Calculating the Overall Percentage Reduction in a Compound Discount Price
The concept of compound discounts or successive discounts is a common scenario in retail, where an initial discount is applied, followed by another discount on the already reduced price. This example will walk you through a detailed calculation to determine the overall percentage reduction when a 20% discount is followed by a 25% discount on a product's original price.
Scenario:
A shop offers a sale with a 20% discount on all prices. On the final day, they further reduce all sale prices by 25%. Alex buys a hair dryer on the final day of the sale. The question is to determine the overall percentage reduction on the price of the hair dryer.
Step-by-Step Calculation
Let P be the original price of the hair dryer.
Step 1: Applying the first discount of 20%
The first discount of 20% on P would result in a sale price:
[ text{Sale Price after 20% Off} P - 0.20P 0.80P ]
Step 2: Applying the second discount of 25% on the sale price
On the final day, Alex buys the item at a further discount of 25% on the already reduced price:
[ text{Final Price} 0.80P - 0.25(0.80P) 0.80P - 0.20P 0.60P ]
Step 3: Determining the overall price reduction
The overall reduction from the original price would be:
[ text{Reduction in Price} P - 0.60P 0.40P ]
Converting this into a percentage, we have:
[ text{Overall Percentage Reduction} frac{0.40P}{P} times 100 40% ]
This shows that the overall percentage reduction on the price of the hair dryer is 40%.
Example with Radio
To further illustrate, let’s use a radio as the example:
Scenario: P is the original price of the radio. The sales price is 20% off (0.80P) and on the final day, the price is further reduced by 25% (0.75 of 0.80P).
Final Price: 0.60P (60% of the original price).
Overall Reduction: 100% - 60% 40%.
This confirms that the overall percentage reduction is indeed 40%.
Understanding Compound Discounts
To better understand compound discounts, consider the following steps:
Step 1: Define the Original Price (O)
Let O be the original price of a product.
After a first discount of 20%, the new price (S1) is:
[ S1 O - 0.20O 0.80O ]
Step 2: Apply the Second Discount of 25%
On the final day, a further reduction of 25% on the already reduced price (S1) is applied:
[ S2 S1 - 0.25S1 0.80O - 0.25(0.80O) 0.80O - 0.20O 0.60O ]
This indicates a final price of 60% of the original price.
Step 3: Calculate the Overall Percentage Reduction
The overall percentage reduction is calculated as follows:
[ text{Overall Percentage Reduction} 100% - 60% 40% ]
This shows that the overall reduction from the original price is 40%.
Conclusion: Compound discounts can lead to significant overall reductions in prices. Understanding how these discounts are calculated is crucial for both consumers and retailers to make informed decisions.