Calculate Equivalent Discount for Successive Discounts

This question requires finding a single discount percentage that is equal to applying two consecutive discounts, 5% and 20%. We can solve this using a hypothetical marked price or a direct formula.

Method 1: Using Hypothetical Marked Price

1. Assume the Marked Price (MP) is $100.
2. Calculate the price after the first discount of 5%.

   Price after 1st discount = MP - (5% of MP)

   Price = $100 - ( \frac{5}{100} \times 100 ) = 100 - 5 = 95$

3. Calculate the price after the second discount of 20% on the intermediate price ($95$).

   Price after 2nd discount = $95 - (20\% \text{ of } 95)$

   Price = $95 - ( \frac{20}{100} \times 95 ) = 95 - (0.20 \times 95) = 95 - 19 = 76$

4. Calculate the total discount amount.

   Total Discount = Marked Price - Final Price

   Total Discount = $100 - 76 = 24$

5. Calculate the equivalent single discount percentage.

   Equivalent Discount = $ \frac{\text{Total Discount}}{\text{Marked Price}} \times 100\% $

   Equivalent Discount = $ \frac{24}{100} \times 100\% = 24\% $

Method 2: Using the Discount Formula

For two successive discounts, $d_1$ and $d_2$, the equivalent single discount ($D$) can be calculated using the formula:

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

Given $d_1 = 5\%$ and $d_2 = 20\%$.

1. Substitute the values into the formula:

   $ D = 5 + 20 - \frac{5 \times 20}{100} $

2. Calculate the result:

   $ D = 25 - \frac{100}{100} $

   $ D = 25 - 1 $

   $ D = 24\% $

Both methods show that the single discount equivalent to successive discounts of 5% and 20% is 24%.