In a family there wre 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 an 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. a. 5
B. b. 12
C. c. 9
D. d. 10

Answer

To solve the problem step by step, we will follow these steps:

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

**Hint:** Remember that "directly proportional" means we can express the relationship as an equation involving multiplication.

### Step 2: Write the equation for the new expenditure after one person leaves
When one elder son leaves, the number of people in the family becomes \( n - 1 \). The new expenditure can be expressed as:
\[ E' = 5(n - 1)^2 \]

**Hint:** When a person leaves, reduce the number of people by 1 and square that value.

### Step 3: Set up the equation based on the decrease in consumption
According to the problem, the decrease in consumption of rice is 95 kg, which gives us the equation:
\[ E - E' = 95 \]
Substituting the expressions for \( E \) and \( E' \):
\[ 5n^2 - 5(n - 1)^2 = 95 \]

**Hint:** Use the information about the decrease to set up an equation that relates the two expenditures.

### Step 4: Expand the equation
Now, we need to expand \( 5(n - 1)^2 \):
\[ 5(n - 1)^2 = 5(n^2 - 2n + 1) = 5n^2 - 10n + 5 \]
Substituting this back into our equation gives:
\[ 5n^2 - (5n^2 - 10n + 5) = 95 \]

**Hint:** Remember to distribute the negative sign when subtracting the second expenditure.

### Step 5: Simplify the equation
Now simplify the equation:
\[ 5n^2 - 5n^2 + 10n - 5 = 95 \]
This simplifies to:
\[ 10n - 5 = 95 \]

**Hint:** Combine like terms carefully to isolate the variable.

### Step 6: Solve for \( n \)
Now, add 5 to both sides:
\[ 10n = 100 \]
Then, divide by 10:
\[ n = 10 \]

**Hint:** To isolate \( n \), perform inverse operations (addition and division).

### Final Answer
The value of \( n \) is 10.