Understanding successive discounts is important in percentage and profit & loss problems. Successive discounts mean that a second discount is applied not on the original price, but on the price after the first discount has been applied. We need to find a single discount percentage that results in the same final price as applying two successive discounts.

Let's calculate the single equivalent discount for successive discounts of 10% and 20% using two common methods:

Method 1: Assuming an Original Price

Let's assume the original price of an item is 100 units. This makes calculating percentages easy.

1. Calculate the price after the first discount (10%).
   • Discount amount = 10% of 100 = \(\frac{10}{100} \times 100 = 10\)
   • Price after 1st discount = Original Price - Discount Amount = \(100 - 10 = 90\)
2. Calculate the price after the second discount (20%) on the *new* price (90).
   • Discount amount = 20% of 90 = \(\frac{20}{100} \times 90 = \frac{1}{5} \times 90 = 18\)
   • Price after 2nd discount = Price after 1st discount - Discount Amount = \(90 - 18 = 72\)
3. Calculate the total discount on the original price.
   • Total discount amount = Original Price - Final Price = \(100 - 72 = 28\)
4. Convert the total discount amount into a single equivalent discount percentage.
   • Single Equivalent Discount % = \(\frac{\text{Total Discount}}{\text{Original Price}} \times 100\)
   • Single Equivalent Discount % = \(\frac{28}{100} \times 100 = 28\%\)

So, successive discounts of 10% and 20% are equivalent to a single discount of 28%.

Method 2: Using a Formula for Successive Discounts

For two successive discounts, say \(D_1\%\) and \(D_2\%\), the single equivalent discount (\(D\%\)) can be calculated using the formula:

\(D = D_1 + D_2 - \frac{D_1 \times D_2}{100}\)

In this problem, \(D_1 = 10\%\) and \(D_2 = 20\%\).

Substitute these values into the formula:

\(D = 10 + 20 - \frac{10 \times 20}{100}\)

\(D = 30 - \frac{200}{100}\)

\(D = 30 - 2\)

\(D = 28\)

Therefore, the single equivalent discount is 28%.

Both methods show that the single discount equivalent to successive discounts of 10% and 20% is 28%.

Successive Discounts Revision Table

Additional Information on Equivalent Single Discount

• The order of successive discounts does not matter. Applying 10% then 20% gives the same result as applying 20% then 10%. Using the formula, \(D_1 + D_2 - \frac{D_1D_2}{100} = D_2 + D_1 - \frac{D_2D_1}{100}\).
• The equivalent single discount is always less than the sum of the individual successive discounts (\(D_1 + D_2\)). In our case, \(28\% < 10\% + 20\% = 30\%\). This is because the second discount is applied to a smaller amount.
• This concept is often used in retail for promotional offers (e.g., "Extra 20% off sale items").
• The formula can be extended for three or more successive discounts, although it becomes more complex. Alternatively, the step-by-step method of calculating the price after each discount can always be used.