A man invested Rs X and Rs x + 400 in two different schemes A & B respectively. The rate of interest offered by schemes A & B is 10% p.a. and 12% p.a. respectively. If the sum of interest received by man from scheme A after four years and from scheme B after two year is Rs 640, then find X.

The correct option is 850.

Let's find the value of $X$ step-by-step:

Step 1: Understand the investments in both schemes.
- Scheme A: Principal investment is Rs $X$, Rate of interest ($R_A$) is 10% per annum, and Time period ($T_A$) is 4 years.
- Scheme B: Principal investment is Rs $(X+400)$, Rate of interest ($R_B$) is 12% per annum, and Time period ($T_B$) is 2 years.

Step 2: Formula for Simple Interest.
The formula for Simple Interest (SI) is:

$SI=\frac{P\times R\times T}{100}$

where $P$ is the principal, $R$ is the rate of interest per annum, and $T$ is the time in years.

Step 3: Calculate the interest from each scheme.
- Interest from Scheme A ($I{I}_A$):

$I{I}_A=\frac{X\times 10\times 4}{100}=\frac{40X}{100}=0.4X$

- Interest from Scheme B ($I{I}_B$):

$I{I}_B=\frac{(X+400)\times 12\times 2}{100}=\frac{24(X+400)}{100}=0.24(X+400)$

Step 4: Set up the equation using the given sum of interests.
We are given that the sum of the interests from both schemes is Rs 640:

$I{I}_A+I{I}_B=640$

Substitute the expressions we obtained:

$0.4X+0.24(X+400)=640$

Multiply both sides of the equation by 100 to clear the decimals:

$40X+24(X+400)=64000$

Step 5: Solve for $X$.
Expand the terms:

$40X+24X+9600=64000$

Combine the $X$ terms:

$64X+9600=64000$

Subtract 9600 from both sides:

$64X=64000-9600$

$64X=54400$

Divide by 64:

$X=\frac{54400}{64}$

Let's simplify by dividing both numerator and denominator by 8:

$X=\frac{6800}{8}$

Divide 6800 by 8:

$X=850$

Thus, the value of $X$ is Rs 850.