If the cost of x metres of wire is d rupees at a constant rate, what is the cost of y metres of wire at the same rate?

Introduction / Context:
This is a symbolic unitary method question that tests your ability to work with algebraic expressions in place of numerical values. Instead of concrete numbers, the length and the cost are expressed using variables. You must derive the general formula for the cost of a different length of wire at the same rate per metre.

Given Data / Assumptions:

• Cost of x metres of wire = d rupees.
• The cost per metre is constant.
• We are asked to find the cost of y metres of wire.
• All variables x, y and d are positive real numbers.

Concept / Approach:
Cost is directly proportional to length when the rate per metre is fixed. First we compute the cost per metre by dividing the total cost by the total length. Then we multiply this rate by the new length y to find the new total cost. The process is identical to numerical unitary method, but carried out using algebraic symbols.

Step-by-Step Solution:
Step 1: Cost of x metres = d rupees.Step 2: Cost per metre = d / x rupees.Step 3: Required length is y metres.Step 4: Cost of y metres = (d / x) * y.Step 5: This simplifies to cost = (y * d) / x = Rs. (yd / x).

Verification / Alternative check:
You can test the expression with simple numbers. Suppose x = 2 metres and d = 10 rupees, then cost per metre is 5 rupees. For y = 3 metres, cost should be 15 rupees. Plugging into (yd / x) gives (3 * 10) / 2 = 30 / 2 = 15 rupees, which matches. This confirms that Rs. (yd / x) is correct.

Why Other Options Are Wrong:
Rs. (xy / d) inverts the relationship and has wrong dimensions. Rs. (xd) or Rs. (yd) ignore division by x and therefore do not preserve the unit rate. Rs. (d / xy) is also dimensionally incorrect and would decrease cost for larger y, which is not logical when the rate is fixed.

Common Pitfalls:

• Not computing cost per metre first and trying to guess a direct formula.
• Mixing up x and y in the numerator and denominator.
• Forgetting that cost must increase linearly with y when the rate is constant, so any expression where cost decreases with y is incorrect.

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