  ## From a handpicked tutor in live 1-to-1 classes 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{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{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{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{(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{(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{(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*}

\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*}

\begin{align*} &\text{Question 1: Given that } \frac{\sqrt{y^4}}{\sqrt{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{z^5}}{\sqrt{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{w^6}}{\sqrt{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{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{m^2 n^{10}} ? \\ &(A) \quad m n^2 \\ &(B) \quad mn \\ &(C) \quad n \sqrt{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{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{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{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{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{3} \text{ and } 3 a^2=\sqrt{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{2} \text{ and } 2 b^3=\sqrt{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 60. Four times the first is 8 less than thrice the second. What are these numbers?} \\ &(A) \quad (16, 44) \\ &(B) \quad (20, 40) \\ &(C) \quad (24, 36) \\ &(D) \quad (28, 32) \end{align*}
\begin{align*} &\text{Question 2: The sum of two numbers is 70. If five times the second number is 10 more than twice the first, what are the numbers?} \\ &(A) \quad (20, 50) \\ &(B) \quad (25, 45) \\ &(C) \quad (30, 40) \\ &(D) \quad (35, 35) \end{align*}
\begin{align*} &\text{Question 3: Two numbers have a sum of 80. The second number is seven times the first minus 10. What are the numbers?} \\ &(A) \quad (10, 70) \\ &(B) \quad (15, 65) \\ &(C) \quad (20, 60) \\ &(D) \quad (30, 50) \end{align*}
\begin{align*} &\text{Question 1: Two numbers add up to 100. The square of the first is 100 more than four times the second. What are these numbers?} \\ &(A) \quad (10, 90) \\ &(B) \quad (20, 80) \\ &(C) \quad (30, 70) \\ &(D) \quad (40, 60) \end{align*}
\begin{align*} &\text{Question 2: The sum of two numbers is 120. If the cube of the second number is 27 times the first, what are the numbers?} \\ &(A) \quad (9, 111) \\ &(B) \quad (18, 102) \\ &(C) \quad (36, 84) \\ &(D) \quad (45, 75) \end{align*}
\begin{align*} &\text{Question 3: Two numbers have a sum of 140. The square of the second number is four times the first plus 100. What are the numbers?} \\ &(A) \quad (20, 120) \\ &(B) \quad (30, 110) \\ &(C) \quad (40, 100) \\ &(D) \quad (50, 90) \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 130?} \\ &(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 375?} \\ &(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 204?} \\ &(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 20 cm, what are the lengths of the sides?} \\ &(A) \quad (5, 6, 9) \text{ cm} \\ &(B) \quad (6, 7, 7) \text{ cm} \\ &(C) \quad (7, 8, 5) \text{ cm} \\ &(D) \quad (8, 9, 3) \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 24 feet, what are the lengths of the sides?} \\ &(A) \quad (4, 6, 14) \text{ feet} \\ &(B) \quad (5, 7, 12) \text{ feet} \\ &(C) \quad (6, 8, 10) \text{ feet} \\ &(D) \quad (7, 9, 8) \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 31 meters, what are the lengths of the sides?} \\ &(A) \quad (8, 4, 19) \text{ meters} \\ &(B) \quad (9, 5, 17) \text{ meters} \\ &(C) \quad (10, 6, 15) \text{ meters} \\ &(D) \quad (11, 7, 13) \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 42 cm, what are the lengths of the sides?} \\ &(A) \quad (12, 8, 22) \text{ cm} \\ &(B) \quad (11, 9, 22) \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 the square of the second side minus 1 m.} \\ &\text{If the perimeter of the triangle is 23 m, what are the lengths of the sides?} \\ &(A) \quad (8, 5, 10) \text{ m} \\ &(B) \quad (7, 6, 10) \text{ m} \\ &(C) \quad (9, 4, 10) \text{ m} \\ &(D) \quad (6, 7, 10) \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 40 feet, what are the lengths of the sides?} \\ &(A) \quad (12, 4, 24) \text{ feet} \\ &(B) \quad (15, 5, 20) \text{ feet} \\ &(C) \quad (18, 6, 16) \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 mph. After three hours, a sports car leaves the same location heading for Miami, 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 1.5 \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 \text{ hours} \\ &(B) \quad 5 \text{ hours} \\ &(C) \quad 6 \text{ hours} \\ &(D) \quad 7 \text{ hours} \end{align*} These questions involve understanding of relative speed and distance.
\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 80 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, how many hours after the second runner starts do the runners 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 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 10 a.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 600 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 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 650 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 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 8 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 7 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: Mr. Williams drives from his house to his farm in 3 hours. Due to a roadblock, he returns home on a longer route with a speed 30 mph less than his speed going. If Mr. Williams' total travel time is 9 hours, how fast did he travel on each leg of the trip?} \\ &(A) \quad 90 \text{ mph and } 60 \text{ mph} \\ &(B) \quad 60 \text{ mph and } 30 \text{ mph} \\ &(C) \quad 75 \text{ mph and } 45 \text{ mph} \\ &(D) \quad 70 \text{ mph and } 40 \text{ mph} \end{align*} \begin{align*} &\text{Question: Mr. Thompson drives from his home to his office in 2 hours. Because of a traffic jam, he returns home at a speed 25 mph less than his speed going. If Mr. Thompson's total travel time is 7 hours, how fast did he travel on each leg of the trip?} \\ &(A) \quad 60 \text{ mph and } 35 \text{ mph} \\ &(B) \quad 70 \text{ mph and } 45 \text{ mph} \\ &(C) \quad 75 \text{ mph and } 50 \text{ mph} \\ &(D) \quad 80 \text{ mph and } 55 \text{ mph} \end{align*} \begin{align*} &\text{Question: Mr. Clark travels from his home to a resort in 4 hours. On his way back, he takes a scenic route at a speed 15 mph less than his speed going. If Mr. Clark's total travel time is 10 hours, how fast did he travel on each leg of the trip?} \\ &(A) \quad 60 \text{ mph and } 45 \text{ mph} \\ &(B) \quad 75 \text{ mph and } 60 \text{ mph} \\ &(C) \quad 65 \text{ mph and } 50 \text{ mph} \\ &(D) \quad 70 \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 12 \text{ mph} \\ &(B) \quad 15 \text{ mph} \\ &(C) \quad 18 \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: 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*}