If the cost of x metres of wire is d rupees, then, at the same uniform rate per metre, what will be the cost (in rupees) of y metres of the same wire?

Introduction / Context:
This question involves direct proportion and algebraic manipulation. It asks you to express the cost of y metres of wire in terms of x, y and d, given that x metres cost d rupees. Such algebraic unit-rate questions are common in aptitude and quantitative exams.

Given Data / Assumptions:
- x metres of wire cost d rupees.
- The cost per metre is constant (uniform rate).
- We need the cost of y metres of the same wire in rupees, expressed in terms of x, y, and d.

Concept / Approach:
The fundamental idea is cost = (length) * (cost per metre). First, we find the cost per metre from the given information. Since x metres cost d rupees, cost per metre is d / x. Then we multiply this rate by y metres to get the cost of y metres.

Step-by-Step Solution:
Step 1: Given: x metres cost d rupees.Step 2: Cost per metre = total cost / total length = d / x rupees per metre.Step 3: We want the cost of y metres at the same rate.Step 4: Cost of y metres = (cost per metre) * (number of metres) = (d / x) * y.Step 5: Rearrange the expression: (d / x) * y = (y d) / x.Step 6: Therefore, cost of y metres = Rs. (yd / x).

Verification / Alternative check:
As a quick check, substitute concrete numbers. Suppose x = 2 metres and d = 10 rupees. Then each metre costs 5 rupees. For y = 3 metres, cost should be 3 * 5 = 15 rupees. Using the formula Rs. (yd / x): (3 * 10) / 2 = 30 / 2 = 15 rupees, which matches perfectly.

Why Other Options Are Wrong:
Rs. (xy / d) inverts the relationship and would give nonsensical units. Rs. (xd) or Rs. (yd) ignore the need to divide by x and would not reduce to the correct numerical values when checked. Rs. (d / xy) is dimensionally wrong because it decreases as y increases, which contradicts the idea that more length costs more. Only Rs. (yd / x) correctly models cost ∝ length and fits the given rate.

Common Pitfalls:
Students often mix up which variables go in the numerator or denominator when setting up the rate. Another frequent mistake is to assume cost is proportional to the product xy without checking units. Always start from cost per unit (here, per metre) and then multiply by the desired quantity. This systematic approach prevents such algebraic errors.

Final Answer:
The cost of y metres of wire is Rs. (yd / x).