Given below is a question followed by three statements. Identify the statements that are necessary to answer the question. Question: What is the percentage profit gained by a vendor when he sells an article? Statements: (i) He gives a 10% discount on the marked price. (ii) He would have earned 25% profit if he did not give any discount. (iii) The cost price of the article is Rs. 8000. Choose the correct option from those given below :
(i) and (ii) only
Analyzing Necessary Statements for Percentage Profit Calculation
The question asks us to identify which of the given statements are essential to determine the percentage profit gained by a vendor when selling an article with a discount. We need to find the relationship between the cost price (CP) and the selling price (SP) under the specified conditions to calculate the percentage profit.
Understanding the Statements
Let's break down what each statement provides:
- Statement (i): "He gives a 10% discount on the marked price."
This statement tells us that the Selling Price (SP) is obtained by reducing the Marked Price (MP) by 10%. Mathematically, this can be written as:
\( \text{SP} = \text{MP} - 0.10 \times \text{MP} \)
\( \text{SP} = \text{MP} (1 - 0.10) \)
\( \text{SP} = 0.90 \times \text{MP} \)
- Statement (ii): "He would have earned 25% profit if he did not give any discount."
This statement describes a hypothetical scenario where the selling price is equal to the marked price (no discount). In this scenario, the profit is 25% of the cost price (CP). This means the marked price (MP) is 25% more than the cost price (CP). Mathematically, this can be written as:
\( \text{MP} = \text{CP} + 0.25 \times \text{CP} \)
\( \text{MP} = \text{CP} (1 + 0.25) \)
\( \text{MP} = 1.25 \times \text{CP} \)
- Statement (iii): "The cost price of the article is Rs. 8000."
This statement provides the exact numerical value of the cost price (CP). \( \text{CP} = 8000 \).
Combining Statements to Find Percentage Profit
We want to find the percentage profit when a 10% discount is given. Percentage profit is calculated as \(\left( \frac{\text{SP} - \text{CP}}{\text{CP}} \right) \times 100\). To find this, we need to express SP in terms of CP.
From statement (i), we have \(\text{SP} = 0.90 \times \text{MP}\).
From statement (ii), we have \(\text{MP} = 1.25 \times \text{CP}\).
We can substitute the expression for MP from statement (ii) into the equation from statement (i):
\( \text{SP} = 0.90 \times (1.25 \times \text{CP}) \)
\( \text{SP} = (0.90 \times 1.25) \times \text{CP} \)
\( \text{SP} = 1.125 \times \text{CP} \)
Now we have the selling price (with the 10% discount) expressed in terms of the cost price. We can calculate the profit:
\( \text{Profit} = \text{SP} - \text{CP} \)
\( \text{Profit} = 1.125 \times \text{CP} - \text{CP} \)
\( \text{Profit} = (1.125 - 1) \times \text{CP} \)
\( \text{Profit} = 0.125 \times \text{CP} \)
Finally, we can calculate the percentage profit:
\( \text{Percentage Profit} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100 \)
\( \text{Percentage Profit} = \left( \frac{0.125 \times \text{CP}}{\text{CP}} \right) \times 100 \)
\( \text{Percentage Profit} = 0.125 \times 100 \)
\( \text{Percentage Profit} = 12.5\% \)
As shown by the calculation, combining statements (i) and (ii) is sufficient to find the percentage profit. The value of the cost price given in statement (iii) (Rs. 8000) is not needed to determine the percentage profit, as the CP term cancels out in the calculation.
Determining Necessary Statements
Statements (i) and (ii) together establish the relationship between SP (with discount) and CP, allowing us to calculate the percentage profit. Statement (iii) provides a specific numerical value for CP, but the percentage profit is a ratio that doesn't depend on the absolute value of CP if the percentages are relative to CP or MP, which are themselves relative to CP.
Therefore, only statements (i) and (ii) are necessary to answer the question about the percentage profit gained by the vendor.
| Statement | Information Provided | Necessary for Calculation? |
|---|---|---|
| (i) | SP in relation to MP (10% discount) | Yes |
| (ii) | MP in relation to CP (25% profit if no discount) | Yes |
| (iii) | Specific value of CP (Rs. 8000) | No |
Conclusion
Statements (i) and (ii) are necessary and sufficient to calculate the percentage profit. Statement (iii) is not required.
Revision Table: Profit and Discount Concepts
| Term | Definition | Relation |
|---|---|---|
| Cost Price (CP) | The price at which the vendor buys the article. | Base for calculating profit/loss. |
| Marked Price (MP) | The price marked on the article. | Base for calculating discount. |
| Selling Price (SP) | The price at which the vendor sells the article. | Result after discount (if any) on MP. Result after adding profit (or subtracting loss) from CP. |
| Discount | Reduction offered on the Marked Price. | Discount % is on MP. \( \text{Discount Amount} = \text{MP} \times \text{Discount Rate} \) |
| Profit | \( \text{SP} - \text{CP} \) (when SP > CP) | Profit % is typically on CP. \( \text{Profit Amount} = \text{CP} \times \text{Profit Rate} \) |
Additional Information: Why CP Value Isn't Always Needed for Percentage Profit
When a question asks for a percentage profit (or loss), and the relationships between CP, MP, and SP are given in terms of percentages or ratios, the final percentage profit often does not depend on the absolute numerical value of the cost price or selling price. This is because the percentage profit is itself a ratio of the profit amount to the cost price, multiplied by 100. If both the profit and the cost price are proportional to some base value (like the CP in this case, where SP = 1.125 * CP), that base value cancels out in the ratio.
In this problem, statements (i) and (ii) together allowed us to express SP as a multiple of CP (SP = 1.125 * CP). The profit is then 0.125 * CP. When calculating the percentage profit, \(\left( \frac{0.125 \times \text{CP}}{\text{CP}} \right) \times 100\), the 'CP' term cancels out, leaving a fixed percentage (12.5%) regardless of whether CP is 8000 or any other positive value.
Therefore, knowing the specific value of the cost price (statement iii) is redundant when the question can be answered using only percentage or ratio relationships provided in other statements.
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'Neigh' is related to 'Horse' in the same way as 'Bleat' is related to ____
Three of the following four letter-clusters are alike in a certain way and one is different. Select the odd one.
Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R) :
Assertion (A): Substance is always the subject of all predicates.
Reason (R): Substance is self-existent and self-conceived.
In the light of the above statements. choose the correct answer from the options given below:
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Statement I: Human reason has this peculiar fate that it reaches at a stage when it can neither answer certain questions nor can it ignore such questions.
Statement II: Human reason, by its own nature, is not competent to resolve certain questions concerning the ultimate reality.
In light of the above statements, choose the most appropriate answer from the options given below
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Assertion (A): In Samanyatodrsta inference the universal relation between the major and middle term depends on causal uniformity.
Reason (R): The middle term of the inference is related to the major term neither as a cause nor as an effect.
In the light of the above statements, choose the correct answer from the options given below:
