Understanding Successive Discounts

Successive discounts, also known as compound discounts or multiple discounts, are applied one after another on the remaining price after the previous discount has been applied. This is different from simply adding the discount percentages together.

To find the single equivalent discount for successive discounts, we can calculate the final price after applying each discount sequentially. Let's assume the original price of an item is \( P \).

Calculating the Final Price After Successive Discounts

We are given three successive discounts: 12%, 20%, and 25%.

Let's calculate the price remaining after each discount:

1. After the first discount of 12%:

   The price becomes \( P \times \left(1 - \frac{12}{100}\right) = P \times (1 - 0.12) = P \times 0.88 \).

2. After the second discount of 20% (applied to the new price \( P \times 0.88 \)):

   The price becomes \( (P \times 0.88) \times \left(1 - \frac{20}{100}\right) = (P \times 0.88) \times (1 - 0.20) = P \times 0.88 \times 0.80 \).

3. After the third discount of 25% (applied to the new price \( P \times 0.88 \times 0.80 \)):

   The final price becomes \( (P \times 0.88 \times 0.80) \times \left(1 - \frac{25}{100}\right) = P \times 0.88 \times 0.80 \times 0.75 \).

Now, let's calculate the product of the multipliers:

\( 0.88 \times 0.80 \times 0.75 \)

First, \( 0.88 \times 0.80 = 0.704 \).

Then, \( 0.704 \times 0.75 = 0.704 \times \frac{3}{4} \).

To multiply \( 0.704 \) by \( 0.75 \):

\( 0.704 \times 0.75 = 0.528 \).

So, the final price is \( P \times 0.528 \), which means the final price is 52.8% of the original price.

Determining the Single Equivalent Discount Percentage

The single equivalent discount is the total reduction in price expressed as a percentage of the original price. If the final price is 52.8% of the original price, the discount is the remaining percentage:

Single Equivalent Discount \( = \text{Original Price Percentage} - \text{Final Price Percentage} \)

Single Equivalent Discount \( = 100\% - 52.8\% \)

Single Equivalent Discount \( = 47.2\% \).

Therefore, the successive discounts of 12%, 20%, and 25% are equivalent to a single discount of 47.2%.

The cumulative multiplier of 0.528 means the final price is 52.8% of the original price. The single equivalent discount is \( (1 - 0.528) \times 100\% = 0.472 \times 100\% = 47.2\% \).

Revision Table: Successive Discounts

Let's quickly recap the steps to find the single equivalent discount:

1. For each discount percentage, calculate the remaining percentage (100% - discount %).
2. Convert these percentages to decimals.
3. Multiply all the decimal values together. This gives the fraction of the original price that remains after all discounts.
4. Subtract this final fraction from 1 and multiply by 100% to get the single equivalent discount percentage.

Additional Information: Why Successive Discounts Aren't Simply Added

It's important to understand why simply adding the discounts (12% + 20% + 25% = 57%) does not give the correct single equivalent discount. Each successive discount is applied to a smaller base (the price after the previous discount), not the original price. This compounding effect results in a smaller total discount than the sum of individual percentages.

For example, a 20% discount followed by a 10% discount:

• Using addition: 20% + 10% = 30%.
• Using successive calculation: Price after 20% is 80% of original. Price after 10% on the new price is \( 0.80 \times (1 - 0.10) = 0.80 \times 0.90 = 0.72 \) of original. Total discount \( = (1 - 0.72) \times 100\% = 28\% \).

The actual single equivalent discount (28%) is less than the simple sum (30%), illustrating that discounts applied successively compound downwards.