What is the total percentage discount applied when 2 successive discounts of 20% and 10% are applicable.
Understanding Successive Discounts
Successive discounts are multiple discounts applied one after another on a price. When a discount is applied, the price is reduced. Any subsequent discount is then applied to this new, reduced price, not the original price. It's important to calculate the impact of each discount sequentially.
Calculating Total Percentage Discount Step-by-Step
Let's consider the problem of finding the total percentage discount when two successive discounts of 20% and 10% are applied. We can solve this in a couple of ways:
Method 1: Assume an Initial Price
Let's assume the original price of an item is $100 for simplicity in calculation.
- Step 1: Apply the first discount. The first discount is 20%. Discount amount = 20% of $100 = \( \frac{20}{100} \times 100 = $20 \).
- Price after the first discount = Original Price - First Discount = $100 - $20 = $80.
- Step 2: Apply the second discount. The second discount is 10%. This discount is applied to the price after the first discount, which is $80. Discount amount = 10% of $80 = \( \frac{10}{100} \times 80 = \frac{800}{100} = $8 \).
- Price after the second discount = Price after first discount - Second Discount = $80 - $8 = $72.
- Step 3: Calculate the total discount amount. Total discount amount = Original Price - Price after second discount = $100 - $72 = $28.
- Step 4: Calculate the total percentage discount. Total percentage discount = \( \frac{\text{Total Discount Amount}}{\text{Original Price}} \times 100\% \).
- Total percentage discount = \( \frac{28}{100} \times 100\% = 28\% \).
Method 2: Using the Formula for Successive Discounts
For two successive discounts \(d_1\%\) and \(d_2\%\), the single equivalent discount percentage (total discount) is given by the formula:
\( \text{Total Discount} = \left( d_1 + d_2 - \frac{d_1 \times d_2}{100} \right)\% \)
In this question, the first discount \(d_1 = 20\%\) and the second discount \(d_2 = 10\%\).
- Substitute the values into the formula:
- Total Discount = \( \left( 20 + 10 - \frac{20 \times 10}{100} \right)\% \)
- Total Discount = \( \left( 30 - \frac{200}{100} \right)\% \)
- Total Discount = \( ( 30 - 2 )\% \)
- Total Discount = \( 28\% \)
Both methods show that the total percentage discount applied is 28%.
Summary of Successive Discount Calculation
Applying successive discounts means each subsequent discount reduces the price from the previously discounted amount. It's not simply adding the percentages together.
| Discount Step | Discount Percentage | Applied On | Calculation (assuming $100) | Resulting Price |
|---|---|---|---|---|
| Initial | - | Original Price | $100 | $100 |
| First Discount | 20% | Original Price | $100 - (20% of $100) = $100 - $20 | $80 |
| Second Discount | 10% | Price after 1st Discount | $80 - (10% of $80) = $80 - $8 | $72 |
The final price is $72 when the original price was $100. The total reduction is $100 - $72 = $28, which is 28% of the original price.
Revision Table: Successive Discounts Concepts
| Concept | Description | Key Takeaway |
|---|---|---|
| Single Discount | One percentage reduction applied to the original price. | Simple calculation. |
| Successive Discounts | Multiple percentage reductions applied one after another to the *current* price. | Order matters (though for two discounts, the final result is the same regardless of order), total discount is less than sum of individual discounts. |
| Equivalent Single Discount | A single discount percentage that gives the same final price as the series of successive discounts. | Useful for comparing different discount schemes. Calculated using formula or step-by-step method. |
Additional Information: Related Percentage Calculations
Understanding successive discounts helps with other percentage-based problems like successive increases or net percentage change.
- Successive Increases: If there are successive increases of \(r_1\%\) and \(r_2\%\), the net percentage increase is \( \left( r_1 + r_2 + \frac{r_1 \times r_2}{100} \right)\% \). Notice the '+' sign in the last term compared to the successive discount formula.
- Net Percentage Change: If there is a percentage increase of \(r_1\%\) followed by a percentage decrease of \(d_2\%\), the net percentage change is \( \left( r_1 - d_2 - \frac{r_1 \times d_2}{100} \right)\% \).
- These formulas are derived from calculating the final value after applying the changes sequentially, similar to the step-by-step method shown for successive discounts.
Important Questions from Discount and MP
If a trader marks the price of an article 50 percent more than the cost price and allows a discount of 30 percent, then what is his profit percentage?
What will be the selling price of an article if two successive discounts of 15% and 12% are offered on its marked price of Rs. 25,500?
Printed price of a mobile is Rs. 6,400. It is sold in Rs. 2,560 with two consecutive discount. If first discount is 20%, then what is the second discount?
Lekhana saves Rs. 15 on the purchase of a book when a discount of 20% is given. How much did she pay for the book?
Marked price of an article is Rs.4000. Selling price of the article is Rs.2600. What is the discount percentage?
