In a family there were n people. The expenditure of rice per month in this family is directly proportional to the 5 times the square of no. of people in the family. If the elder son left the family to study in USA there was decrease in consumption of 95 kg rice per month. What is the value of n?

Options

A. 5
B. 12
C. 9
D. 10

Answer

To solve the problem, we will follow these steps:

1. **Understand the relationship**: The expenditure of rice per month (E) is directly proportional to 5 times the square of the number of people in the family (n). We can express this relationship mathematically as:
\[ E = k \cdot 5n^2 \]
where \(k\) is a constant of proportionality.

2. **Set up the equation before and after the elder son leaves**:
- Before the elder son leaves, the expenditure is:
\[ E_{initial} = 5n^2 \]
- After the elder son leaves, the number of people becomes \(n - 1\), and the new expenditure is:
\[ E_{final} = 5(n - 1)^2 \]

3. **Express the decrease in expenditure**: According to the problem, the decrease in consumption of rice is 95 kg. Therefore, we can set up the equation:
\[ E_{initial} - E_{final} = 95 \]

4. **Substituting the expressions for expenditure**:
\[ 5n^2 - 5(n - 1)^2 = 95 \]

5. **Expanding the equation**:
\[ 5n^2 - 5(n^2 - 2n + 1) = 95 \]
Simplifying this gives:
\[ 5n^2 - 5n^2 + 10n - 5 = 95 \]
Which reduces to:
\[ 10n - 5 = 95 \]

6. **Solving for n**:
\[ 10n = 95 + 5 \]
\[ 10n = 100 \]
\[ n = \frac{100}{10} = 10 \]

Thus, the value of \(n\) is **10**.