1. Suppose A and B invest Rs. [latex]x[/latex] and Rs. [latex]y[/latex] respectively for a year in a business, then at the end of the year:

(A's share of profit) : (B's share of profit) = [latex]x[/latex] : [latex]y[/latex]

Here investment of all partners are for same time, and the gain or loss is distributed among them in the ratios of their investments.

2. Suppose A invests Rs. [latex]x[/latex] for 'p' months and B invests Rs. [latex]y[/latex] for 'q' months, then

(A's share of profit) : (B's share of profit) = [latex]x[/latex]p : [latex]x[/latex]q

Here investments are for different time periods, equivalent capitals are calculated for a unit of time by taking,

(capital x number of units of time).

Profit or loss is divided in the ratio of these capitals.

Here first person invested amount A for [latex]t_{1} [/latex] period, second persons invested amount B for [latex]t_{2} [/latex] period and so on.

[latex] 4000 \times 8 + 4400 \times 4 : 6000 \times 8 + 5600 \times 4[/latex]

31 : 44

So B gets = [latex](\frac{44}{75}) \times 14,250[/latex] = Rs 8,360

Let B invested for y months, then A for [latex](\frac {y}{2})[/latex] months, and C for [latex](\frac {y}{3})[/latex] months

So ratio of profit shares-

A : B : C

[latex]x \times (\frac {y}{2}) : (\frac {x}{2}) \times (y) : (\frac {x}{3}) \times (\frac {y}{3})[/latex]

[latex]\frac{1}{2} : \frac{1}{2} : \frac{1}{9} = 9 : 9 : 2[/latex]

Both A and B gets equal and highest profits.

Then A’s = [latex](\frac{1}{4}) \times x[/latex]

Remaining = [latex]{(\frac {3}{4})}^{th} [/latex] of x

So B’s investment = [latex](\frac {1}{3}) \times (\frac {3x}{4}) = \frac {x}{4} [/latex]

And C’s = x – [latex](\frac {x}{4} +\frac {x}{4}) = \frac {x}{2}[/latex]

so ratio of profits = [latex]\frac {x}{4}: \frac {x}{4} : \frac {x}{2} = 1 : 1 : 2[/latex]

so [latex]\frac {1}{4} \times x = 10,000[/latex]

x = 40,000

So

[latex][\frac {x}{(9600 + x)}] \times 19,600 = 10,000[/latex]

Solve, x = 10,000

4x + 6x = 60,000

x = 6,000

So C got 9 × 6000 = 54,000 and A got 4 × 6000 = 24000

So total = 78,000

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