Initially, 'A' invested Rs. 'x' and 'B' invested Rs. 'x - y'. After six months, they both increased their investments by Rs. 'y'. Therefore, 'A' has a total investment of 'x + y' and 'B' has a total investment of '(x - y) + y'.
At the end of the year, the business made a profit of Rs. '10x'. To determine the profit share received by 'A' and 'B', we need to calculate their respective profit ratios based on their investments.
The profit share received by 'A' can be calculated as follows:
Profit share of 'A' = (Investment of 'A' / Total Investment) * Total Profit
= (x + y) / ((x + y) + (x - y)) * 10x
= (x + y) / (2x) * 10x
= (x + y) / 2 * 10x
= (x + y) * 5x
Similarly, the profit share received by 'B' can be calculated as follows:
Profit share of 'B' = (Investment of 'B' / Total Investment) * Total Profit
= ((x - y) + y) / ((x + y) + (x - y)) * 10x
= (x / (2x)) * 10x
= 1/2 * 10x
= 5x
Now, to find the difference between the profit shares received by 'A' and 'B', we subtract the profit share of 'B' from the profit share of 'A':
Difference = Profit share of 'A' - Profit share of 'B'
= (x + y) * 5x - 5x
= 5x^2 + 5xy - 5x
= 5x(x + y - 1)
Therefore, the profit share received by 'A' exceeds that of 'B' by 5x(x + y - 1)