In March Rohan's monthly expenditure was `90%` of his monthly income. His monthyl income increased by `30%` and his monthly expenditure increased by `20%` when compared to the previous month. Find the percentage increase in his montly savings.

Options

A. `130%`
B. `120%`
C. `110%`
D. `125%`

Answer

To solve the problem step by step, let's define the variables and calculate the required values.

### Step 1: Define Rohan's Monthly Income and Expenditure
Let Rohan's monthly income in March be \( I \).
According to the problem, Rohan's monthly expenditure in March was \( 90\% \) of his monthly income.

\[ \text{Expenditure in March} = 0.90 \times I = \frac{9}{10} I \]

### Step 2: Calculate Rohan's Monthly Savings in March
Rohan's savings in March can be calculated as:

\[ \text{Savings in March} = \text{Income} - \text{Expenditure} = I - \frac{9}{10} I = \frac{1}{10} I \]

### Step 3: Calculate Rohan's New Monthly Income in April
Rohan's monthly income increased by \( 30\% \) in April. Therefore, his new income is:

\[ \text{New Income in April} = I + 0.30 \times I = 1.30 I \]

### Step 4: Calculate Rohan's New Monthly Expenditure in April
Rohan's monthly expenditure increased by \( 20\% \) in April. Therefore, his new expenditure is:

\[ \text{New Expenditure in April} = \text{Expenditure in March} + 0.20 \times \text{Expenditure in March} \]
\[ = \frac{9}{10} I + 0.20 \times \frac{9}{10} I = \frac{9}{10} I + \frac{18}{100} I = \frac{9}{10} I + \frac{9}{50} I \]

To add these fractions, we need a common denominator:

\[ \frac{9}{10} I = \frac{45}{50} I \]
Thus,
\[ \text{New Expenditure in April} = \frac{45}{50} I + \frac{9}{50} I = \frac{54}{50} I = \frac{27}{25} I \]

### Step 5: Calculate Rohan's New Monthly Savings in April
Rohan's savings in April can be calculated as:

\[ \text{Savings in April} = \text{New Income} - \text{New Expenditure} = 1.30 I - \frac{27}{25} I \]

To perform this subtraction, we need a common denominator:

\[ 1.30 I = \frac{130}{100} I = \frac{65}{50} I \]

Now, we can subtract:

\[ \text{Savings in April} = \frac{65}{50} I - \frac{54}{50} I = \frac{11}{50} I \]

### Step 6: Calculate the Percentage Increase in Monthly Savings
To find the percentage increase in savings, we use the formula:

\[ \text{Percentage Increase} = \left( \frac{\text{New Savings} - \text{Old Savings}}{\text{Old Savings}} \right) \times 100 \]

Substituting the values we calculated:

\[ \text{Percentage Increase} = \left( \frac{\frac{11}{50} I - \frac{1}{10} I}{\frac{1}{10} I} \right) \times 100 \]

Convert \( \frac{1}{10} I \) to a fraction with a denominator of 50:

\[ \frac{1}{10} I = \frac{5}{50} I \]

Now substituting back:

\[ = \left( \frac{\frac{11}{50} I - \frac{5}{50} I}{\frac{5}{50} I} \right) \times 100 = \left( \frac{\frac{6}{50} I}{\frac{5}{50} I} \right) \times 100 = \left( \frac{6}{5} \right) \times 100 = 120\% \]

### Final Answer
The percentage increase in Rohan's monthly savings is \( 120\% \).

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