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 answer is 850 (Option 1).

Let's break down the problem and solve it step-by-step using the Simple Interest formula:

The formula for Simple Interest (SI) is:
$SI=\frac{P\times R\times T}{100}$
where:
- $P$ is the principal amount invested.
- $R$ is the rate of interest per annum.
- $T$ is the time period in years.

Step 1: Calculate the interest from Scheme A
- Principal invested in Scheme A, $P_A=X$
- Rate of interest for Scheme A, $R_A=10\%$ p.a.
- Time period for Scheme A, $T_A=4$ years
The interest received from Scheme A ($I_A$) is:
$I_A=\frac{X\times 10\times 4}{100}=\frac{40X}{100}=0.4X$

Step 2: Calculate the interest from Scheme B
- Principal invested in Scheme B, $P_B=X+400$
- Rate of interest for Scheme B, $R_B=12\%$ p.a.
- Time period for Scheme B, $T_B=2$ years
The interest received from Scheme B ($I_B$) is:
$I_B=\frac{(X+400)\times 12\times 2}{100}=\frac{24(X+400)}{100}=0.24(X+400)=0.24X+96$

Step 3: Solve for X using the sum of interests
The sum of the interests from both schemes is given as Rs 640. Therefore:
$I_A+I_B=640$
Substitute the expressions we found in Step 1 and Step 2:
$0.4X+(0.24X+96)=640$
Combine the terms containing $X$:
$0.64X+96=640$
Subtract 96 from both sides:
$0.64X=640-96$
$0.64X=544$
Divide by 0.64 to find $X$:
$X=\frac{544}{0.64}$
To simplify, multiply the numerator and denominator by 100:
$X=\frac{54400}{64}$
Dividing 54400 by 64 gives:
$X=850$

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