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.

After paying to charity, A's share = [latex] (\frac {95 \times 3}{5}) = Rs. 57[/latex]

If A's share is Rs. 57, total profit = Rs. 100.

If A's share is Rs. 855, total profit = [latex](\frac {100}{57 \times 855}) [/latex] = 1500.

A : B : C = [latex] (x \times 4 + 2x \times 8) : (3x \times 4 + (\frac {3x}{2}) \times 8) : (5x \times 4 + (\frac {5x}{2}) \times 8)[/latex]

20x : 24x : 40x = 5 : 6 : 10

= 70 : 60 : 45

= 14 : 12 : 9

C's rent = Rs.[latex] (\frac {175 \times 9}{35})[/latex]

= Rs. 45

Then, [latex] \frac {(85000 \times 12)} {(42500 \times x)} = 3. [/latex]

Or x = [latex]\frac{(85000 \times 12)}{(42500 \times 3)} = 8[/latex]

So, B joined for 8 months.

A : B :C = [latex]\frac {k}{4} : \frac {k}{6} : \frac {k}{10}[/latex] = 15 : 10 : 6.

Hence, C's share [latex](\frac {4650 \times 6}{31})[/latex] = Rs, 900.

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