A and B started a business with investments of Rs.5X & and Rs. 5X -400 respectively. After six months, B added Rs.200 and at the end of the year profit received by B is Rs.510 out of a total profit of Rs.1110. Find the value of X.

The correct answer is 400 (Option 400).

Step-by-step Explanation:

Let's analyze the investments of A and B and the duration for which their investments were kept in the business.

1. Investment of A:

A invested Rs. 5X for the entire year (12 months).

Total equivalent investment for A = 5X × 12 = 60X

2. Investment of B:

For the first 6 months, B's investment was Rs. (5X - 400).

After 6 months, B added Rs. 200, making the new investment = (5X - 400) + 200 = Rs. (5X - 200) for the remaining 6 months.

Total equivalent investment for B = (5X - 400) × 6 + (5X - 200) × 6

= 6 × [(5X - 400) + (5X - 200)]

= 6 × (10X - 600)

= 60X - 3600

3. Ratio of Profit Sharing:

The ratio of profit sharing between A and B is equal to the ratio of their total equivalent investments:

Ratio (A : B) = 60X : (60X - 3600)

Simplifying by dividing both sides by 60:

Ratio (A : B) = X : (X - 60)

4. Calculating the Value of X:

Total profit = Rs. 1110

Profit received by B = Rs. 510

Profit received by A = Total Profit - B's Profit = 1110 - 510 = Rs. 600

Now, set up the proportion between the profit ratio and the actual profits:

$\frac{\text{Profit of A}}{\text{Profit of B}}=\frac{\text{X}}{\text{X}-60}$

$\frac{600}{510}=\frac{\text{X}}{\text{X}-60}$

Simplify the fraction 600 / 510 by dividing numerator and denominator by 30:

$\frac{20}{17}=\frac{\text{X}}{\text{X}-60}$

Cross-multiply to solve for X:

20 × (X - 60) = 17 × X

20X - 1200 = 17X

20X - 17X = 1200

3X = 1200

X = 400

Thus, the value of X is 400.