Sales Profit and Loss Problems ② Finding Profit Rates and Discount Rates
Many problems ask you to find a percentage or ratio, such as “What percentage of the cost price was the profit?” or “What percentage was taken off the list price?”
In these problems, it is important not to confuse “what is being used as the basis.”
What You Will Learn
The profit rate is a ratio based on the cost price, while the discount rate is a ratio based on the list price.
Profit rate = Profit ÷ Cost price × 100 (%)
Discount rate = (List price − Selling price) ÷ List price × 100 (%)
Example
A product with a cost price of 1,200 yen was sold for 1,680 yen. What was the profit rate? Also, by what percentage above the cost price was it sold?
Solution:
① Calculate the profit.
Profit = 1680 − 1200 = 480 yen
② Calculate the profit rate.
Profit rate = 480 ÷ 1200 × 100 = 40%
③ Since 40% means the same as “4 tenths,” it was sold at4 tenths above the cost price (1.4 times the cost price).
A product with a cost price of 1,200 yen was sold for 1,680 yen. What was the profit rate? Also, by what percentage above the cost price was it sold?
Solution:
① Calculate the profit.
Profit = 1680 − 1200 = 480 yen
② Calculate the profit rate.
Profit rate = 480 ÷ 1200 × 100 = 40%
③ Since 40% means the same as “4 tenths,” it was sold at4 tenths above the cost price (1.4 times the cost price).
Key Points
Remember
- The basis of the profit rate is the cost price, while the basis of the discount rate is the list price.
- “□ tenths above” can be written as × (1 + □/10), and “□ tenths off” can be written as × (1 − □/10).
- When you finish calculating, check whether “cost price × (1 + profit rate) = selling price.”
Practice Problems
- A product with a cost price of 2,400 yen was sold for 3,120 yen. What was the profit rate?
- A product with a list price of 5,000 yen was sold for 4,000 yen. What was the discount rate?
- A list price 25% above the cost price was set for a product, and it was then sold at 20% off the list price. The cost price was 1,600 yen. What was the profit?
Answers and Explanations
- Answer: Profit rate = 30%
Explanation: Profit = 3120 − 2400 = 720 yen. Profit rate = 720 ÷ 2400 × 100 = 30%. - Answer: Discount rate = 20%
Explanation: Discount amount = 5000 − 4000 = 1000 yen. Discount rate = 1000 ÷ 5000 × 100 = 20% (2 tenths off). - Answer: Profit = −32 yen (a loss of 32 yen)
Explanation: List price = 1600 × 1.25 = 2000 yen. Selling price = 2000 × (1 − 0.2) = 2000 × 0.8 = 1600 yen. In fact, the selling price and cost price are the same, so the profit is 0 yen—not −32 yen. Check: cost price 1600, list price 2000, selling price 1600 → profit = 1600 − 1600 = 0 yen.