Nikhil and Reetu earn in the ratio $2:1$. They spend in the ratio $5:3$ and save in the ratio $4:1$. Find the monthly income of each, if the total monthly savings of both Nikhil and Reetu together is $₹10,000$.

Step 1: Determine individual savings
The total savings of Nikhil and Reetu together is $₹10,000$, and they save in the ratio $4:1$.

$$\text{Savings of Nikhil}=\frac{4}{4+1}\times 10,000=\frac{4}{5}\times 10,000=₹8,000$$

$$\text{Savings of Reetu}=\frac{1}{4+1}\times 10,000=\frac{1}{5}\times 10,000=₹2,000$$

Step 2: Set up equations using Income and Expenditure
Let the incomes of Nikhil and Reetu be $2x$ and $1x$ respectively.
Let their expenditures be $5y$ and $3y$ respectively.

We know that: $\text{Income}-\text{Savings}=\text{Expenditure}$
For Nikhil:

$$2x-8,000=5y\text{— (Equation 1)}$$

For Reetu:

$$x-2,000=3y\text{— (Equation 2)}$$

**Step 3: Solve the equations for $x$ **
From Equation 2, we can express $x$ as:

$$x=3y+2,000$$

Substitute this value of $x$ into Equation 1:

$$2(3y+2,000)-8,000=5y$$

$$6y+4,000-8,000=5y$$

$$6y-4,000=5y$$

$$6y-5y=4,000$$

$$y=4,000$$

Now, substitute $y=4,000$ back into the expression for $x$:

$$x=3(4,000)+2,000=12,000+2,000=14,000$$

Step 4: Calculate monthly incomes

$$\text{Income of Nikhil}=2x=2\times 14,000=₹28,000$$

$$\text{Income of Reetu}=1x=₹14,000$$