Understanding Income, Expenditure, and Savings Ratios

This problem involves understanding the relationship between income, expenditure, and savings, and how changes in income affect the total amount. The key is to use the initial ratio and the given initial savings to find the initial expenditure and income. Then, we can calculate the income after the percentage increase.

Step-by-Step Solution for Soham's Income

Let's break down the problem into manageable steps to find Soham's income after the increment.

Given Information:

• Ratio of initial expenditure to savings = 5 ∶ 3
• Initial savings = ₹4,500
• Income increases by 25%

Step 1: Find the value of one ratio unit.

The initial expenditure and savings are in the ratio 5:3. This means for every 3 units of savings, there are 5 units of expenditure. We are given that the initial savings represent 3 units and are equal to ₹4,500.

Let the value of one ratio unit be $x$.

So, Initial Savings = $3x = \text{₹}4,500$

To find the value of one unit ($x$), we divide the initial savings by 3:

$x = \frac{\text{₹}4,500}{3} = \text{₹}1,500$

The value of one ratio unit is ₹1,500.

Step 2: Calculate the initial expenditure.

The initial expenditure is represented by 5 units in the ratio.

Initial Expenditure = $5x = 5 \times \text{₹}1,500$

Initial Expenditure = $\text{₹}7,500$

Step 3: Calculate the initial income.

Income is the sum of expenditure and savings.

Initial Income = Initial Expenditure + Initial Savings

Initial Income = $\text{₹}7,500 + \text{₹}4,500$

Initial Income = $\text{₹}12,000$

Soham's initial income was ₹12,000.

Step 4: Calculate the income increase.

His income increases by 25%. We need to find 25% of the initial income.

Income Increase = 25% of Initial Income

Income Increase = $0.25 \times \text{₹}12,000$

Income Increase = $\frac{25}{100} \times \text{₹}12,000$

Income Increase = $\frac{1}{4} \times \text{₹}12,000$

Income Increase = $\text{₹}3,000$

Step 5: Calculate the income after the increment.

The income after the increment is the initial income plus the income increase.

Income after Increment = Initial Income + Income Increase

Income after Increment = $\text{₹}12,000 + \text{₹}3,000$

Income after Increment = $\text{₹}15,000$

Soham's income after the 25% increment is ₹15,000.

Summary of Calculations

The final income after the increment is ₹15,000.

Revision Table: Key Concepts

Additional Information on Income Calculations

When dealing with problems involving income, expenditure, and savings, it's important to remember the fundamental relationship:

$\text{Income} = \text{Expenditure} + \text{Savings}$

This equation holds true in all scenarios unless stated otherwise. Ratios provide a proportional relationship between expenditure and savings. If the income changes, either the expenditure, savings, or both must change to maintain the relationship.

In this specific problem, we assumed the expenditure-to-savings ratio was for the initial amounts. The income increase is applied to the initial income to find the new income. The problem does not specify how the new income is distributed between expenditure and savings, and we were only asked for the new income value itself.

Percentage increase can also be calculated by multiplying the original value by $(1 + \text{percentage increase as decimal})$. For example, a 25% increase on ₹12,000 is $\text{₹}12,000 \times (1 + 0.25) = \text{₹}12,000 \times 1.25 = \text{₹}15,000$. This is a quicker way to get the final value directly after an increase.