The selling price of an article marked for Rs. 10000 after giving three discounts, 20%, 10% and k% is Rs. 6120. What will be selling price (in Rs.) of the same article if a single discount of (k + 20)% is allowed?
6500
Solving the Discount Problem: Step-by-Step
This problem involves calculating the selling price of an article after applying multiple successive discounts and then finding the selling price under a single combined discount. We are given the marked price, the final selling price after three discounts (two known percentages and one unknown 'k' percent), and we need to find 'k' first. Once 'k' is found, we will calculate the selling price with a single discount of (k + 20)%.
Understanding Successive Discounts
When successive discounts are applied, each subsequent discount is calculated on the price remaining after the previous discount. If the marked price is MP and discounts are \(d_1\%\), \(d_2\%\), and \(d_3\%\), the final selling price (SP) is given by:
\[ \text{SP} = \text{MP} \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right) \times \left(1 - \frac{d_3}{100}\right) \]
Calculating the Unknown Discount 'k'
Given:
- Marked Price (MP) = Rs. 10000
- First discount = 20%
- Second discount = 10%
- Third discount = k%
- Final Selling Price (SP1) = Rs. 6120
Using the formula for successive discounts, we can set up the equation:
\[ 6120 = 10000 \times \left(1 - \frac{20}{100}\right) \times \left(1 - \frac{10}{100}\right) \times \left(1 - \frac{k}{100}\right) \]
Let's simplify the terms:
\[ 1 - \frac{20}{100} = 1 - 0.20 = 0.80 \]
\[ 1 - \frac{10}{100} = 1 - 0.10 = 0.90 \]
Substitute these values back into the equation:
\[ 6120 = 10000 \times (0.80) \times (0.90) \times \left(1 - \frac{k}{100}\right) \]
\[ 6120 = 10000 \times (0.72) \times \left(1 - \frac{k}{100}\right) \]
\[ 6120 = 7200 \times \left(1 - \frac{k}{100}\right) \]
Now, isolate the term containing 'k':
\[ 1 - \frac{k}{100} = \frac{6120}{7200} \]
Simplify the fraction \(\frac{6120}{7200}\):
\[ \frac{6120}{7200} = \frac{612}{720} = \frac{306}{360} = \frac{153}{180} = \frac{51}{60} = \frac{17}{20} \]
So, the equation becomes:
\[ 1 - \frac{k}{100} = \frac{17}{20} \]
Now, solve for \(\frac{k}{100}\):
\[ \frac{k}{100} = 1 - \frac{17}{20} = \frac{20 - 17}{20} = \frac{3}{20} \]
Finally, solve for 'k':
\[ k = \frac{3}{20} \times 100 = 3 \times 5 = 15 \]
So, the value of k is 15%.
Calculating Selling Price with a Single Discount of (k + 20)%
We found that k = 15%. The new single discount is (k + 20)%.
Single Discount = (15 + 20)% = 35%.
We need to find the selling price (SP2) of the same article with a single discount of 35% on the marked price of Rs. 10000.
The selling price with a single discount is given by:
\[ \text{SP} = \text{MP} \times \left(1 - \frac{\text{Discount \%}}{100}\right) \]
Substitute the values:
\[ \text{SP2} = 10000 \times \left(1 - \frac{35}{100}\right) \]
\[ \text{SP2} = 10000 \times (1 - 0.35) \]
\[ \text{SP2} = 10000 \times (0.65) \]
\[ \text{SP2} = 6500 \]
Therefore, the selling price of the same article if a single discount of (k + 20)% is allowed is Rs. 6500.
Summary of Calculations
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Price after 20% discount | \(10000 \times (1 - 0.20)\) | Rs. 8000 |
| 2 | Price after 10% discount (on Rs. 8000) | \(8000 \times (1 - 0.10)\) | Rs. 7200 |
| 3 | Equation for k% discount | \(7200 \times (1 - k/100) = 6120\) | \(1 - k/100 = 17/20\) |
| 4 | Solve for k | \(k/100 = 3/20 \implies k = 15\) | k = 15% |
| 5 | New single discount | (k + 20)% = (15 + 20)% | 35% |
| 6 | Selling price with 35% discount | \(10000 \times (1 - 0.35)\) | Rs. 6500 |
Final Selling Price Result
The selling price of the article with a single discount of (k + 20)% or 35% is Rs. 6500.
Revision Table: Key Concepts
| Concept | Explanation |
|---|---|
| Marked Price (MP) | The price printed on the article, before any discounts. |
| Selling Price (SP) | The price at which the article is sold after discounts or including profit. |
| Discount | A reduction in the marked price. Usually expressed as a percentage. |
| Successive Discounts | Multiple discounts applied one after another, with each discount calculated on the reduced price. |
| Single Equivalent Discount | A single discount percentage that results in the same final selling price as a series of successive discounts. |
Additional Information: Discount Formulas
Here are some useful formulas related to discounts:
- Discount Amount = Marked Price - Selling Price
- Discount Percentage = \(\left(\frac{\text{Discount Amount}}{\text{Marked Price}}\right) \times 100\)
- Selling Price = Marked Price \(\times \left(1 - \frac{\text{Discount \%}}{100}\right)\)
- For two successive discounts \(d_1\%\) and \(d_2\%\), the single equivalent discount is \(\left(d_1 + d_2 - \frac{d_1 d_2}{100}\right)\%\). This can be extended for more discounts.
In this problem, we used the formula for calculating the final selling price after successive discounts to find the value of k. Then, we used the basic discount formula to find the selling price with a single combined discount.
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