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Basics of Algebra

Multiplication and Division by Exponents


\[ \begin{align*} &\text{Simplify the following expression: } \frac{3 x^4}{\left(x^{-2}\right)^2} \\ &(A) \quad 3x^{10} \\ &(B) \quad 3x^{8} \\ &(C) \quad 3x^{2} \\ &(D) \quad 3x^{-4} \end{align*} \]
\[ \begin{align*} &\text{Simplify the following expression: } \frac{5 y^{6}}{(y^{-3})^2} \\ &(A) \quad 5y^{12} \\ &(B) \quad 5y^{9} \\ &(C) \quad 5y^{3} \\ &(D) \quad 5y^{0} \end{align*} \]
\[ \begin{align*} &\text{Simplify the following expression: } \frac{7 z^{2}}{(z^{-5})^3} \\ &(A) \quad 7z^{17} \\ &(B) \quad 7z^{15} \\ &(C) \quad 7z^{11} \\ &(D) \quad 7z^{1} \end{align*} \]
\[ \begin{align*} &\text{Simplify the following expression: } \frac{6 p^{8}}{(p^{-4})^2} \\ &(A) \quad 6p^{16} \\ &(B) \quad 6p^{10} \\ &(C) \quad 6p^{4} \\ &(D) \quad 6p^{-4} \end{align*} \]
\[ \begin{align*} &\text{Simplify the following expression: } \frac{4 q^{3}}{(q^{-1})^4} \\ &(A) \quad 4q^{7} \\ &(B) \quad 4q^{4} \\ &(C) \quad 4q^{2} \\ &(D) \quad 4q^{-4} \end{align*} \]
\[ \begin{align*} &\text{Question 1: Simplify the following expression: } \frac{\left(a^{-3} b\right)^4}{\left(a b^2\right)^3} \\ &(A) \quad a^{-12}b \\ &(B) \quad a^{-5}b^{-2} \\ &(C) \quad a^{1}b^{-2} \\ &(D) \quad a^{3}b^{-4} \end{align*} \]
\[ \begin{align*} &\text{Question 2: Simplify the following expression: } \frac{\left(c^{-4} d\right)^2}{\left(c d^3\right)^3} \\ &(A) \quad c^{-8}d^{-7} \\ &(B) \quad c^{-5}d^{-6} \\ &(C) \quad c^{1}d^{-5} \\ &(D) \quad c^{3}d^{-4} \end{align*} \]
\[ \begin{align*} &\text{Question 3: Simplify the following expression:} \frac{\left(e^{-1} f\right)^5}{\left(e f^2\right)^4} \\ &(A) \quad e^{-5}f^{3} \\ &(B) \quad e^{-4}f^{1} \\ &(C) \quad e^{1}f^{-1} \\ &(D) \quad e^{3}f^{-3} \end{align*} \]
\[ \begin{align*} &\text{Question 4: Simplify the following expression with positive exponents: } \frac{\left(g^{-2} h\right)^3}{\left(g h^4\right)^2} \\ &(A) \quad g^{-6}h^{-5} \\ &(B) \quad g^{-4}h^{-3} \\ &(C) \quad g^{2}h^{-1} \\ &(D) \quad g^{4}h^{1} \end{align*} \]
\[ \begin{align*} &\text{Question 5: Simplify the following expression: } \frac{\left(i^{-3} j\right)^2}{\left(i j^5\right)^3} \\ &(A) \quad i^{-6}j^{-13} \\ &(B) \quad i^{-4}j^{-10} \\ &(C) \quad i^{2}j^{-7} \\ &(D) \quad i^{4}j^{-5} \end{align*} \]
\[ \begin{align*} &\text{Simplify the following expression: } \frac{\left(a^{-3} b^2 c^{-1}\right)^4}{\left(a b^2 c^{-2}\right)^3} \\ &(A) \quad a^{-12}b^{5}c^{2} \\ &(B) \quad a^{-7}b^{8}c^{1} \\ &(C) \quad a^{2}b^{11}c^{-3} \\ &(D) \quad a^{5}b^{14}c^{-6} \end{align*} \]
\[ \begin{align*} &\text{Simplify the following expression: } \frac{\left(d^{-4} e^2 f^{-1}\right)^2}{\left(d e^2 f^{-3}\right)^3} \\ &(A) \quad d^{-8}e^{1}f^{7} \\ &(B) \quad d^{-6}e^{4}f^{5} \\ &(C) \quad d^{-2}e^{7}f^{3} \\ &(D) \quad d^{1}e^{10}f^{1} \end{align*} \]
\[ \begin{align*} &\text{Simplify the following expression: } \frac{\left(g^{-1} h^2 i^{-1}\right)^5}{\left(g h^2 i^{-3}\right)^4} \\ &(A) \quad g^{-5}h^{6}i^{7} \\ &(B) \quad g^{-3}h^{8}i^{5} \\ &(C) \quad g^{-1}h^{10}i^{3} \\ &(D) \quad g^{1}h^{12}i^{1} \end{align*} \]
\[ \begin{align*} &\text{Simplify the following expressions: } \frac{\left(j^{-2} k^2 l^{-1}\right)^3}{\left(j k^2 l^{-4}\right)^2} \\ &(A) \quad j^{-6}k^{2}l^{5} \\ &(B) \quad j^{-4}k^{4}l^{3} \\ &(C) \quad j^{-2}k^{6}l^{1} \\ &(D) \quad j^{1}k^{8}l^{-1} \end{align*} \]
\[ \begin{align*} &\text{Simplify the following expression: } \frac{\left(m^{-3} n^2 o^{-1}\right)^2}{\left(m n^2 o^{-5}\right)^3} \\ &(A) \quad m^{-6}n^{-4}o^{13} \\ &(B) \quad m^{-4}n^{-2}o^{11} \\ &(C) \quad m^{-2}n^{0}o^{9} \\ &(D) \quad m^{1}n^{2}o^{7} \end{align*} \]

Comparing Exponents


\[ \begin{align*} &\text{Question 1: If } 2^{b+2}=2^{-b+8}, \text{ what is the value of } b? \\ &(A) \quad -3 \\ &(B) \quad -2 \\ &(C) \quad 2 \\ &(D) \quad 3 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If } 4^{c+3}=4^{-c+9}, \text{ what is the value of } c? \\ &(A) \quad -2 \\ &(B) \quad -1 \\ &(C) \quad 1 \\ &(D) \quad 2 \end{align*} \]
\[ \begin{align*} &\text{Question 3: If } 5^{d+4}=5^{-d+10}, \text{ what is the value of } d? \\ &(A) \quad -2 \\ &(B) \quad -1 \\ &(C) \quad 1 \\ &(D) \quad 2 \end{align*} \]
\[ \begin{align*} &\text{Question 1: If } 2^{2b+1}=8^{b-2}, \text{ what is the value of } b? \\ &(A) \quad -1 \\ &(B) \quad 1 \\ &(C) \quad 2 \\ &(D) \quad 3 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If } 4^{3c+2}=16^{2c-1}, \text{ what is the value of } c? \\ &(A) \quad -1 \\ &(B) \quad 1 \\ &(C) \quad 2 \\ &(D) \quad 3 \end{align*} \]
\[ \begin{align*} &\text{Question 3: If } 5^{2d+3}=125^{d-1}, \text{ what is the value of } d? \\ &(A) \quad -1 \\ &(B) \quad 1 \\ &(C) \quad 2 \\ &(D) \quad 3 \end{align*} \]

Converting Related Exponents


\[ \begin{align*} &\text{Question 1: If } 3b-c=5, \text{ what is the value of } \frac{9^b}{3^c} ? \\ &(A) \quad 3 \\ &(B) \quad 9 \\ &(C) \quad 27 \\ &(D) \quad 81 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If } 2d-e=3, \text{ what is the value of } \frac{8^d}{2^e} ? \\ &(A) \quad 2 \\ &(B) \quad 4 \\ &(C) \quad 8 \\ &(D) \quad 16 \end{align*} \]
\[ \begin{align*} &\text{Question 3: If } 4f-g=6, \text{ what is the value of } \frac{16^f}{4^g} ? \\ &(A) \quad 4 \\ &(B) \quad 16 \\ &(C) \quad 64 \\ &(D) \quad 256 \end{align*} \]

Converting to Fractional Exponents


\[ \begin{align*} &\text{Question 1: Which of the following is equal to } \sqrt[3]{y^7} ? \\ &(A) \quad y^{2} \\ &(B) \quad y^{7}-y^{3} \\ &(C) \quad y^{\frac{7}{3}} \\ &(D) \quad y^{3} \end{align*} \]
\[ \begin{align*} &\text{Question 2: Which of the following is equal to } \sqrt[5]{z^{8}} ? \\ &(A) \quad z^{1} \\ &(B) \quad z^{8}-z^{5} \\ &(C) \quad z^{\frac{8}{5}} \\ &(D) \quad z^{5} \end{align*} \]
\[ \begin{align*} &\text{Question 3: Which of the following is equal to } \sqrt[2]{w^{6}} ? \\ &(A) \quad w^{2} \\ &(B) \quad w^{6}-w^{2} \\ &(C) \quad w^{3} \\ &(D) \quad w^{4} \end{align*} \]
\[ \begin{align*} &\text{Question 1: Which of the following is equal to } \sqrt[3]{(2y^2)^7} ? \\ &(A) \quad 2^{7}y^{14} \\ &(B) \quad 2^{2}y^{7} \\ &(C) \quad 2^{\frac{7}{3}}y^{\frac{14}{3}} \\ &(D) \quad 2^{7}y^{21} \end{align*} \]
\[ \begin{align*} &\text{Question 2: Which of the following is equal to } \sqrt[5]{(3z^3)^8} ? \\ &(A) \quad 3^{8}z^{24} \\ &(B) \quad 3^{2}z^{8} \\ &(C) \quad 3^{\frac{8}{5}}z^{\frac{24}{5}} \\ &(D) \quad 3^{8}z^{40} \end{align*} \]
\[ \begin{align*} &\text{Question 3: Which of the following is equal to } \sqrt[2]{(4w^4)^6} ? \\ &(A) \quad 4^{6}w^{24} \\ &(B) \quad 4^{2}w^{6} \\ &(C) \quad 4^{3}w^{12} \\ &(D) \quad 4^{6}w^{48} \end{align*} \]

Solving Radical Equations


\[ \begin{align*} &\text{Question 1: Given that } 4 \sqrt{10}=\sqrt{2 y}, \text{ what is the value of } y ? \\ &(A) \quad 20 \\ &(B) \quad 40 \\ &(C) \quad 80 \\ &(D) \quad 160 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } 5 \sqrt{15}=\sqrt{3 z}, \text{ then what is } z ? \\ &(A) \quad 75 \\ &(B) \quad 150 \\ &(C) \quad 225 \\ &(D) \quad 300 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } 6 \sqrt{21}=\sqrt{7 w}, \text{ what could be the possible value of } w ? \\ &(A) \quad 126 \\ &(B) \quad 252 \\ &(C) \quad 504 \\ &(D) \quad 1008 \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } 3 \sqrt{5y+7}=6 \sqrt{2y-3}, \text{ what is the value of } y ? \\ &(A) \quad 3 \\ &(B) \quad 5 \\ &(C) \quad 7 \\ &(D) \quad 9 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } 4 \sqrt{7z-10}=8 \sqrt{3z+2}, \text{ then what is } z ? \\ &(A) \quad 2 \\ &(B) \quad 4 \\ &(C) \quad 6 \\ &(D) \quad 8 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } 5 \sqrt{6w+11}=10 \sqrt{2w+3}, \text{ what could be the possible value of } w ? \\ &(A) \quad 1 \\ &(B) \quad 3 \\ &(C) \quad 5 \\ &(D) \quad 7 \end{align*} \]

Balancing the Exponents


\[ \begin{align*} &\text{Question 1: Given } b^{-\frac{1}{2}}=2, \text{ what could be the possible value of } b ? \\ &(A) \quad -4 \\ &(B) \quad \frac{1}{4} \\ &(C) \quad \frac{1}{2} \\ &(D) \quad 4 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } c^{-\frac{1}{3}}=4, \text{ then what is } c ? \\ &(A) \quad -64 \\ &(B) \quad \frac{1}{64} \\ &(C) \quad \frac{1}{4} \\ &(D) \quad 64 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } d^{-\frac{1}{4}}=5, \text{ what could be the possible value of } d ? \\ &(A) \quad -625 \\ &(B) \quad \frac{1}{625} \\ &(C) \quad \frac{1}{5} \\ &(D) \quad 625 \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } (2p+1)^{-\frac{1}{2}}=3, \text{ what could be the possible value of } p ? \\ &(A) \quad 8 \\ &(B) \quad 4 \\ &(C) \quad 2 \\ &(D) \quad 1 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } (3q-2)^{-\frac{1}{3}}=2, \text{ then what is } q ? \\ &(A) \quad 6 \\ &(B) \quad 8 \\ &(C) \quad 10 \\ &(D) \quad 12 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } (4r+3)^{-\frac{1}{4}}=5, \text{ what could be the possible value of } r ? \\ &(A) \quad 155 \\ &(B) \quad 156 \\ &(C) \quad 157 \\ &(D) \quad 158 \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } a^3=2, \text{ what could be the possible value of } a^{12} ? \\ &(A) \quad 8 \\ &(B) \quad 16 \\ &(C) \quad 64 \\ &(D) \quad 128 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } b^4=3, \text{ then what is } b^{16} ? \\ &(A) \quad 27 \\ &(B) \quad 81 \\ &(C) \quad 243 \\ &(D) \quad 729 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } c^2=4, \text{ what could be the possible value of } c^{10} ? \\ &(A) \quad 512 \\ &(B) \quad 1024 \\ &(C) \quad 2048 \\ &(D) \quad 4096 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } (3q-4)^2=9, \text{ then what is } (3q-4)^{10} ? \\ &(A) \quad 81 \\ &(B) \quad 243 \\ &(C) \quad 729 \\ &(D) \quad 2187 \end{align*} \]

Radicals Simplification


\[ \begin{align*} &\text{Question 1: Given that } \frac{\sqrt[3]{y^4}}{\sqrt[6]{y}}=y^d \text{ for all positive values of } y, \text{ what could be the possible value of } d ? \\ &(A) \quad 1 \\ &(B) \quad \frac{2}{3} \\ &(C) \quad \frac{5}{6} \\ &(D) \quad \frac{7}{6} \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } \frac{\sqrt[4]{z^5}}{\sqrt[8]{z}}=z^e \text{ for all positive values of } z, \text{ then what is } e ? \\ &(A) \quad 1 \\ &(B) \quad \frac{3}{2} \\ &(C) \quad \frac{7}{4} \\ &(D) \quad 2 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } \frac{\sqrt[5]{w^6}}{\sqrt[10]{w}}=w^f \text{ for all positive values of } w, \text{ what could be the possible value of } f ? \\ &(A) \quad 1 \\ &(B) \quad \frac{11}{10} \\ &(C) \quad \frac{6}{5} \\ &(D) \quad \frac{7}{5} \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } p>0 \text{ and } q>0, \text{ the expression } \sqrt[3]{p^3 q^6} \text{ is equivalent to which of the following?} \\ &(A) \quad p^2 q \\ &(B) \quad pq^2 \\ &(C) \quad \frac{1}{pq} \\ &(D) \quad pq \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } m>0 \text{ and } n>0, \text{ then what is the equivalent form of } \sqrt[5]{m^2 n^{10}} ? \\ &(A) \quad m n^2 \\ &(B) \quad mn \\ &(C) \quad n \sqrt[5]{m} \\ &(D) \quad m^2 n \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } r>0 \text{ and } s>0, \text{ what could be the equivalent form of } \sqrt[6]{r^4 s^{12}} ? \\ &(A) \quad r s^2 \\ &(B) \quad rs \\ &(C) \quad \frac{1}{rs} \\ &(D) \quad rs^2 \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } p>0 \text{ and } q>0, \text{ the expression } \sqrt[3]{p^4 q^2 + p^3 q} \text{ is equivalent to which of the following for sufficiently large values of } p \text{ and } q? \\ &(A) \quad pq \\ &(B) \quad p^2 q \\ &(C) \quad p q^2 \\ &(D) \quad p^{\frac{4}{3}} q^{\frac{2}{3}} \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } m>0 \text{ and } n>0, \text{ then what is the equivalent form of } \sqrt[5]{m^3 n^7 - m^2 n^6} \text{ for sufficiently large values of } m \text{ and } n? \\ &(A) \quad mn \\ &(B) \quad m^{\frac{3}{5}} n^{\frac{7}{5}} \\ &(C) \quad m^2 n \\ &(D) \quad m n^2 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } r>0 \text{ and } s>0, \text{ what could be the equivalent form of } \sqrt[6]{r^5 s^4 - r^4 s^3} \text{ for sufficiently large values of } r \text{ and } s? \\ &(A) \quad rs \\ &(B) \quad r^2 s \\ &(C) \quad r s^2 \\ &(D) \quad r^{\frac{5}{6}} s^{\frac{4}{6}} \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } a^3=b^2, \text{ for which value of } p \text{ does } a^{2 p}=b^6 \text{ hold true?} \\ &(A) \quad -2 \\ &(B) \quad -1 \\ &(C) \quad 1 \\ &(D) \quad 2 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } m^4=n^3, \text{ then for which value of } q \text{ does } m^{3 q}=n^9 \text{ hold true?} \\ &(A) \quad -3 \\ &(B) \quad -2 \\ &(C) \quad 2 \\ &(D) \quad 3 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } r^5=s^4, \text{ for which value of } a \text{ does } r^{4 a}=s^{10} \text{ hold true?} \\ &(A) \quad -5 \\ &(B) \quad -4 \\ &(C) \quad 4 \\ &(D) \quad 5 \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } a^{3/2}=b^2, \text{ for which value of } p \text{ does } a^{2 p}=b^{9/2} \text{ hold true?} \\ &(A) \quad 1/4 \\ &(B) \quad 1/2 \\ &(C) \quad 3/2 \\ &(D) \quad 3 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } m^{4/3}=n^2, \text{ then for which value of } q \text{ does } m^{3 q}=n^{18/3} \text{ hold true?} \\ &(A) \quad -2/3 \\ &(B) \quad -1/3 \\ &(C) \quad 2/3 \\ &(D) \quad 1 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } r^{5/2}=s^3, \text{ for which value of } a \text{ does } r^{4 a}=s^{15/2} \text{ hold true?} \\ &(A) \quad -5/4 \\ &(B) \quad -2/3 \\ &(C) \quad 2/3 \\ &(D) \quad 5/4 \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } a^3 b^2=12 \text{ and } a^2 b^3=15, \text{ what is the value of } a^5 b^5 ? \\ &(A) \quad 60 \\ &(B) \quad 80 \\ &(C) \quad 180 \\ &(D) \quad 225 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } m^2 n^3=6 \text{ and } m^3 n^2=9, \text{ then what is the value of } m^5 n^5 ? \\ &(A) \quad 15 \\ &(B) \quad 27 \\ &(C) \quad 54 \\ &(D) \quad 108 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } r^3 s^2=8 \text{ and } r^2 s^3=12, \text{ what is the value of } r^5 s^5 ? \\ &(A) \quad 24 \\ &(B) \quad 48 \\ &(C) \quad 96 \\ &(D) \quad 192 \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } a^4 b^3 c = 15 \text{, } a^3 b^4 c = 20 \text{ and } a^2 b^2 c^3 = 25, \text{ what is the value of } a^5 b^5 c^5 ? \\ &(A) \quad 100 \\ &(B) \quad 150 \\ &(C) \quad 200 \\ &(D) \quad 250 \end{align*} \]
\[ \begin{align*} &\text{Question 2: If it's known that } m^3 n^2 p^4 = 8 \text{, } m^2 n^3 p^4 = 12 \text{ and } m^2 n^2 p^6 = 16, \text{ then what is the value of } m^5 n^5 p^10 ? \\ &(A) \quad 48 \\ &(B) \quad 96 \\ &(C) \quad 192 \\ &(D) \quad 384 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } r^4 s^3 t^2 = 20 \text{, } r^3 s^4 t^2 = 30 \text{ and } r^2 s^2 t^5 = 40, \text{ what is the value of } r^5 s^5 t^5 ? \\ &(A) \quad 240 \\ &(B) \quad 480 \\ &(C) \quad 960 \\ &(D) \quad 1920 \end{align*} \]
\[ \begin{align*} &\text{Question 1: If } b=3 \sqrt{3} \text{ and } 4 b=\sqrt{3 y}, \text{ what is the value of } y ? \\ &(A) \quad 9 \\ &(B) \quad 12 \\ &(C) \quad 36 \\ &(D) \quad 48 \end{align*} \]
\[ \begin{align*} &\text{Question 2: Given that } c=4 \sqrt{5} \text{ and } 5 c=\sqrt{5 z}, \text{ then what is the value of } z ? \\ &(A) \quad 16 \\ &(B) \quad 20 \\ &(C) \quad 80 \\ &(D) \quad 100 \end{align*} \]
\[ \begin{align*} &\text{Question 3: Suppose } d=6 \sqrt{7} \text{ and } 3 d=\sqrt{7 w}, \text{ what is the value of } w ? \\ &(A) \quad 36 \\ &(B) \quad 42 \\ &(C) \quad 126 \\ &(D) \quad 252 \end{align*} \]
\[ \begin{align*} &\text{Question 1: Given that } a=5 \sqrt[4]{3} \text{ and } 3 a^2=\sqrt[4]{3x}, \text{ what is the value of } x ? \\ &(A) \quad 25 \\ &(B) \quad 100 \\ &(C) \quad 225 \\ &(D) \quad 400 \end{align*} \]
\[ \begin{align*} &\text{Question 2: Suppose } b=3 \sqrt[3]{2} \text{ and } 2 b^3=\sqrt[3]{2y}, \text{ then what is the value of } y ? \\ &(A) \quad 27 \\ &(B) \quad 54 \\ &(C) \quad 81 \\ &(D) \quad 108 \end{align*} \]
Numbers
\[ \begin{align*} &\text{Question 1: Two numbers add up to 54. If the second number is three times the first, what are these numbers?} \\ &(A) \quad (13.5, 40.5) \\ &(B) \quad (18, 36) \\ &(C) \quad (27, 27) \\ &(D) \quad (36, 18) \end{align*} \]
\[ \begin{align*} &\text{Question 2: Two numbers have a sum of 90. If one number is four times as large as the other, what are the numbers?} \\ &(A) \quad (18, 72) \\ &(B) \quad (22.5, 67.5) \\ &(C) \quad (30, 60) \\ &(D) \quad (45, 45) \end{align*} \]
\[ \begin{align*} &\text{Question 3: The sum of two numbers is 36. The second number is five times the first. What are the numbers?} \\ &(A) \quad (6, 30) \\ &(B) \quad (9, 27) \\ &(C) \quad (12, 24) \\ &(D) \quad (18, 18) \end{align*} \]
\[ \begin{align*} &\text{Question 3: Two numbers add up to 360. If one number is half of the other plus 30, what are the numbers?} \\ &(A) \quad (110, 250) \\ &(B) \quad (120, 240) \\ &(C) \quad (130, 230) \\ &(D) \quad (140, 220) \end{align*} \]
\[ \begin{align*} &\text{Question 3: Two numbers add up to 360. If one number is half of the other plus 30, what are the numbers?} \\ &(A) \quad (110, 250) \\ &(B) \quad (120, 240) \\ &(C) \quad (130, 230) \\ &(D) \quad (140, 220) \end{align*} \]
\[ \begin{align*} &\text{Question 1: Two numbers add up to 64. Four times the first is 8 less than thrice the second. What are these numbers?} \\ &(A) \quad (4, 60) \\ &(B) \quad (8, 56) \\ &(C) \quad (16, 24) \\ &(D) \quad (28, 32) \end{align*} \]
\[ \begin{align*} &\text{Question 2: The sum of two numbers is 37. If five times the second number is 10 more than twice the first, what are the numbers?} \\ &(A) \quad (25, 12) \\ &(B) \quad (20, 17) \\ &(C) \quad (15, 22) \\ &(D) \quad (10, 27) \end{align*} \]
\[ \begin{align*} &\text{Question 3: Two numbers have a sum of 32. The second number is seven times the first minus 10. What are the numbers?} \\ &(A) \quad (2, 30) \\ &(B) \quad (4, 28) \\ &(C) \quad (6, 32) \\ &(D) \quad (8, 24) \end{align*} \]
\[ \begin{align*} &\text{Question 1: What are the three consecutive integers that sum to 66?} \\ &(A) \quad (21, 22, 23) \\ &(B) \quad (22, 23, 24) \\ &(C) \quad (23, 24, 25) \\ &(D) \quad (24, 25, 26) \end{align*} \]
\[ \begin{align*} &\text{Question 2: Which three consecutive integers have a total of 33?} \\ &(A) \quad (10, 11, 12) \\ &(B) \quad (11, 12, 13) \\ &(C) \quad (12, 13, 14) \\ &(D) \quad (13, 14, 15) \end{align*} \]
\[ \begin{align*} &\text{Question 3: Which consecutive integers form a sequence whose sum equals 111?} \\ &(A) \quad (36, 37, 38) \\ &(B) \quad (37, 38, 39) \\ &(C) \quad (38, 39, 40) \\ &(D) \quad (39, 40, 41) \end{align*} \]
\[ \begin{align*} &\text{Question 1: Which set of four consecutive integers has a sum of 138?} \\ &(A) \quad (31, 32, 33, 34) \\ &(B) \quad (32, 33, 34, 35) \\ &(C) \quad (33, 34, 35, 36) \\ &(D) \quad (34, 35, 36, 37) \end{align*} \]
\[ \begin{align*} &\text{Question 2: What is the set of five consecutive integers that sum to 390?} \\ &(A) \quad (73, 74, 75, 76, 77) \\ &(B) \quad (74, 75, 76, 77, 78) \\ &(C) \quad (75, 76, 77, 78, 79) \\ &(D) \quad (76, 77, 78, 79, 80) \end{align*} \]
\[ \begin{align*} &\text{Question 3: What are the four consecutive even integers whose sum is 220?} \\ &(A) \quad (50, 52, 54, 56) \\ &(B) \quad (51, 53, 55, 57) \\ &(C) \quad (52, 54, 56, 58) \\ &(D) \quad (53, 55, 57, 59) \end{align*} \]
\[ \begin{align*} &\text{Question 2: One side of a triangle is 3 cm longer than the second} \\ &\text{side, and the third side is 2 cm shorter than twice the second side.} \\ &\text{If the perimeter of the triangle is 35 cm, what are the lengths of the sides?} \\ &(A) \quad (5, 7, 9) \text{ cm} \\ &(B) \quad (6, 9, 16) \text{ cm} \\ &(C) \quad (7, 10, 18) \text{ cm} \\ &(D) \quad (8, 9, 16) \text{ cm} \end{align*} \]
\[ \begin{align*} &\text{Question 3: A triangle has one side that is 4 feet shorter than the second} \\ &\text{side, and the third side is twice as long as the second side.} \\ &\text{If the perimeter of the triangle is 28 feet, what are the lengths of the sides?} \\ &(A) \quad (2, 6, 12) \text{ feet} \\ &(B) \quad (3, 7, 14) \text{ feet} \\ &(C) \quad (4, 8, 16) \text{ feet} \\ &(D) \quad (5, 9, 18) \text{ feet} \end{align*} \]
\[ \begin{align*} &\text{Question 4: In a triangle, the first side is twice as long as the second side,} \\ &\text{and the third side is 7 meters longer than the second side.} \\ &\text{If the perimeter of the triangle is 35 meters, what are the lengths of the sides?} \\ &(A) \quad (8, 4, 19) \text{ meters} \\ &(B) \quad (10, 5, 17) \text{ meters} \\ &(C) \quad (12, 6, 15) \text{ meters} \\ &(D) \quad (14, 7, 14) \text{ meters} \end{align*} \]
\[ \begin{align*} &\text{Question 1: The first side of a triangle is twice the length of the second side} \\ &\text{minus 2 cm. The third side is three times the length of the second side} \\ &\text{plus 1 cm. If the perimeter of the triangle is 48 cm, what are the lengths of the sides?} \\ &(A) \quad (16, 8, 24) \text{ cm} \\ &(B) \quad (18, 9, 27) \text{ cm} \\ &(C) \quad (13, 7, 22) \text{ cm} \\ &(D) \quad (10, 10, 22) \text{ cm} \end{align*} \]
\[ \begin{align*} &\text{Question 2: In a triangle, the first side is 3 m less than twice the length of the second side.} \\ &\text{The third side is twice the second side minus 1 m.} \\ &\text{If the perimeter of the triangle is 26 m, what are the lengths of the sides?} \\ &(A) \quad (8, 5, 9) \text{ m} \\ &(B) \quad (9, 6, 11) \text{ m} \\ &(C) \quad (5, 4, 7) \text{ m} \\ &(D) \quad (6, 7, 13) \text{ m} \end{align*} \]
\[ \begin{align*} &\text{Question 3: The first side of a triangle is three times the second side,} \\ &\text{and the third side is the difference between four times the second side and 2 feet.} \\ &\text{If the perimeter of the triangle is 46 feet, what are the lengths of the sides?} \\ &(A) \quad (12, 4, 14) \text{ feet} \\ &(B) \quad (15, 5, 18) \text{ feet} \\ &(C) \quad (18, 6, 22) \text{ feet} \\ &(D) \quad (20, 8, 12) \text{ feet} \end{align*} \]
\[ \begin{align*} &\text{Question 1: A truck leaves from Boston heading towards Miami at a speed of 50}\\ &\text{mph. After three hours, a sports car leaves the same location heading for Miami}\\ &\text{, driving at 70 mph. How long will it take for the sports car to overtake the truck?} \\ &(A) \quad 3.5 \text{ hours} \\ &(B) \quad 5 \text{ hours} \\ &(C) \quad 7.5 \text{ hours} \\ &(D) \quad 10 \text{ hours} \end{align*} \]
\[ \begin{align*} &\text{Question 2: A cyclist departs from city A and heads towards city B at 15 mph. Two hours later, a motorcyclist leaves from the same city towards city B, traveling at 25 mph. How long will it take for the motorcyclist to catch up with the cyclist?} \\ &(A) \quad 1.2 \text{ hours} \\ &(B) \quad 3 \text{ hours} \\ &(C) \quad 4.5 \text{ hours} \\ &(D) \quad 6 \text{ hours} \end{align*} \]
\[ \begin{align*} &\text{Question 3: A ship departs from port A and sails towards port B at a speed of 20 knots. Three hours later, a speedboat leaves the same port heading in the same direction, traveling at 30 knots. How many hours will it take for the speedboat to catch up with the ship?} \\ &(A) \quad 2 \text{ hours} \\ &(B) \quad 4 \text{ hours} \\ &(C) \quad 6 \text{ hours} \\ &(D) \quad 8 \text{ hours} \end{align*} \]
\[ \begin{align*} &\text{Question 1: A car departs from city A heading towards city B at 60 mph. Simultaneously, another car departs from city B heading towards city A at 50 mph. If cities A and B are 660 miles apart, how long before the two cars meet, given that they maintain their average speeds?} \\ &(A) \quad 5 \text{ hours} \\ &(B) \quad 6 \text{ hours} \\ &(C) \quad 7 \text{ hours} \\ &(D) \quad 8 \text{ hours} \end{align*} \] \[ \begin{align*} &\text{Question 2: A motorcycle leaves from city C heading for city D at 75 mph. At the same time, another motorcycle leaves city D for city C at 65 mph. If the distance between city C and city D is 700 miles, how long before the two motorcycles meet, assuming they maintain their average speeds?} \\ &(A) \quad 4 \text{ hours} \\ &(B) \quad 5 \text{ hours} \\ &(C) \quad 6 \text{ hours} \\ &(D) \quad 7 \text{ hours} \end{align*} \] \[ \begin{align*} &\text{Question 3: A train leaves from station E heading for station F at 45 mph. At the same time, another train leaves station F for station E at 35 mph. If the distance between station E and station F is 360 miles, how long before the two trains meet, given they maintain their average speeds?} \\ &(A) \quad 4.5 \text{ hours} \\ &(B) \quad 5.5 \text{ hours} \\ &(C) \quad 6.5 \text{ hours} \\ &(D) \quad 7.5 \text{ hours} \end{align*} \]
\[ \begin{align*} &\text{Question 1: Two cities are 700 miles apart. A car and a bus start moving towards each other simultaneously from these cities. If the car moves with a speed of 50 mph and the bus with a speed of 40 mph, but the bus stops for a total of 0.5 hour, how long does it take for them to meet?} \\ &(A) \quad 7.5 \text{ hours} \\ &(B) \quad 8 \text{ hours} \\ &(C) \quad 8.5 \text{ hours} \\ &(D) \quad 9 \text{ hours} \end{align*} \] \[ \begin{align*} &\text{Question 2: A train departs from station A to station B, traveling at a speed of 60 mph. Two hours later, a second train departs from station B to station A, traveling at 100 mph. If the distance between the stations is 520 miles, how long after the second train starts does it take for the two trains to meet?} \\ &(A) \quad 2 \text{ hours} \\ &(B) \quad 2.5 \text{ hours} \\ &(C) \quad 3 \text{ hours} \\ &(D) \quad 3.5 \text{ hours} \end{align*} \] \[ \begin{align*} &\text{Question 3: A runner begins a marathon from city C and heads to city D at 8 mph. One hour later, another runner starts from city D and heads towards city C at 10 mph. If the distance between city C and city D is 26.2 miles, approximately how many hours after the second runner starts do the runners meet?} \\ &(A) \quad 0.4 \text{ hours} \\ &(B) \quad 0.5 \text{ hours} \\ &(C) \quad 0.6 \text{ hours} \\ &(D) \quad 0.7 \text{ hours} \end{align*} \]
\[ \begin{align*} &\text{Question 1: Two cars leave City A at the same time, one driving north at 70 mph and the other south at 50 mph. At what time will they be 600 miles apart?} \\ &(A) \quad 4:00 \text{ p.m.} \\ &(B) \quad 5:00 \text{ p.m.} \\ &(C) \quad 6:00 \text{ p.m.} \\ &(D) \quad 7:00 \text{ p.m.} \end{align*} \] \[ \begin{align*} &\text{Question 2: Two boats depart from the same harbor at 9 a.m., one sailing east at 20 knots and the other west at 15 knots. At what time will they be 175 nautical miles apart?} \\ &(A) \quad 11:00 \text{ a.m.} \\ &(B) \quad 12:00 \text{ p.m.} \\ &(C) \quad 1:00 \text{ p.m.} \\ &(D) \quad 2:00 \text{ p.m.} \end{align*} \] \[ \begin{align*} &\text{Question 3: Two trains leave the same station at 1 p.m., one heading east at 60 mph and the other heading west at 40 mph. At what time will they be 500 miles apart?} \\ &(A) \quad 5:00 \text{ p.m.} \\ &(B) \quad 6:00 \text{ p.m.} \\ &(C) \quad 7:00 \text{ p.m.} \\ &(D) \quad 8:00 \text{ p.m.} \end{align*} \]
\[ \begin{align*} &\text{Question 1: Two planes leave from the same airport, one heading north at 500 mph and the other heading east at 600 mph. At what time will they be 2200 miles apart if they leave at 1 p.m.?} \\ &(A) \quad 2:30 \text{ p.m.} \\ &(B) \quad 3:00 \text{ p.m.} \\ &(C) \quad 3:30 \text{ p.m.} \\ &(D) \quad 4:00 \text{ p.m.} \end{align*} \] \[ \begin{align*} &\text{Question 2: A car and a bus leave from City A at the same time. The car drives east at 70 mph and the bus drives west at 50 mph. If there is a 30-minute delay in the bus's departure, at what time will they be 635 miles apart assuming they leave at 2 p.m.?} \\ &(A) \quad 6:00 \text{ p.m.} \\ &(B) \quad 6:30 \text{ p.m.} \\ &(C) \quad 7:00 \text{ p.m.} \\ &(D) \quad 7:30 \text{ p.m.} \end{align*} \] \[ \begin{align*} &\text{Question 3: Two trains start from the same station at 12:30 p.m., one heading north at 60 mph and the other heading south at 70 mph. If the southern train has a 15-minute delay in departure, at what time will they be 665 miles apart?} \\ &(A) \quad 5:15 \text{ p.m.} \\ &(B) \quad 5:30 \text{ p.m.} \\ &(C) \quad 5:45 \text{ p.m.} \\ &(D) \quad 6:00 \text{ p.m.} \end{align*} \]
\[ \begin{align*} &\text{Question: Mrs. Johnson drove from her house to her office in 1.5 hours. After working for few hours, she returns home in rush hour traffic at a rate 15 mph less than her rate going. If Mrs. Johnson's total travel time is 3.5 hours, how fast did she travel on each leg of the trip?} \\ &(A) \quad 45 \text{ mph and } 30 \text{ mph} \\ &(B) \quad 60 \text{ mph and } 45 \text{ mph} \\ &(C) \quad 55 \text{ mph and } 40 \text{ mph} \\ &(D) \quad 50 \text{ mph and } 35 \text{ mph} \end{align*} \] \[ \begin{align*} &\text{Question: Mr. Smith drives from his home to a friend's place in 3 hours. Later, he returns home in traffic at a speed 10 mph less than his speed going. If Mr. Smith's total travel time is 7 hours, how fast did he travel on each leg of the trip?} \\ &(A) \quad 60 \text{ mph and } 50 \text{ mph} \\ &(B) \quad 50 \text{ mph and } 40 \text{ mph} \\ &(C) \quad 40 \text{ mph and } 30 \text{ mph} \\ &(D) \quad 70 \text{ mph and } 60 \text{ mph} \end{align*} \] \[ \begin{align*} &\text{Question: Miss Taylor travels from her home to her university in 2 hours. After attending her classes, she returns home in traffic at a rate 25 mph less than her rate going. If Miss Taylor's total travel time is 5 hours, how fast did she travel on each leg of the trip?} \\ &(A) \quad 65 \text{ mph and } 40 \text{ mph} \\ &(B) \quad 75 \text{ mph and } 50 \text{ mph} \\ &(C) \quad 70 \text{ mph and } 45 \text{ mph} \\ &(D) \quad 80 \text{ mph and } 55 \text{ mph} \end{align*} \]
\[ \begin{align*} &\text{Question: In his motorboat, a man can go downstream in 1 hour less time than he can go upstream the same distance. If the current is 5 mph, how fast can he travel in still water if it takes him 2 hours to travel upstream the given distance?} \\ &(A) \quad 10 \text{ mph} \\ &(B) \quad 15 \text{ mph} \\ &(C) \quad 20 \text{ mph} \\ &(D) \quad 25 \text{ mph} \end{align*} \] \[ \begin{align*} &\text{Question: In her motorboat, a woman can travel downstream in 2 hours less time than she can travel upstream the same distance. If the current is 3 mph, how fast can she travel in still water if it takes her 5 hours to travel upstream the given distance?} \\ &(A) \quad 10 \text{ mph} \\ &(B) \quad 12 \text{ mph} \\ &(C) \quad 15 \text{ mph} \\ &(D) \quad 21 \text{ mph} \end{align*} \] \[ \begin{align*} &\text{Question: In his sailboat, a man can go downstream in 1.5 hours less time than he can go upstream the same distance. If the current is 4 mph, how fast can he travel in still water if it takes him 3 hours to travel upstream the given distance?} \\ &(A) \quad 12 \text{ mph} \\ &(B) \quad 16 \text{ mph} \\ &(C) \quad 20 \text{ mph} \\ &(D) \quad 24 \text{ mph} \end{align*} \] \[ \begin{align*} &\text{Question: In her kayak, a woman can go downstream in 3 hours less time than she can go upstream the same distance. If the current is 6 mph, how fast can she travel in still water if it takes her 6 hours to travel upstream the given distance?} \\ &(A) \quad 12 \text{ mph} \\ &(B) \quad 18 \text{ mph} \\ &(C) \quad 24 \text{ mph} \\ &(D) \quad 30 \text{ mph} \end{align*} \]
\[ \begin{align*} &\text{Question: A police officer notices a motorbike going over the speed limit at 50 mph and starts chasing it from 1 mile behind. If the officer travels at an average rate of 60 mph, how long will it take to catch up to the motorbike?} \\ &(A) \quad 0.1 \text{ hours} \\ &(B) \quad 0.2 \text{ hours} \\ &(C) \quad 0.3 \text{ hours} \\ &(D) \quad 0.4 \text{ hours} \end{align*} \] \[ \begin{align*} &\text{Question: A security guard spots a trespasser running away at 8 mph and starts pursuing him from 1.8 miles behind. If the guard can run at an average speed of 10 mph, how long will it be before he catches up?} \\ &(A) \quad 0.3 \text{ hours} \\ &(B) \quad 0.6 \text{ hours} \\ &(C) \quad 0.9 \text{ hours} \\ &(D) \quad 1.2 \text{ hours} \end{align*} \] \[ \begin{align*} &\text{Question: A park ranger spots a bear moving at 5 mph and follows it from 5 miles behind. If the ranger travels at an average speed of 7 mph, how long will it take for the ranger to reach the bear?} \\ &(A) \quad 1.5 \text{ hours} \\ &(B) \quad 2.0 \text{ hours} \\ &(C) \quad 2.5 \text{ hours} \\ &(D) \quad 3.0 \text{ hours} \end{align*} \]
\[ \begin{align*} &\text{Question: How many gallons of a solution that contains 30% salt should be mixed with 7 gallons of a 10% salt solution to achieve a final solution with 16% salt concentration?} \\ &(A) \quad 2 \text{ gallons} \\ &(B) \quad 3 \text{ gallons} \\ &(C) \quad 4 \text{ gallons} \\ &(D) \quad 5 \text{ gallons} \end{align*} \]
\[ \begin{align*} &\text{Question: If you have a 30% sugar mixture and you wish to blend it with 6 liters of an 10% sugar mixture to yield a 18% sugar solution, how many liters of the 30% solution will you need?} \\ &(A) \quad 3 \text{ liters} \\ &(B) \quad 4 \text{ liters} \\ &(C) \quad 5 \text{ liters} \\ &(D) \quad 6 \text{ liters} \end{align*} \]
\[ \begin{align*} &\text{Question: To produce a 7% alcohol solution, how many pints of a 5% alcohol mixture would be required to mix with 2 pints of a 10% alcohol mixture?} \\ &(A) \quad 2.5 \text{ pints} \\ &(B) \quad 3 \text{ pints} \\ &(C) \quad 3.5 \text{ pints} \\ &(D) \quad 4 \text{ pints} \end{align*} \]
\[ \begin{align*} &\text{Question: Jessica has some coins in her wallet consisting of quarters, dimes, and pennies. She has three more dimes than quarters, and twice as many pennies as dimes. How many of each kind of coin does she have if the total value is 136 cents?} \\ &(A) \quad \text{2 quarters, 5 dimes, 10 pennies} \\ &(B) \quad \text{3 quarters, 6 dimes, 12 pennies} \\ &(C) \quad \text{1 quarter, 4 dimes, 8 pennies} \\ &(D) \quad \text{4 quarters, 7 dimes, 14 pennies} \end{align*} \] \[ \begin{align*} &\text{Question: Richard has some coins in his bag consisting of quarters, nickels, and pennies. He has four more nickels than quarters, and twice as many pennies as nickels. How many of each kind of coin does he have if the total value is 1.56 dollars?} \\ &(A) \quad \text{3 quarters, 7 nickels, 14 pennies} \\ &(B) \quad \text{2 quarters, 6 nickels, 12 pennies} \\ &(C) \quad \text{4 quarters, 8 nickels, 16 pennies} \\ &(D) \quad \text{1 quarter, 5 nickels, 10 pennies} \end{align*} \] \[ \begin{align*} &\text{Question: Samantha has some coins in her purse consisting of dimes, nickels, and pennies. She has one more dime than nickels, and five times as many pennies as nickels. How many of each kind of coin does she have if the total value is 70 cents?} \\ &(A) \quad \text{2 dimes, 1 nickel, 5 pennies} \\ &(B) \quad \text{3 dimes, 2 nickels, 10 pennies} \\ &(C) \quad \text{1 dime, 0 nickels, 0 pennies} \\ &(D) \quad \text{4 dimes, 3 nickels, 15 pennies} \end{align*} \]
\[ \begin{align*} &\text{Question: Sam has a total of \$95. If he has two more \$5 bills than \$10 bills, and three more \$1 bills than \$5 bills, how many of each does he have?} \\ &(A) \quad \text{4 \$10 bills, 6 \$5 bills, 5 \$1 bills} \\ &(B) \quad \text{3 \$10 bills, 5 \$5 bills, 7 \$1 bills} \\ &(C) \quad \text{2 \$10 bills, 4 \$5 bills, 9 \$1 bills} \\ &(D) \quad \text{5 \$10 bills, 7 \$5 bills, 10 \$1 bills} \end{align*} \] \[ \begin{align*} &\text{Question: Monica has a total of \$77. If she has two more \$5 bills than \$10 bills, and one more \$1 bill than \$5 bills, how many of each does she have?} \\ &(A) \quad \text{3 \$10 bills, 5 \$5 bills, 4 \$1 bills} \\ &(B) \quad \text{2 \$10 bills, 4 \$5 bills, 6 \$1 bills} \\ &(C) \quad \text{4 \$10 bills, 6 \$5 bills, 7 \$1 bills} \\ &(D) \quad \text{1 \$10 bill, 3 \$5 bills, 8 \$1 bills} \end{align*} \] \[ \begin{align*} &\text{Question: Harry has a total of \$116. If he has three more \$5 bills than \$10 bills, and two more \$1 bills than \$5 bills, how many of each does he have?} \\ &(A) \quad \text{5 \$10 bills, 8 \$5 bills, 7 \$1 bills} \\ &(B) \quad \text{4 \$10 bills, 7 \$5 bills, 9 \$1 bills} \\ &(C) \quad \text{6 \$10 bills, 9 \$5 bills, 11 \$1 bills} \\ &(D) \quad \text{3 \$10 bills, 6 \$5 bills, 11 \$1 bills} \end{align*} \]
\[ \begin{align*} &\text{Question: Sarah’s father is six times as old as Sarah. Five years ago he was nine times as old. How old is each now?} \\ &(A) \quad \text{Sarah is 10 years old, Father is 60 years old} \\ &(B) \quad \text{Sarah is 5 years old, Father is 30 years old} \\ &(C) \quad \text{Sarah is 21 years old, Father is 126 years old} \\ &(D) \quad \text{Sarah is 24 years old, Father is 144 years old} \end{align*} \] \[ \begin{align*} &\text{Question: David’s mother is three times as old as David. Five years ago she was five times as old. How old is each now?} \\ &(A) \quad \text{David is 8 years old, Mother is 24 years old} \\ &(B) \quad \text{David is 10 years old, Mother is 30 years old} \\ &(C) \quad \text{David is 16 years old, Mother is 48 years old} \\ &(D) \quad \text{David is 20 years old, Mother is 60 years old} \end{align*} \]
\[ \begin{align*} &\text{Question: Mary’s father is four times as old as Mary. Five years ago he was seven times as old. How old is each now?} \\ &(A) \quad \text{Mary is 10 years old, Father is 40 years old} \\ &(B) \quad \text{Mary is 15 years old, Father is 60 years old} \\ &(C) \quad \text{Mary is 20 years old, Father is 80 years old} \\ &(D) \quad \text{Mary is 25 years old, Father is 100 years old} \end{align*} \] \[ \begin{align*} &\text{Question: John’s mother is five times as old as John. Six years ago she was nine times as old. How old is each now?} \\ &(A) \quad \text{John is 12 years old, Mother is 60 years old} \\ &(B) \quad \text{John is 15 years old, Mother is 75 years old} \\ &(C) \quad \text{John is 18 years old, Mother is 90 years old} \\ &(D) \quad \text{John is 21 years old, Mother is 105 years old} \end{align*} \] \[ \begin{align*} &\text{Question: Sarah’s father is six times as old as Sarah. Three years ago he was nine times as old. How old is each now?} \\ &(A) \quad \text{Sarah is 5 years old, Father is 30 years old} \\ &(B) \quad \text{Sarah is 8 years old, Father is 48 years old} \\ &(C) \quad \text{Sarah is 21 years old, Father is 126 years old} \\ &(D) \quad \text{Sarah is 24 years old, Father is 144 years old} \end{align*} \] \[ \begin{align*} &\text{Question: David’s mother is three times as old as David. Eight years ago she was five times as old. How old is each now?} \\ &(A) \quad \text{David is 8 years old, Mother is 24 years old} \\ &(B) \quad \text{David is 12 years old, Mother is 36 years old} \\ &(C) \quad \text{David is 16 years old, Mother is 48 years old} \\ &(D) \quad \text{David is 20 years old, Mother is 60 years old} \end{align*} \]
\[ \begin{align*} &\text{Question: The width of a rectangular garden is 3 meters less than half the length. If the perimeter is 48 meters, what are the garden's dimensions?} \\ &(A) \quad \text{Length is 14 m, Width is 7 m} \\ &(B) \quad \text{Length is 16 m, Width is 5 m} \\ &(C) \quad \text{Length is 18 m, Width is 6 m} \\ &(D) \quad \text{Length is 15 m, Width is 6 m} \end{align*} \] \[ \begin{align*} &\text{Question: The length of a rectangular field is 5 m more than twice the width. If the perimeter is 52 m, what are the field's dimensions?} \\ &(A) \quad \text{Length is 9 m, Width is 2 m} \\ &(B) \quad \text{Length is 19 m, Width is 7 m} \\ &(C) \quad \text{Length is 25 m, Width is 10 m} \\ &(D) \quad \text{Length is 45 m, Width is 20 m} \end{align*} \] \[ \begin{align*} &\text{Question: The width of a rectangular parking lot is 4 m less than the length. If the perimeter is 32 m, what are the parking lot's dimensions?} \\ &(A) \quad \text{Length is 12 m, Width is 8 m} \\ &(B) \quad \text{Length is 11 m, Width is 7 m} \\ &(C) \quad \text{Length is 10 m, Width is 6 m} \\ &(D) \quad \text{Length is 9 m, Width is 5 m} \end{align*} \]
\[ \begin{align*} &\text{Question: The third angle of a triangle is 3 times the first angle, and the second angle is 10° more than the first. Find the three angles.} \\ &(A) \quad 30°, 40°, 110° \\ &(B) \quad 40°, 50°, 90° \\ &(C) \quad 20°, 30°, 130° \\ &(D) \quad 35°, 45°, 100° \end{align*} \] \[ \begin{align*} &\text{Question: The second angle of a triangle is 4 times the first, and the third angle is 5° less than the first. Find the three angles.} \\ &(A) \quad 15°, 60°, 105° \\ &(B) \quad 20°, 80°, 80° \\ &(C) \quad 25°, 100°, 55° \\ &(D) \quad 34°, 102°, 44° \end{align*} \] \[ \begin{align*} &\text{Question: The first angle of a triangle is half the second, and the third is 10° smaller than the second. Find the three angles.} \\ &(A) \quad 40°, 80°, 70° \\ &(B) \quad 35°, 70°, 75° \\ &(C) \quad 38°, 76°, 66° \\ &(D) \quad 45°, 90°, 45° \end{align*} \]
\[ \begin{align*} &\text{Question: The length of a rectangle is three times the width. If the length is increased by 4 feet and the width is decreased by 4 feet, the area is decreased by 64 square feet. Find the dimensions of the original rectangle.} \\ &(A) \quad 6 \text{ ft} \times 18 \text{ ft} \\ &(B) \quad 14 \text{ ft} \times 42 \text{ ft} \\ &(C) \quad 10 \text{ ft} \times 30 \text{ ft} \\ &(D) \quad 11 \text{ ft} \times 33 \text{ ft} \end{align*} \] \[ \begin{align*} &\text{Question: The width of a rectangle is half the length. If the width is increased by 3 feet and the length is decreased by 3 feet, the area is increased by 18 square feet. Find the dimensions of the original rectangle.} \\ &(A) \quad 9 \text{ ft} \times 18 \text{ ft} \\ &(B) \quad 12 \text{ ft} \times 24 \text{ ft} \\ &(C) \quad 15 \text{ ft} \times 30 \text{ ft} \\ &(D) \quad 8 \text{ ft} \times 16 \text{ ft} \end{align*} \] \[ \begin{align*} &\text{Question: The length of a rectangle is four times the width. If the length is decreased by 2 feet and the width is increased by 2 feet, the area is increased by 8 square feet. Find the dimensions of the original rectangle.} \\ &(A) \quad 8 \text{ ft} \times 32 \text{ ft} \\ &(B) \quad 10 \text{ ft} \times 40 \text{ ft} \\ &(C) \quad 7 \text{ ft} \times 28 \text{ ft} \\ &(D) \quad 2 \text{ ft} \times 8 \text{ ft} \end{align*} \]
\[ \begin{align*} &\text{Question: The hundreds digit of a certain number is 2 less than the units digit. The sum of the digits is 14. What is the number?} \\ &(A) \quad 368 \\ &(B) \quad 842 \\ &(C) \quad 533 \\ &(D) \quad 761 \end{align*} \] \[ \begin{align*} &\text{Question: The tens digit of a certain two-digit number is 4 more than the units digit. The sum of the digits is 10. What is the number?} \\ &(A) \quad 73 \\ &(B) \quad 64 \\ &(C) \quad 19 \\ &(D) \quad 28 \end{align*} \] \[ \begin{align*} &\text{Question: The tens digit of a certain number is 5 less than the units digit. The sum of the digits is 13. What is the number?} \\ &(A) \quad 89 \\ &(B) \quad 76 \\ &(C) \quad 94 \\ &(D) \quad 38 \end{align*} \]
\[ \begin{align*} &\text{Question: The tens digit of a two-digit number is three times the units digit. If the digits are reversed, the new number is 27 less than the original number. What is the number?} \\ &(A) \quad 14 \\ &(B) \quad 31 \\ &(C) \quad 63 \\ &(D) \quad 74 \end{align*} \] \[ \begin{align*} &\text{Question: The hundreds digit of a three-digit number is twice the units digit. If the digits are reversed, the new number is 198 less than the original number. What is the number?} \\ &(A) \quad 204 \\ &(B) \quad 408 \\ &(C) \quad 315 \\ &(D) \quad 520 \end{align*} \] \[ \begin{align*} &\text{Question: The tens digit of a two-digit number is four times the units digit. If the digits are reversed, the new number is 27 less than the original number. What is the number?} \\ &(A) \quad 24 \\ &(B) \quad 41 \\ &(C) \quad 82 \\ &(D) \quad 81 \end{align*} \]
\[ \begin{align*} &\text{Question: The ratio of the units digit to the tens digit of a two-digit number is 3. The tens digit is 6 less than the units digit. What is the number?} \\ &(A) \quad 13 \\ &(B) \quad 31 \\ &(C) \quad 39 \\ &(D) \quad 26 \end{align*} \] \[ \begin{align*} &\text{Question: The ratio of the units digit to the tens digit of a two-digit number is 4. The tens digit is 6 less than the units digit. What is the number?} \\ &(A) \quad 28 \\ &(B) \quad 42 \\ &(C) \quad 36 \\ &(D) \quad 12 \end{align*} \] \[ \begin{align*} &\text{Question: The ratio of the units digit to the tens digit of a two-digit number is one-third. The tens digit is 4 more than the units digit. What is the number?} \\ &(A) \quad 31 \\ &(B) \quad 62 \\ &(C) \quad 93 \\ &(D) \quad 84 \end{align*} \]