In a family a person saves 30% of his income . If his salary is increased by 20% and saving are decreased by 20% then find the percentage increase in the expenditure.

Options

A. `34.(1)/(4)%`
B. `37.(1)/(7)%`
C. `41.(1)/(7)%`
D. `43.(1)/(3)%`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the initial income and savings
Let the initial income of the person be \( I \).

Given that the person saves 30% of his income, we can calculate the initial savings:
\[ \text{Initial Savings} = 30\% \text{ of } I = 0.3I \]

### Step 2: Calculate the initial expenditure
The expenditure can be calculated as:
\[ \text{Expenditure} = \text{Income} - \text{Savings} = I - 0.3I = 0.7I \]

### Step 3: Calculate the new income after the salary increase
The salary is increased by 20%. Therefore, the new income \( I' \) is:
\[ I' = I + 20\% \text{ of } I = I + 0.2I = 1.2I \]

### Step 4: Calculate the new savings after the decrease
The savings are decreased by 20%. The new savings \( S' \) can be calculated as:
\[ S' = \text{Initial Savings} - 20\% \text{ of Initial Savings = } 0.3I - 0.2(0.3I) = 0.3I - 0.06I = 0.24I \]

### Step 5: Calculate the new expenditure
Now, we can calculate the new expenditure \( E' \):
\[ E' = I' - S' = 1.2I - 0.24I = 0.96I \]

### Step 6: Calculate the increase in expenditure
The increase in expenditure can be calculated as:
\[ \text{Increase in Expenditure} = E' - \text{Initial Expenditure} = 0.96I - 0.7I = 0.26I \]

### Step 7: Calculate the percentage increase in expenditure
To find the percentage increase in expenditure, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase in Expenditure}}{\text{Initial Expenditure}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{0.26I}{0.7I} \right) \times 100 = \left( \frac{0.26}{0.7} \right) \times 100 \approx 37.14\% \]

### Final Answer
The percentage increase in the expenditure is approximately **37.14%**.

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