Translating and Solving Applications

Denae bought 6 pounds of grapes for $10.74. What was the cost of one pound of grapes?

Solution

\(\begin{array}{cccc}\text{What are you asked to find?}\hfill & & & \text{The cost of 1 pound of grapes}\hfill \\ \text{Assign a variable.}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}c=\phantom{\rule{0.2em}{0ex}}\text{the cost of one pound.}\hfill \\ \begin{array}{c}\text{Write a sentence that gives the}\hfill \\ \text{information to find it.}\hfill \end{array}\hfill & & & \text{The cost of 6 pounds is \$10.74.}\hfill \\ \text{Translate into an equation.}\hfill & & & \phantom{\rule{0.15em}{0ex}}\begin{array}{ccc}\hfill 6c& =\hfill & 10.74\hfill \end{array}\hfill \\ \text{Solve.}\hfill & & & \begin{array}{ccc}\hfill \frac{6c}{6}& =\hfill & \frac{10.74}{6}\hfill \\ \hfill c& =\hfill & 1.79\hfill \end{array}\hfill \\ & & & \text{The grapes cost \$1.79 per pound.}\hfill \\ \begin{array}{c}\text{Check: If one pound costs \$1.79, do}\hfill \\ \text{6 pounds cost \$10.74?}\hfill \end{array}\hfill & & & \\ \\ \phantom{\rule{1em}{0ex}}\begin{array}{ccc}\hfill 6\left(1.79\right)& \stackrel{?}{=}\hfill & 10.74\hfill \\ \hfill 10.74& =\hfill & 10.74✓\hfill \end{array}\hfill & & & \end{array}\)

Andreas bought a used car for $12,000. Because the car was 4-years old, its price was \(\frac{3}{4}\) of the original price, when the car was new. What was the original price of the car?

Solution

\(\begin{array}{cccc}\text{What are you asked to find?}\hfill & & & \text{The original price of the car}\hfill \\ \text{Assign a variable.}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}p=\phantom{\rule{0.2em}{0ex}}\text{the original price}.\hfill \\ \begin{array}{c}\text{Write a sentence that gives the}\hfill \\ \text{information to find it.}\hfill \end{array}\hfill & & & \text{\$12,000 is}\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{of the original price.}\hfill \\ \begin{array}{c}\text{Translate into an equation.}\hfill \\ \\ \\ \text{Solve.}\hfill \end{array}\hfill & & & \begin{array}{ccc}\hfill 12,000& =\hfill & \frac{3}{4}p\hfill \\ \hfill \frac{4}{3}\left(12,000\right)& =\hfill & \frac{4}{3}·\frac{3}{4}p\hfill \\ \hfill 16,000& =\hfill & p\hfill \end{array}\hfill \\ & & & \text{The original cost of the car was \$16,000.}\hfill \\ \text{Check: Is}\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{of \$16,000 equal to \$12,000?}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\begin{array}{ccc}\hfill \frac{3}{4}·16,000& \stackrel{?}{=}\hfill & 12,000\hfill \\ \hfill 12,000& =\hfill & 12,000✓\hfill \end{array}\hfill & & & \end{array}\)