Decoding the Price of Oranges: A Mathematical Puzzle Resolved

In today's world, the ability to think critically and solve mathematical puzzles can be both fun and practical. One such intriguing puzzle involves Rama purchasing fruits, which involves a clever exchange of items to determine their respective costs. Let's delve into this problem and understand the steps involved in finding the price of an orange.

Problem Statement

Rama bought 5 mangoes and 10 oranges for 40 rupees. Subsequently, he returned one mango and got two oranges in exchange. What would the price of an orange be?

Step-by-Step Solution

Let's break down the problem step by step. First, we need to denote the price of one mango and one orange algebraically.

Denote the price of one mango as m and the price of one orange as o. Based on the information given, we can set up the following equation:

5m 10o 40

We can simplify this equation by dividing everything by 5, giving us:

m 2o 8 emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp;emsp; emsp; emsp; (1)

Next, we consider the scenario where Rama returns one mango and gets two oranges in exchange. This means that the value of one mango in rupees is m, and the value of two oranges is 2o. The transaction can be represented as:

m 2o emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp;emsp; emsp; emsp; emsp; emsp;emsp; emsp;emsp;emsp; (2)

Substituting equation (2) into equation (1), we get:

2o 2o 8

This simplifies to:

4o 8

Dividing both sides by 4, we find:

o 2

The price of one orange is 2 rupees.

To find the price of one mango using o 2, we substitute back into equation (2):

m 2o 2 x 2 4

The price of one mango is 4 rupees.

Therefore, the final answer is:

Price of an orange: 2 rupees.

Statement of the Given Problem

Let PM and PO denote the price of a mango and an orange, respectively. From the given data, we get the following relation:

1PM 2PO emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; (3)

This implies:

PM 2PO / 1 emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; (4)

From equation (3) and the given relation 5PM 10PO 40, we can derive:

2PO 1/28 [given: 5PM 10PO 40 hence: PM 2PO 8]

Substituting 2PO 1/28 into equation (3), we get:

PO 2 Rs emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; emsp; (5)

The price of one orange is 2 rupees.

Conclusion

The puzzle involves a series of logical steps and algebraic manipulations to determine the price of an orange. Understanding these steps not only helps in solving the puzzle but also enhances one's problem-solving skills in real-world applications. For instance, learners and professionals can apply similar logical reasoning in analyzing data, optimizing algorithms, and making rational decisions in their respective fields.

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