A person can save 25% of his income. If his income increases by 20% and still he saves the same amount as before, the percentage increase in his expenditure is ...........

Options

A. `26_(3)^(2)`
B. `24`
C. `25_(3)^(1)`
D. 25

Answer

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

### Step 1: Understand the Savings and Income
The person saves 25% of his income. If we denote the total income as \( I \), then the savings \( S \) can be expressed as:
\[ S = 0.25I \]
This means that the expenditure \( E \) (which is the remaining part of the income after savings) can be expressed as:
\[ E = I - S = I - 0.25I = 0.75I \]

### Step 2: Calculate the New Income After Increase
The income increases by 20%. Therefore, the new income \( I' \) can be calculated as:
\[ I' = I + 0.20I = 1.20I \]

### Step 3: Determine the New Savings
According to the problem, the savings remain the same even after the income increase. Thus, the new savings \( S' \) is still:
\[ S' = S = 0.25I \]

### Step 4: Calculate the New Expenditure
The new expenditure \( E' \) can be calculated using the new income and the unchanged savings:
\[ E' = I' - S' = 1.20I - 0.25I = 0.95I \]

### Step 5: Calculate the Increase in Expenditure
Now, we need to find the increase in expenditure. The initial expenditure was \( E = 0.75I \) and the new expenditure is \( E' = 0.95I \). The increase in expenditure \( \Delta E \) is:
\[ \Delta E = E' - E = 0.95I - 0.75I = 0.20I \]

### Step 6: Calculate the Percentage Increase in Expenditure
To find the percentage increase in expenditure, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta E}{E} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{0.20I}{0.75I} \right) \times 100 = \left( \frac{0.20}{0.75} \right) \times 100 \]
Calculating this gives:
\[ \text{Percentage Increase} = \left( \frac{20}{75} \right) \times 100 = \frac{2000}{75} \approx 26.67\% \]

### Conclusion
Thus, the percentage increase in his expenditure is approximately \( 26.67\% \).

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