Understanding Successive Discounts

When multiple discounts are applied one after another on an original price, they are called successive discounts or series discounts. The key thing to remember is that each subsequent discount is applied to the price remaining after the previous discount, not on the original price.

Finding a single discount equivalent to a series of successive discounts is a common problem. It helps in understanding the overall reduction in price. Let's calculate the single equivalent discount for the given successive discounts of 7%, 12%, and 5%.

Calculating the Single Equivalent Discount

We can calculate the single equivalent discount by assuming an initial price. A simple approach is to assume the original price is $100.

Step 1: Apply the First Discount

The first discount is 7%. We apply this discount to the original price ($100).

Price after 7% discount = Original Price \($ \times \) (1 - Discount Rate)

Price after 7% discount = $100 \times (1 - 0.07) = 100 \times 0.93 = 93$

So, after the first discount, the price becomes $93.

Step 2: Apply the Second Discount

The second discount is 12%. This discount is applied to the price after the first discount, which is $93.

Price after 12% discount = Price after first discount \($ \times \) (1 - Second Discount Rate)

Price after 12% discount = $93 \times (1 - 0.12) = 93 \times 0.88$

Calculation:

So, after the second successive discount, the price becomes $81.84.

Step 3: Apply the Third Discount

The third discount is 5%. This discount is applied to the price after the second discount, which is $81.84.

Price after 5% discount = Price after second discount \($ \times \) (1 - Third Discount Rate)

Price after 5% discount = $81.84 \times (1 - 0.05) = 81.84 \times 0.95$

Calculation:

After all three successive discounts, the final price is $77.748.

Step 4: Calculate the Total Discount

The total discount is the difference between the original price and the final price after all successive discounts.

Total Discount = Original Price - Final Price

Total Discount = $100 - 77.748 = 22.252$

Step 5: Calculate the Equivalent Single Discount Percentage

The equivalent single discount percentage is the total discount expressed as a percentage of the original price.

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

Equivalent Single Discount % = \(\frac{22.252}{100} \times 100 = 22.252\%\)

Therefore, a single discount equivalent to successive discounts of 7%, 12%, and 5% is 22.252%.

Summary of Successive Discount Calculation

Let's summarize the process for finding the single equivalent discount:

1. Assume an initial price (e.g., 100).
2. Calculate the price after the first discount: Price₁ = Price_initial \($ \times \) (1 - d₁/100).
3. Calculate the price after the second discount: Price₂ = Price₁ \($ \times \) (1 - d₂/100).
4. Calculate the price after the third discount: Price₃ = Price₂ \($ \times \) (1 - d₃/100).
5. The final price after all discounts is Price_final = Price_initial \($ \times \) (1 - d₁/100) \($ \times \) (1 - d₂/100) \($ \times \) (1 - d₃/100).
6. Total discount amount = Price_initial - Price_final.
7. Equivalent single discount percentage = \(\frac{\text{Price}_{initial} - \text{Price}_{final}}{\text{Price}_{initial}} \times 100\).
8. This can be simplified to: Equivalent Single Discount % = \([1 - (1 - d_1/100)(1 - d_2/100)(1 - d_3/100)] \times 100\).

Using the formula with our values (d₁=7, d₂=12, d₃=5):

Equivalent Single Discount % = \([1 - (1 - 0.07)(1 - 0.12)(1 - 0.05)] \times 100\)

Equivalent Single Discount % = \([1 - (0.93)(0.88)(0.95)] \times 100\)

Equivalent Single Discount % = \([1 - 0.77748] \times 100\)

Equivalent Single Discount % = \(0.22252 \times 100 = 22.252\%\)

Both methods confirm the equivalent single discount is 22.252%.

Revision Table: Successive Discounts

Additional Information: Discounts and Percentages

Understanding percentages is crucial for discount calculations. A percentage represents a part out of 100. A discount is a reduction in price, usually expressed as a percentage.

• Calculating a price after a discount of d% is equivalent to multiplying the original price by \((1 - d/100)\). For example, a 10% discount means you pay \(1 - 0.10 = 0.90\) times the original price.
• Successive discounts always result in an equivalent single discount that is less than the sum of the individual discount percentages. In this case, the sum is \(7\% + 12\% + 5\% = 24\%\), but the equivalent discount is 22.252%. This is because each subsequent discount is applied to a smaller base amount.
• Discounts and markups are inverse operations when calculating final prices, but the percentages cannot be simply added or subtracted directly in successive applications.