The average monthly income of a person in a certain family of 5 is  ₹10,000. What will be the average monthly income of a person in the same family if the income of one person increased by  ₹1,20,000 per year ?

The correct option is ₹12,000.

Let us break down the solution step-by-step to understand how we arrive at this answer:

Step 1: Understand the initial scenario
The family has 5 members. The average monthly income of a person in this family is ₹10,000.
Using the formula for average:
$$\text{Average}=\frac{\text{Total Income}}{\text{Number of People}}$$
We can calculate the initial total monthly income of the family as:
$$\text{Initial Total Monthly Income}=\text{Average Monthly Income}\times \text{Number of People}$$
$$\text{Initial Total Monthly Income}=10,000\times 5=\text{₹}50,000$$

Step 2: Calculate the monthly increase in income
We are given that the income of one person increases by ₹1,20,000 per year.
Since we need to find the new average monthly income, we must convert this annual increase into a monthly increase:
$$\text{Monthly Increase}=\frac{\text{Annual Increase}}{12}$$
$$\text{Monthly Increase}=\frac{1,20,000}{12}=\text{₹}10,000$$

Step 3: Calculate the new total monthly income
The new total monthly income of the family is the sum of the initial total monthly income and the monthly increase:
$$\text{New Total Monthly Income}=50,000+10,000=\text{₹}60,000$$

Step 4: Find the new average monthly income
Now, we divide the new total monthly income by the number of family members (which remains 5):
$$\text{New Average Monthly Income}=\frac{\text{New Total Monthly Income}}{\text{Number of People}}$$
$$\text{New Average Monthly Income}=\frac{60,000}{5}=\text{₹}12,000$$

Alternative Method (Change in Average):
We can also solve this by finding how much the average itself increases.
The total monthly increase for the family is ₹10,000.
Since there are 5 people, this increase is distributed equally among them to find the change in the average:
$$\text{Increase in Average}=\frac{\text{Total Monthly Increase}}{\text{Number of People}}=\frac{10,000}{5}=\text{₹}2,000$$
Adding this to the original average:
$$\text{New Average}=10,000+2,000=\text{₹}12,000$$