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.

Let us break down the logical steps to find the value of X step-by-step.

Step 1: Understand the given details for Scheme A
Principal invested in Scheme A = Rs X
Rate of interest for Scheme A = 10% per annum
Time period for Scheme A = 4 years

The Simple Interest (SI) formula is:

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

Interest from Scheme A (${\text{SI}}_{\text{A}}$):

${\text{SI}}_{\text{A}}=\frac{\text{X}\times 10\times 4}{100}=\frac{40\text{X}}{100}=0.4\text{X}$

Step 2: Understand the given details for Scheme B
Principal invested in Scheme B = Rs (X + 400)
Rate of interest for Scheme B = 12% per annum
Time period for Scheme B = 2 years

Interest from Scheme B (${\text{SI}}_{\text{B}}$):

${\text{SI}}_{\text{B}}=\frac{(\text{X}+400)\times 12\times 2}{100}=\frac{24(\text{X}+400)}{100}=0.24\text{X}+96$

Step 3: Set up the total interest equation
We are given that the sum of interest received from both schemes is Rs 640.

${\text{SI}}_{\text{A}}+{\text{SI}}_{\text{B}}=640$

$0.4\text{X}+0.24\text{X}+96=640$

$0.64\text{X}+96=640$

Step 4: Solve for X
Subtract 96 from both sides:

$0.64\text{X}=640-96$

$0.64\text{X}=544$

Divide by 0.64:

$\text{X}=\frac{544}{0.64}=\frac{54400}{64}=850$

Therefore, the value of X is 850.