Understanding Successive Discounts

Successive discounts, also known as compound discounts, are applied one after another on the reduced price. This means the second discount is calculated on the price after the first discount has already been applied, not on the original price.

When you see successive discounts like 10% and 10%, it's important to know that the combined effect is not simply the sum of the discounts (10% + 10% = 20%). The second discount reduces the price further based on the new, lower price.

Calculating Equivalent Single Discount

Let's find the single equivalent discount for successive discounts of 10% and 10% using a simple example. Assume the original marked price of an item is ₹100.

1. First Discount: A 10% discount is applied to the original price of ₹100.
   Discount amount = \(10\% \text{ of } ₹100 = \frac{10}{100} \times 100 = ₹10\).
   Price after first discount = \(₹100 - ₹10 = ₹90\).
2. Second Discount: A 10% discount is applied to the reduced price of ₹90.
   Discount amount = \(10\% \text{ of } ₹90 = \frac{10}{100} \times 90 = ₹9\).
   Final price after both discounts = \(₹90 - ₹9 = ₹81\).
3. Total Discount: The total reduction in price from the original ₹100 to the final ₹81 is the total discount.
   Total discount = \(₹100 - ₹81 = ₹19\).
4. Equivalent Single Discount Percentage: To find the single discount that would result in the same final price, we express the total discount as a percentage of the original price.
   Equivalent single discount percentage = \(\frac{\text{Total discount}}{\text{Original price}} \times 100\)
   Equivalent single discount percentage = \(\frac{₹19}{₹100} \times 100 = 19\%\).

Therefore, successive discounts of 10% and 10% are equivalent to a single discount of 19%.

Formula for Equivalent Single Discount

Alternatively, you can use a formula to quickly calculate the equivalent single discount for two successive discounts. If two successive discounts are \(d_1\%\) and \(d_2\%\), the equivalent single discount \(D\%\) is given by the formula:

\(D = d_1 + d_2 - \frac{d_1 \times d_2}{100}\)

Using this formula for \(d_1 = 10\%\) and \(d_2 = 10\%\):

\(D = 10 + 10 - \frac{10 \times 10}{100}\)

\(D = 20 - \frac{100}{100}\)

\(D = 20 - 1\)

\(D = 19\%\)

Both methods confirm that the equivalent single discount is 19%.

Comparing Options for Successive Discounts

Let's look at the given options:

Based on our calculations, the correct option is 19%.

Revision Table: Successive Discounts Key Concepts

Additional Information on Successive Discounts

Understanding successive discounts is important in real-world scenarios like shopping sales. Retailers often offer multiple discounts (e.g., "20% off, plus an additional 10% off"). Knowing how to calculate the single equivalent discount helps you determine the actual total percentage reduction you receive.

The formula \(D = d_1 + d_2 - \frac{d_1 \times d_2}{100}\) clearly shows why the equivalent discount is always less than the sum \(d_1 + d_2\), provided \(d_1\) and \(d_2\) are positive. The term \(\frac{d_1 \times d_2}{100}\) represents the reduction in the discount amount due to the second discount being applied to a smaller base.

This concept can be extended to three or more successive discounts by applying the formula iteratively. For example, to find the equivalent of three discounts \(d_1, d_2, d_3\), first find the equivalent of \(d_1\) and \(d_2\) (let's call it \(D_{12}\)), and then find the equivalent of \(D_{12}\) and \(d_3\).