A man's expenditure for 8 months is equal to his income for 6 months. Over one full year he saves a total of ₹6000. What is his average monthly income in rupees?

Introduction / Context:
This question involves a typical income and expenditure scenario based on simple arithmetic and ratio reasoning. The man spends in eight months what he earns in six months, and we are told his annual savings. By relating income and expenditure per month, we can determine his monthly income.

Given Data / Assumptions:

• Let monthly income be I rupees.
• Let monthly expenditure be E rupees.
• Total expenditure for 8 months equals total income for 6 months: 8E = 6I.
• Total savings in 12 months = ₹6000.
• Saving in each month is I − E.

Concept / Approach:
Key ideas:

• Use the relation 8E = 6I to express E in terms of I.
• Compute monthly saving S = I − E using this relationship.
• Total annual saving is 12S, which is given as 6000, so solve for I.
• This leads to a straightforward linear equation in I.

Step-by-Step Solution:
Step 1: From 8E = 6I, divide both sides by 2 to get 4E = 3I. Step 2: Express E in terms of I: E = (3/4)I. Step 3: Monthly saving S = I − E = I − (3/4)I = (1/4)I. Step 4: Total saving for 12 months is 12S = 12 × (1/4)I = 3I. Step 5: We are told that 3I = 6000. Step 6: Solve for I: I = 6000 / 3 = 2000. Step 7: Therefore the average monthly income is ₹2000.
Verification / Alternative check:
Step 1: With income I = 2000, expenditure E = (3/4) × 2000 = 1500 per month. Step 2: Income in 6 months = 6 × 2000 = 12000. Expenditure in 8 months = 8 × 1500 = 12000, which matches the condition. Step 3: Monthly saving S = 2000 − 1500 = 500. Annual saving over 12 months is 12 × 500 = 6000, which matches the given figure.
Why Other Options Are Wrong:
Option Rs. 2400 would give E = 1800, so annual saving becomes 12 × 600 = 7200, not 6000. Option Rs. 1800 would lead to a lower saving and would not satisfy both 8E = 6I and the given annual saving. Option Rs. 2150 or Rs. 3000 similarly do not satisfy both the ratio condition and the exact annual saving amount.
Common Pitfalls:
Students sometimes misinterpret the statement and set 8I = 6E instead of 8E = 6I, reversing the relationship. Another mistake is to forget that savings accumulate over 12 months, not just 6 or 8 months. Careless manipulation of fractions, especially when computing E = (3/4)I, can lead to slight numerical errors.
Final Answer:
The man's average monthly income is Rs. 2000.