The incomes of A and B are in the ratio $3:2$ and their expenditures are in the ratio $5:3$. If each of them manages to save $\text{Rs. }1,000$, then B's income is:

Let the incomes of A and B be $3x$ and $2x$ respectively.
Let the expenditures of A and B be $5y$ and $3y$ respectively.

We know that: $\text{Income}-\text{Expenditure}=\text{Savings}$
Therefore, we get two linear equations:

$$3x-5y=1000\text{— (Equation 1)}$$

$$2x-3y=1000\text{— (Equation 2)}$$

To eliminate $y$, multiply Equation 1 by $3$ and Equation 2 by $5$:

$$9x-15y=3000\text{— (Equation 3)}$$

$$10x-15y=5000\text{— (Equation 4)}$$

Subtracting Equation 3 from Equation 4:

$$(10x-15y)-(9x-15y)=5000-3000$$

$$x=2000$$

Now, B's income is represented by $2x$:

$$\text{B’s income}=2\times 2000=\text{Rs. }4,000$$