3.2 Solve percent applications  (Page 6/18)

There may be times when we know the original price and the sale price, and we want to know the discount rate    . To find the discount rate, first we will find the amount of discount and then use it to compute the rate as a percent of the original price. [link] will show this case.

Jeannette bought a swimsuit at a sale price of $13.95. The original price of the swimsuit was $31. Find the ⓐ amount of discount and ⓑ discount rate.

Solution

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

ⓑ
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%.

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Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the original cost    . The retailer then adds a mark-up    to the original cost to get the list price    , the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.

We summarize the mark-up model in the box below.

Adam’s art gallery bought a photograph at original cost $250. Adam marked the price up 40%. Find the ⓐ amount of mark-up and ⓑ the list price of the photograph.

Solution

ⓐ

Step 1. Read the problem.
Step 2. Identify what we are looking for.		the amount of mark-up
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let $m=$ the amount of markup.
Step 4. Translate into an equation.
Write a sentence that gives the information to find it.
Translate into an equation.
Step 5. Solve the equation.
Step 6. Check. Does this make sense?
Yes, 40% is less than one-half and 100 is less than half of 250.
Step 7. Answer the question with a complete sentence.		The mark-up on the phtograph was $100.

ⓑ

Step 1. Read the problem again.
Step 2. Identify what we are looking for.		the list price
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let $p=$ the list price.
Step 4. Translate into an equation.
Write a sentence that gives the information to find it.
Translate into an equation.
Step 5. Solve the equation.
Step 6. Check. Does this make sense?
Is the list price more than the net price? Is $350 more than $250? Yes.
Step 7. Answer the question with a complete sentence.		The list price of the photograph was $350.

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