The monthly earnings of Alex are 22.5% greater than those of Brian, whereas Alex's monthly expenditures exceed Brian's expenditures by Rs.15000. If Alex manages to save Rs.3000 more than Brian each month, calculate Brian's monthly earnings.

The correct answer is Rs.80000.

To calculate Brian's monthly earnings, we can set up algebraic equations relating earnings, expenditures, and savings for both Alex and Brian.

Step 1: Define the variables for earnings and expenditures.

Let Brian's monthly earnings be $E$.

Since Alex's monthly earnings are 22.5% greater than Brian's earnings, we express Alex's earnings as:
$E_{\text{Alex}}=E+0.225E=1.225E$

Let Brian's monthly expenditure be $X$.

Since Alex's monthly expenditure exceeds Brian's by Rs.15000, we express Alex's expenditure as:
$X_{\text{Alex}}=X+15000$

Step 2: Express savings in terms of earnings and expenditures.

Monthly savings are given by the formula:
$\text{Savings}=\text{Earnings}-\text{Expenditure}$

Brian's monthly savings:
$S_{\text{Brian}}=E-X$

Alex's monthly savings:
$S_{\text{Alex}}=E_{\text{Alex}}-X_{\text{Alex}}=1.225E-(X+15000)$

Step 3: Set up the equation using the given savings difference.

We are given that Alex manages to save Rs.3000 more than Brian each month:
$S_{\text{Alex}}-S_{\text{Brian}}=3000$

Substitute the expressions for Alex's and Brian's savings into the equation:
$[1.225E-(X+15000\left)\right]-(E-X)=3000$

Step 4: Simplify and solve for $E$.

Remove the parentheses:
$1.225E-X-15000-E+X=3000$

Notice that the expenditure variable $X$ cancels out:
$(1.225E-E)-15000=3000$
$0.225E-15000=3000$

Add 15000 to both sides of the equation:
$0.225E=3000+15000$
$0.225E=18000$

Divide by 0.225 to solve for $E$:
$E=\frac{18000}{0.225}$

Converting 0.225 into a fraction ($0.225=\frac{9}{40}$):
$E=18000\times \frac{40}{9}$
$E=2000\times 40=80000$

Therefore, Brian's monthly earnings are Rs.80000.