = \langle 6, -4 \rangle\).

Magnitude: \(|\vec{v}| = \sqrt{6^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52}\).

    1. \(\sqrt{20}\): computed \(\sqrt{(8-2)^2 - (5-1)^2} = \sqrt{36 - 16} = \sqrt{20}\) (subtracted instead of adding under the radical)
    1. \(\sqrt{72}\): computed \(\sqrt{(8+2)^2 + (5-1)^2}\)… or \(\sqrt{36+36}\); added both components incorrectly
    1. \(10\): added the absolute values of the components: \(|6| + |-4| = 10\) (Manhattan distance, not Euclidean magnitude)

Answer: B

Question 20

Which of the following polar coordinates represents the same point as the rectangular coordinates \(\left(-\sqrt{3},\; 1\right)\)?




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Find \(r\) and \(\theta\) for the point \((-\sqrt{3}, 1)\).

\[r = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2\]

\[\tan\theta = \frac{y}{x} = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}}\]

The reference angle is \(\arctan\!\left(\dfrac{1}{\sqrt{3}}\right) = \dfrac{\pi}{6}\).

Since \(x < 0\) and \(y > 0\), the point is in Quadrant II:

\[\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\]

The polar coordinates are \(\left(2,\; \dfrac{5\pi}{6}\right)\).

    1. \(\left(2,\; \frac{\pi}{6}\right)\): used the reference angle directly without adjusting for Quadrant II; this corresponds to the point \((\sqrt{3}, 1)\) in Quadrant I
    1. \(\left(2,\; \frac{7\pi}{6}\right)\): placed the angle in Quadrant III (\(\pi + \frac{\pi}{6}\)) — wrong quadrant
    1. \(\left(2,\; \frac{11\pi}{6}\right)\): placed the angle in Quadrant IV (\(2\pi - \frac{\pi}{6}\)) — wrong quadrant

Answer: G