Solving Applications With Discount Or Mark-up

Solution

\(\begin{array}{ccc}\hfill \text{Original price}& =\hfill & \text{\$}140\hfill \\ \hfill \text{Discount rate}& =\hfill & 35\text{%}\hfill \\ \hfill \text{Discount}& =\hfill & ?\hfill \end{array}\)

\(\begin{array}{} \mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \text{the amount of discount}\hfill \\ \begin{array}{c}\mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill \\ \text{Choose a variable to represent that quantity}.\hfill \end{array}\hfill & & & \begin{array}{}\\ \text{Let}\phantom{\rule{0.2em}{0ex}}d=\text{the amount of discount.}\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 4. Translate}}\phantom{\rule{0.2em}{0ex}}\text{into an equation. Write a}\hfill \\ \text{sentence that gives the information to find it.}\hfill \end{array}\hfill & & & \text{The discount is 35% of \$140.}\hfill \\ \text{Translate into an equation.}\hfill & & & d=0.35\left(140\right)\hfill \\ \mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the equation.}\hfill & & & d=49\hfill \\ \mathbf{\text{Step 6. Check}}\text{:}\phantom{\rule{0.2em}{0ex}}\text{Does this make sense?}\hfill & & & \\ \begin{array}{}\\ \phantom{\rule{2.5em}{0ex}}\text{Is a \$49 discount reasonable for a}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{\$140 dress? Yes.}\hfill \end{array}\hfill & & & \\ \begin{array}{c}\mathbf{\text{Step 7. Write}}\phantom{\rule{0.2em}{0ex}}\text{a complete sentence to answer}\hfill \\ \text{the question.}\hfill \end{array}\hfill & & & \text{The amount of discount was \$49.}\hfill \end{array}\)

Read the problem again.

Step 1. Identify what we are looking for.		the sale price of the dress
Step 2. Name what we are looking for.
Choose a variable to represent that quantity.		Let \(s=\) the sale price.
Step 3. Translate into an equation.
Write a sentence that gives the information to find it.
Translate into an equation.
Step 4. Solve the equation.
Step 5. Check. Does this make sense?
Is the sale price less than the original price?
Yes, $91 is less than $140.
Step 6. Answer the question with a complete sentence.		The sale price of the dress was $91.

Solution

\(\begin{array}{ccc}\hfill \text{Original price}& =\hfill & \text{\$}31\hfill \\ \hfill \text{Discount}& =\hfill & ?\hfill \\ \hfill \text{Sale Price}& =\hfill & \text{\$}13.95\hfill \end{array}\)

\(\begin{array}{} \mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \text{the amount of discount}\hfill \\ \begin{array}{c}\mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill \\ \text{Choose a variable to represent that quantity}.\hfill \end{array}\hfill & & & \begin{array}{}\\ \text{Let}\phantom{\rule{0.2em}{0ex}}d=\text{the amount of discount.}\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 4. Translate}}\phantom{\rule{0.2em}{0ex}}\text{into an equation.}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{Write a sentence that gives the}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{information to find it.}\hfill \end{array}\hfill & & & \begin{array}{c}\text{The discount is the difference between the original}\hfill \\ \text{price and the sale price.}\hfill \end{array}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{Translate into an equation.}\hfill & & & d=31-13.95\hfill \\ \mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the equation.}\hfill & & & d=17.05\hfill \\ \mathbf{\text{Step 6. Check}}\text{:}\phantom{\rule{0.2em}{0ex}}\text{Does this make sense?}\hfill & & & \\ \phantom{\rule{2.5em}{0ex}}\text{Is 17.05 less than 31? Yes.}\hfill & & & \\ \mathbf{\text{Step 7. Answer}}\phantom{\rule{0.2em}{0ex}}\text{the question with a complete sentence.}\hfill & & & \text{The amount of discount was \$17.05.}\hfill \end{array}\)

Read the problem again.

Step 1. Identify what we are looking for.		the discount rate
Step 2. Name what we are looking for.
Choose a variable to represent it.		Let \(r=\) the discount rate.
Step 3. Translate into an equation.
Write a sentence that gives the information to find it.
Translate into an equation.
Step 4. Solve the equation.
Divide both sides by 31.
Change to percent form.
Step 5. Check. Does this make sense?
Is $17.05 equal to 55% of $31?
\(17.05\stackrel{?}{=}0.55\left(31\right)\)
\(17.05=17.05✓\)
Step 6. Answer the question with a complete sentence.		The rate of discount was 55%.