Three successive discounts of 15%, 20% and 25% are given. What will be the net discount in percentage?
49%
Calculating Net Discount Percentage
Understanding successive discounts is important in percentage calculations. Successive discounts mean that the discount is applied one after another on the reduced price, not on the original price each time.
Let's calculate the net discount percentage for the given three successive discounts of 15%, 20%, and 25%.
Step-by-Step Calculation of Successive Discounts
To find the net discount, we can assume an original price and see what the final price becomes after applying all the discounts successively. A common approach is to assume the original price is 100 units.
Step 1: Apply the First Discount (15%)
The first discount is 15%. The price after the first discount will be 100 minus 15% of 100.
Price after 1st discount $= 100 - (15\% \text{ of } 100)$
Price after 1st discount $= 100 - (0.15 \times 100)$
Price after 1st discount $= 100 - 15 = 85$
Alternatively, the remaining price after a 15% discount is \(100\% - 15\% = 85\%\) of the original price.
Price after 1st discount $= 100 \times (1 - 0.15) = 100 \times 0.85 = 85$
Step 2: Apply the Second Discount (20%)
The second discount is 20%, but it is applied to the price after the first discount (which is 85). The price after the second discount will be 85 minus 20% of 85.
Price after 2nd discount $= 85 - (20\% \text{ of } 85)$
Price after 2nd discount $= 85 - (0.20 \times 85)$
Price after 2nd discount $= 85 - 17 = 68$
Alternatively, the remaining price after a 20% discount is \(100\% - 20\% = 80\%\) of the previous price.
Price after 2nd discount $= 85 \times (1 - 0.20) = 85 \times 0.80 = 68$
Step 3: Apply the Third Discount (25%)
The third discount is 25%, applied to the price after the second discount (which is 68). The price after the third discount will be 68 minus 25% of 68.
Price after 3rd discount $= 68 - (25\% \text{ of } 68)$
Price after 3rd discount $= 68 - (0.25 \times 68)$
Price after 3rd discount $= 68 - 17 = 51$
Alternatively, the remaining price after a 25% discount is \(100\% - 25\% = 75\%\) of the previous price.
Price after 3rd discount $= 68 \times (1 - 0.25) = 68 \times 0.75 = 51$
Step 4: Calculate the Net Discount
The original price was 100, and the final price after all three successive discounts is 51. The total discount is the difference between the original price and the final price.
Net Discount Amount $= \text{Original Price} - \text{Final Price}$
Net Discount Amount $= 100 - 51 = 49$
Step 5: Calculate the Net Discount Percentage
The net discount percentage is the net discount amount as a percentage of the original price.
Net Discount Percentage $= \frac{\text{Net Discount Amount}}{\text{Original Price}} \times 100\%$
Net Discount Percentage $= \frac{49}{100} \times 100\% = 49\%$
So, the net discount after three successive discounts of 15%, 20%, and 25% is 49%.
Summary of Successive Discount Calculation
| Discount | Calculation | Price Remaining |
|---|---|---|
| Original Price | 100 | |
| 15% | $100 \times (1 - 0.15) = 100 \times 0.85$ | 85 |
| 20% | $85 \times (1 - 0.20) = 85 \times 0.80$ | 68 |
| 25% | $68 \times (1 - 0.25) = 68 \times 0.75$ | 51 |
The final price is 51 when the original price is 100. This means a total discount of $100 - 51 = 49$ has been applied, which is 49% of the original price.
Formula for Successive Discounts
Alternatively, you can use a formula to find the single equivalent discount for successive discounts \(d_1\%, d_2\%, d_3\%, \dots, d_n\%\). The remaining price percentage after all discounts is:
\(\text{Remaining Price Percentage} = 100 \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right) \times \left(1 - \frac{d_3}{100}\right) \times \dots \times \left(1 - \frac{d_n}{100}\right)\)
The net discount percentage is then:
\(\text{Net Discount Percentage} = 100 - \text{Remaining Price Percentage}\)
Using this for 15%, 20%, and 25%:
Remaining Price Percentage $= 100 \times \left(1 - \frac{15}{100}\right) \times \left(1 - \frac{20}{100}\right) \times \left(1 - \frac{25}{100}\right)$
Remaining Price Percentage $= 100 \times (1 - 0.15) \times (1 - 0.20) \times (1 - 0.25)$
Remaining Price Percentage $= 100 \times 0.85 \times 0.80 \times 0.75$
Remaining Price Percentage $= 100 \times 0.51 = 51\% $
Net Discount Percentage $= 100\% - 51\% = 49\%$
This confirms the net discount is 49%.
Revision Table: Key Concepts in Discounts
| Concept | Description | Calculation Note |
|---|---|---|
| Discount | Reduction in the original price of a product or service. | Usually expressed as a percentage or amount. |
| Successive Discounts | Multiple discounts applied one after another on the reducing price. | Each subsequent discount is on the price remaining after the previous discount. |
| Net Discount | The single equivalent discount that results from applying a series of successive discounts. | Calculated as the total reduction from the original price. |
| Selling Price | The final price after applying all discounts to the marked price. | Original Price - Net Discount Amount. |
Additional Information: Percentage Calculations and Discounts
Understanding percentages is fundamental to solving problems involving discounts, profit, loss, interest, etc. A percentage is a fraction out of 100. For example, 15% means 15 out of 100, or \(\frac{15}{100} = 0.15\).
- When a discount of \(d\%\) is given on a price \(P\), the discount amount is \(P \times \frac{d}{100}\).
- The price after the discount is \(P - P \times \frac{d}{100} = P \times \left(1 - \frac{d}{100}\right)\).
- For successive discounts, this multiplicative factor \(\left(1 - \frac{d}{100}\right)\) is applied sequentially for each discount.
It is crucial not to simply add the percentages of successive discounts (15% + 20% + 25% = 60% in this case), as this would incorrectly imply the discounts are all applied to the original price simultaneously. Successive application on the reduced price leads to a lower net discount than the sum of individual percentages.
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