Expenses vary with square of income: In January, Arun’s income and expenses were ₹ 15000 and ₹ 9000, respectively. His monthly expenses vary directly as the square of his monthly income. Find his income at which expenses exactly equal income.

Introduction / Context:
This models a nonlinear cost pattern: expenses ∝ (income)^2. With one observed data point, we can find the proportionality constant and then compute the break-even income where expense equals income.

Given Data / Assumptions:

• Income I₁ = 15000; Expense E₁ = 9000.
• E = k * I^2 for some constant k > 0.
• Find I such that E = I.

Concept / Approach:
Use the January data to solve for k. Then set I = k * I^2 and solve the quadratic I^2 − (1/k) I = 0 for the positive, nonzero root.

Step-by-Step Solution:
k = E₁ / I₁^2 = 9000 / (15000^2) = 9000 / 225000000 = 1 / 25000.At break-even: I = k * I^2 ⇒ I = (1/25000) * I^2.I^2 − 25000 I = 0 ⇒ I(I − 25000) = 0.Nonzero solution: I = ₹ 25000.

Verification / Alternative check:
At I = 25000, E = (1/25000) * 25000^2 = 25000, matching the break-even requirement exactly.

Why Other Options Are Wrong:

• ₹ 15000 is the observed income, not the break-even point.
• ₹ 2000 and ₹ 35000 do not satisfy E = I under the derived k.

Common Pitfalls:

• Treating expenses as proportional to income (linear) instead of income squared (quadratic).
• Arithmetic errors when computing k or solving the quadratic.

Final Answer:
₹ 25000