Calculating Single Equivalent Discount for Successive Discounts

When multiple discounts are applied one after another on an item's price, they are called successive discounts. These successive discounts are not simply added together to find the total discount. Instead, each discount is applied to the price remaining after the previous discount.

To find a single equivalent discount that represents the effect of successive discounts, we can use a specific formula or calculate the final price after applying each discount step-by-step.

Method 1: Using the Equivalent Discount Formula

For two successive discounts, $d_1\%$ 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, the successive discounts are 33% and 30%.

• First discount ($d_1$) = 33%
• Second discount ($d_2$) = 30%

Let's substitute these values into the formula:

\( D = 33 + 30 - \frac{33 \times 30}{100} \)

First, calculate the sum of the individual discounts:

\( 33 + 30 = 63 \)

Next, calculate the product of the discounts divided by 100:

\( \frac{33 \times 30}{100} = \frac{990}{100} = 9.9 \)

Now, substitute these results back into the formula:

\( D = 63 - 9.9 \)

Performing the subtraction:

\( D = 53.1 \)

So, the single equivalent discount is 53.1%.

Method 2: Step-by-Step Calculation (Assuming an Initial Price)

We can also assume an initial price for the item, for example, ₹100, and apply the discounts successively.

• Assume Initial Price = ₹100

Apply the first discount of 33%:

Discount Amount 1 = 33% of ₹100 = \( \frac{33}{100} \times 100 = \) ₹33

Price after first discount = Initial Price - Discount Amount 1 = ₹100 - ₹33 = ₹67

Now, apply the second discount of 30% on the price after the first discount (₹67):

Discount Amount 2 = 30% of ₹67 = \( \frac{30}{100} \times 67 = 0.30 \times 67 = \) ₹20.10

Price after second discount = Price after first discount - Discount Amount 2 = ₹67 - ₹20.10 = ₹46.90

The total discount received is the difference between the initial price and the final price:

Total Discount = Initial Price - Price after second discount = ₹100 - ₹46.90 = ₹53.10

To find the single equivalent discount percentage, we calculate the total discount as a percentage of the initial price:

Single Equivalent Discount % = \( \frac{\text{Total Discount}}{\text{Initial Price}} \times 100 \)

Single Equivalent Discount % = \( \frac{53.10}{100} \times 100 = 53.1 \)

This method also shows that the single equivalent discount is 53.1%.

Both methods confirm that successive discounts of 33% and 30% are equivalent to a single discount of 53.1%.

Revision Table: Successive Discounts

Additional Information: Calculating with Successive Discounts

Understanding successive discounts is important for various calculations involving sales, taxes, and commissions. Here are a few key points:

• The order of successive discounts does not affect the final price. Applying 33% then 30% gives the same result as applying 30% then 33%.
• The single equivalent discount is always less than the sum of the individual discounts ($d_1 + d_2$), because the second discount is applied to a smaller amount.
• The formula \(D = d_1 + d_2 - \frac{d_1 d_2}{100}\) can be extended for more than two successive discounts, but it becomes more complex. Alternatively, it's often easier to calculate the final price using the step-by-step method or by calculating the percentage of the price remaining after each discount. For example, a 33% discount means 67% of the price remains, and a 30% discount means 70% of the price remains. The final price is \( \text{Initial Price} \times (1 - \frac{d_1}{100}) \times (1 - \frac{d_2}{100}) \). For ₹100, this would be \( 100 \times (1 - 0.33) \times (1 - 0.30) = 100 \times 0.67 \times 0.70 = 100 \times 0.469 = 46.9 \). The total discount is \( 100 - 46.9 = 53.1 \), which is 53.1% of 100.