Ratio of Income and Expenditure Problem Solution

This solution details how to calculate the respective incomes of Aamir and Ali by solving a system of equations derived from given ratios and savings.

Income and Expenditure Ratios

Let the income of Aamir be $I_A$ and Ali be $I_{Al}$.

Let the expenditure of Aamir be $E_A$ and Ali be $E_{Al}$.

The problem states:

• Income Ratio: $I_A : I_{Al} = 5 : 3$. This means $I_A = 5x$ and $I_{Al} = 3x$ for some value $x$.
• Expenditure Ratio: $E_A : E_{Al} = 9 : 5$. This means $E_A = 9y$ and $E_{Al} = 5y$ for some value $y$.

Savings Equations

Savings are calculated as Income minus Expenditure.

• Aamir's Savings: $I_A - E_A = ₹2,600$
• Ali's Savings: $I_{Al} - E_{Al} = ₹1,800$

Substituting the ratio expressions, we get:

1. For Aamir: $5x - 9y = 2600$
2. For Ali: $3x - 5y = 1800$

Solving for x and y

We solve this system of two linear equations.

1. Multiply the first equation by 3 and the second equation by 5 to make the coefficients of $x$ equal:
   • Equation 1 (multiplied by 3): $15x - 27y = 7800$
   • Equation 2 (multiplied by 5): $15x - 25y = 9000$
2. Subtract the new Equation 2 from the new Equation 1:

   $(15x - 27y) - (15x - 25y) = 7800 - 9000$

   $-2y = -1200$

   $y = 600$

3. Substitute $y = 600$ into the original second equation ($3x - 5y = 1800$):

   $3x - 5(600) = 1800$

   $3x - 3000 = 1800$

   $3x = 4800$

   $x = 1600$

Determining Respective Incomes

Using the calculated value of $x = 1600$:

• Aamir's Income: $I_A = 5x = 5 \times 1600 = ₹8,000$
• Ali's Income: $I_{Al} = 3x = 3 \times 1600 = ₹4,800$

The respective incomes of Aamir and Ali are ₹8,000 and ₹4,800.