Mathematical Puzzle: Analyzing Ram and Shyam's Bookstore Purchase

In today's post, we will explore a thought-provoking puzzle involving a trip to the bookstore. Two friends, Ram and Shyam, visit a bookstore with a certain amount of money. After their purchases, we find that Shyam spends 50% more than Ram. This article will guide you through the step-by-step mathematical process to determine the percentage of the money Ram spends.

Setting Up the Problem

Let's introduce the variables for the prices of the items purchased:

Let x be the price of a pen. Let y be the price of a notebook. Let z be the price of a pencil.

Given the purchases:

Purchase made by Ram: 5 pens, 3 notebooks, 8 pencils. Purchase made by Shyam: 6 pens, 6 notebooks, 16 pencils.

And the fact that Shyam spends 50% more than Ram, let's denote Ram's total spending by R and Shyam's total spending by S. According to the problem, S 1.5R.

Formulating Equations

Mathematically, we can write:

Equation Formulation

Here, the left-hand side of the first equation represents the purchases made by Ram, and the right-hand side is Ram's total spending. The second equation represents Shyam's purchases, and the right-hand side is 1.5 times Ram's spending.

Algebraic Manipulation

Let's simplify and solve the equations:

Start by substituting the total purchases into the equations. Simplify the equations to isolate common variables.

From the equation 6x 6y 16z 1.5(5x 3y 8z), let's simplify:

Simplifying further:

6x 6y 16z 1.5(5x 3y 8z)

Step 1: Distribute 1.5 on the right-hand side:
6x 6y 16z 7.5x 4.5y 12z

Step 2: Rearrange the terms to isolate unknowns on one side:
6x 6y 16z - 7.5x - 4.5y - 12z 0

Step 3: Combine like terms:
-1.5x 1.5y 4z 0

Let's represent the combined term as u 3y 9z (not previously seen in our variable breakdown).

So the equation becomes:

6x 6u 1.5(5x u)

Solving for x:
6x 6u 7.5x 1.5u
6u - 1.5u 7.5x - 6x
4.5u 1.5x
x 3u

Since we have established that Shyam spends 50% more than Ram, let's calculate Shyam's spending on pens specifically:

Solution: Calculation of Shyam's Spending on Pens

From the solution, it's clear that Shyam spent 50% of his total money on pens. Therefore, we find that:

Pens cost: 50

Calculating the Percentage of Spent Money by Ram

To find the percentage of money Ram spent, let's reframe the question:

If the question is asking, 'What percentage of his money did Ram spend?', the answer is straightforward. By rechecking the calculations:

Given that S 1.5R, and substituting the simplified terms, we establish that Ram spent 50% of his total spending on his purchases.

Conclusion

The key takeaway from this puzzle is the importance of carefully stating the question. Whether the question asks about Shyam's spending or Ram's spending, the calculations and percentages can vary based on the phrasing. Ram spent a significant portion of his money, reflecting the proportional relationship between his and Shyam's purchases.

Wrap-Up

By breaking down the problem methodically and applying algebraic manipulation, we can solve complex puzzles involving purchases and spending. This exercise not only enhances our problem-solving skills but also highlights the importance of precise mathematical formulation.