Calculating Equivalent Discount for Successive Reductions

The question asks for the single equivalent discount when a 10% discount is followed by a 20% discount. It's important to understand that successive discounts are applied one after another, not added together.

Understanding Successive Discounts

When you get a 10% discount, the price is reduced. The second discount (20%) is then calculated on this already reduced price, not the original price. Simply adding 10% and 20% to get 30% would be incorrect.

Step-by-Step Calculation

Let's assume the original price of an item is $P$.

1. First Discount (10%):

   The amount of the first discount is $10\%$ of $P$, which can be written as $0.10 \times P$. The price after this discount is:

   New Price = Original Price - Discount Amount

   New Price = $P - (0.10 \times P) = P \times (1 - 0.10) = P \times 0.90$.

2. Second Discount (20%):

   The 20% discount is applied to the new price ($P \times 0.90$). The amount of the second discount is $20\%$ of $(P \times 0.90)$, which is $0.20 \times (P \times 0.90)$.

   Final Price = Price after first discount - Second Discount Amount

   Final Price = $(P \times 0.90) - (0.20 \times P \times 0.90)$

   Factor out $(P \times 0.90)$: Final Price = $(P \times 0.90) \times (1 - 0.20)$

   Final Price = $(P \times 0.90) \times 0.80 = P \times (0.90 \times 0.80)$

   Final Price = $P \times 0.72$.

3. Calculating Total Equivalent Discount:

   The final price is $0.72 \times P$, which means the final price is 72% of the original price.

   The total discount is the difference between the original price and the final price:

   Total Discount Amount = Original Price - Final Price

   Total Discount Amount = $P - (P \times 0.72) = P \times (1 - 0.72) = P \times 0.28$.

   To express this as a percentage of the original price:

   Total Discount Percentage = $\frac{\text{Total Discount Amount}}{\text{Original Price}} \times 100\%

   Total Discount Percentage = $\frac{P \times 0.28}{P} \times 100\% = 0.28 \times 100\% = 28\%$.

Example Calculation

Let's use an example with an original price of $100.

Step 1: Apply 10% discount

• Discount amount = $10\%$ of $100 = $10$.
• Price after first discount = $100 - $10 = $90$.

Step 2: Apply 20% discount to the new price

• Discount amount = $20\%$ of $90 = 0.20 \times 90 = $18$.
• Final Price = $90 - $18 = $72$.

Step 3: Calculate total equivalent discount

• Total discount = Original Price - Final Price = $100 - $72 = $28$.
• Total discount percentage = $\frac{$28}{$100} \times 100\% = 28\%$.

Conclusion

Applying a 10% discount and then a 20% discount in succession is equivalent to a single total discount of 28%.