Understanding the Income, Expenditure, and Savings Relationship

This problem deals with the relationship between income, expenditure, and savings. The fundamental equation is:

\(\text{Income} = \text{Expenditure} + \text{Savings}\)

We are given initial conditions and how these quantities change over time. We need to find the initial percentage of income saved.

Setting up the Initial Situation

Let's assume Rishu's initial income is \(I\). We are told that she saves \(x\%\) of her income. So, her initial savings, \(S\), can be expressed as:

\(S = \frac{x}{100} \times I\)

Her initial expenditure, \(E\), is the remaining part of her income after saving. So:

\(E = I - S = I - \frac{x}{100}I = I \left(1 - \frac{x}{100}\right)\)

Analyzing the Changes in Income, Expenditure, and Savings

The problem states that her income increases by 26%, her expenditure increases by 20%, and her savings increase by 50%. Let's denote the new values with a prime symbol (').

• New Income (I'): Income increases by 26%.

  \(I' = I + 26\%\text{ of } I = I + \frac{26}{100}I = I(1 + 0.26) = 1.26I\)

• New Expenditure (E'): Expenditure increases by 20%.

  \(E' = E + 20\%\text{ of } E = E + \frac{20}{100}E = E(1 + 0.20) = 1.20E\)

  Substitute the initial expenditure \(E = I \left(1 - \frac{x}{100}\right)\):

  \(E' = 1.20 \times I \left(1 - \frac{x}{100}\right)\)

• New Savings (S'): Savings increase by 50%.

  \(S' = S + 50\%\text{ of } S = S + \frac{50}{100}S = S(1 + 0.50) = 1.50S\)

  Substitute the initial savings \(S = \frac{x}{100}I\):

  \(S' = 1.50 \times \frac{x}{100}I\)

Applying the Relationship to the New Situation

The relationship \( \text{Income} = \text{Expenditure} + \text{Savings} \) must hold true for the new values as well:

\(I' = E' + S'\)

Substitute the expressions we found for \(I'\), \(E'\), and \(S'\) in terms of \(I\) and \(x\):

\(1.26I = 1.20 \times I \left(1 - \frac{x}{100}\right) + 1.50 \times \frac{x}{100}I\)

Solving for the Value of x

We need to solve this equation for \(x\). Notice that \(I\) is a common factor on both sides (assuming income is not zero, which is reasonable for this problem). We can divide both sides by \(I\):

\(1.26 = 1.20 \left(1 - \frac{x}{100}\right) + 1.50 \frac{x}{100}\)

Now, let's simplify and solve for \(x\):

\(1.26 = 1.20 - 1.20 \times \frac{x}{100} + 1.50 \times \frac{x}{100}\)

\(1.26 = 1.20 + \left(1.50 - 1.20\right) \times \frac{x}{100}\)

\(1.26 = 1.20 + 0.30 \times \frac{x}{100}\)

Subtract 1.20 from both sides:

\(1.26 - 1.20 = 0.30 \times \frac{x}{100}\)

\(0.06 = 0.30 \times \frac{x}{100}\)

Divide both sides by 0.30:

\(\frac{0.06}{0.30} = \frac{x}{100}\)

\(0.2 = \frac{x}{100}\)

Multiply both sides by 100:

\(x = 0.2 \times 100\)

\(x = 20\)

So, the initial percentage of income Rishu saves is 20%.

Verification (Optional but Recommended)

Let's verify the result with an example. Assume initial income \(I = 1000\).

Initial savings (\(x=20\%\)): \(S = 20\%\text{ of } 1000 = 200\)

Initial expenditure: \(E = 1000 - 200 = 800\)

New income (26% increase): \(I' = 1000 + 26\%\text{ of } 1000 = 1000 + 260 = 1260\)

New expenditure (20% increase): \(E' = 800 + 20\%\text{ of } 800 = 800 + 160 = 960\)

New savings (50% increase): \(S' = 200 + 50\%\text{ of } 200 = 200 + 100 = 300\)

Check if \(I' = E' + S'\):

\(1260 = 960 + 300\)

\(1260 = 1260\)

The equation holds true, so our value of \(x=20\) is correct.

The value of x, Rishu's initial savings percentage, is 20.

Revision Table: Income, Expenditure, and Savings Concepts

Additional Information on Percentage Changes

When a quantity \(Q\) increases by \(P\%\), the new quantity \(Q'\) is given by:

\(Q' = Q + \frac{P}{100}Q = Q\left(1 + \frac{P}{100}\right)\)

This formula was used to calculate the new income, new expenditure, and new savings in this problem.

• An increase of 26% means multiplying by \(1 + \frac{26}{100} = 1.26\).
• An increase of 20% means multiplying by \(1 + \frac{20}{100} = 1.20\).
• An increase of 50% means multiplying by \(1 + \frac{50}{100} = 1.50\).