Understanding Successive Discounts

When an item is subjected to two or more discounts one after another, these are called successive discounts or series discounts. The key point is that the second discount is applied not to the original price, but to the price after the first discount has been applied.

We are asked to find the single equivalent discount that represents two successive discounts of 15% and 25%.

Calculating the Single Equivalent Discount

Let's find the price after each discount to determine the total reduction. A simple way to do this is to assume an initial price for the item, for example, $100.00.

Step 1: Apply the First Discount (15%)

The first discount is 15%. We apply this to the initial price.

Price after 15% discount = Original Price × (1 - Discount Percentage)

Discount Percentage as a decimal = $$ \frac{15}{100} = 0.15 $$

Price after 15% discount = $$ $100.00 \times (1 - 0.15) $$

Price after 15% discount = $$ $100.00 \times 0.85 $$

Price after 15% discount = $$ $85.00 $$

So, after the first discount of 15%, the price becomes $85.00.

Step 2: Apply the Second Discount (25%)

The second discount of 25% is applied to the price after the first discount ($85.00), not the original price.

Price after 25% discount = Price after 1st Discount × (1 - Second Discount Percentage)

Second Discount Percentage as a decimal = $$ \frac{25}{100} = 0.25 $$

Price after 25% discount = $$ $85.00 \times (1 - 0.25) $$

Price after 25% discount = $$ $85.00 \times 0.75 $$

To calculate $85.00 × 0.75:

Price after 25% discount = $$ $63.75 $$

After both successive discounts, the final price is $63.75.

Step 3: Calculate the Total Discount and Equivalent Discount Percentage

The total discount is the difference between the original price and the final price.

Total Discount Amount = Original Price - Final Price

Total Discount Amount = $$ $100.00 - $63.75 $$

Total Discount Amount = $$ $36.25 $$

To find the single equivalent discount percentage, we compare the total discount amount to the original price.

Equivalent Discount Percentage = $$ \frac{\text{Total Discount Amount}}{\text{Original Price}} \times 100\% $$

Equivalent Discount Percentage = $$ \frac{$36.25}{$100.00} \times 100\% $$

Equivalent Discount Percentage = $$ 0.3625 \times 100\% $$

Equivalent Discount Percentage = $$ 36.25\% $$

Thus, the single discount equivalent to two successive discounts of 15% and 25% is 36.25%.

Alternative Method: Using the Formula

For two successive discounts, $$ D_1 $$ and $$ D_2 $$, the single equivalent discount ($$ D_{eq} $$) can be calculated using the formula:

$$ D_{eq} = D_1 + D_2 - \frac{D_1 \times D_2}{100} $$

Here, $$ D_1 = 15\% $$ and $$ D_2 = 25\% $$.

$$ D_{eq} = 15 + 25 - \frac{15 \times 25}{100} $$

$$ D_{eq} = 40 - \frac{375}{100} $$

$$ D_{eq} = 40 - 3.75 $$

$$ D_{eq} = 36.25\% $$

Both methods yield the same result, confirming the equivalent discount is 36.25%.

Revision Table: Key Steps for Successive Discounts

Additional Information: Successive Changes

The concept of successive discounts is a specific case of successive percentage changes. If a value is changed by $$ x\% $$ and then by $$ y\% $$, the net percentage change is not simply $$ x + y $$.

• If both are increases, the net increase is $$ x + y + \frac{x \times y}{100} $$.
• If both are decreases (like successive discounts), the net decrease (equivalent discount) is $$ x + y - \frac{x \times y}{100} $$.
• If one is an increase ($$ +x\% $$) and the other a decrease ($$ -y\% $$), the net change is $$ x - y - \frac{x \times y}{100} $$. A positive result indicates a net increase, a negative result a net decrease.

Understanding the base on which the percentage is applied is crucial. For successive discounts, the base for the second discount is the reduced price after the first discount.