Question

A girl buys 2 pigeons for Rs. \(182 .\) She sells one at a loss of \(5 \%\) and another at a profit if \(8 \% .\) But she neither gains nor loses on the whole. Find the price of pigeon which has sold at a profit: (a) Rs. 112 (b) Rs. 85 (c) Rs. 70 (d) can't be determined

Short answer

Answer: The cost price of the pigeon sold at a profit was Rs. 70.

Step-by-step solution

Determine and represent the cost prices of the pigeons

Let the cost price of the pigeon sold at a loss be \(x\) and the cost price of the pigeon sold at a profit be \(y\). The total cost price of both pigeons combined is given as Rs. 182. This can be written as an equation: \(x + y = 182\).

Write the equations for selling prices of both pigeons

The pigeon sold at a loss of \(5 \%\) would have a selling price of \(0.95x\) (since it is sold at \(100 \% - 5 \% = 95 \%\) of the cost price). Similarly, the pigeon sold at a profit of \(8 \%\) would have a selling price of \(1.08y\) (because it is sold at \(100 \% + 8 \% = 108 \%\) of the cost price).

Use the condition that she neither gains nor loses on the whole

Since there is no overall gain or loss, the total selling price of both pigeons should equal the total cost price, that is, \(0.95x + 1.08y = 182\).

Solve the equations for x and y

We have two equations: 1. \(x + y = 182\) 2. \(0.95x + 1.08y = 182\) Let's multiply the first equation by \(0.95\), so that we can eliminate term x by subtracting the equations: 1. \(0.95x + 0.95y = 172.9\) Now, subtract equation 1 from equation 2: 2 - 1: \((1.08y - 0.95y) = (182 - 172.9)\) This simplifies to: \(0.13y = 9.1\) Now, divide by \(0.13\) to find the value of \(y\): \(y = \frac{9.1}{0.13} = 70\)

Find the cost price of the pigeon sold at a profit

We found that \(y = 70\), which means that the cost price of the pigeon sold at a profit was Rs. 70. Therefore, the correct answer is (c) Rs. 70.

Key concepts

Quantitative Aptitude

Quantitative aptitude forms the cornerstone of many competitive exams and it includes problems on profit and loss, which are an integral part of business mathematics. These problems test one's ability to calculate and understand how financial transactions affect profit margins. For instance, in our exercise, a girl buys two pigeons for a total and sells them at different rates of profit and loss. The ability to work out the individual prices from a combined total comes under quantitative aptitude.

In order to enhance quantitative aptitude, it is advisable to practice a variety of problems involving different scenarios of buying and selling, as in our example. Additionally, one could apply the real-world implications of profit and loss, such as in the stock market or in small business management, to better understand how these calculations play out in various contexts.

Problem Solving

Problem solving is a logical process that involves a step-by-step approach to finding the solution to a particular issue. In our textbook problem, the steps included are identifying the total cost, deciphering the individual cost prices, and applying the given information to set up equations that reflect the situation. Problem-solving skills are enhanced when students learn to break down complex issues into smaller, more manageable parts.

For this problem, it is recommended to analyze the given data, which are the total cost and the percentages of profit and loss, and to formulate the equations accordingly. Once the equations are set, problem solvers must use appropriate mathematical methods, like equation solving, to find the unknown values. This approach can be applied to a variety of contexts beyond this specific problem, making it a valuable skill for students.

Percentage Calculations

Percentage calculations are an essential aspect of many mathematical problems, including profit and loss questions. Being proficient in calculating percentages allows for a better understanding of how values compare to each other. In the exercise, understanding how to translate a percentage into its decimal form (5% as 0.05 and 8% as 0.08) is crucial for setting up the problem equations correctly.

When providing exercise improvement advice, it is effective to demonstrate how a percentage can be used to calculate both increases and decreases in a given value. For example, selling at a 5% loss means multiplying the original cost by \(0.95\) and making a profit of 8% translates to multiplying the original cost by \(1.08\). Practicing these conversions helps students to quickly and efficiently solve a range of percentage-based problems.