Partnership

$${\text{Ratio of shares}}$$
$$ = 1500:2500$$
$$ = 3:5$$
$${\text{Share of first friend}}$$
$$ = {\text{₹}}{\text{.}}\left[ {\frac{{400}}{2} + \left( {400 \times \frac{3}{8}} \right)} \right]$$
$$ = {\text{₹}}.\left( {200 + 150} \right)$$
$$ = {\text{₹}}{\text{. 350}}$$
$${\text{Share of second friend}}$$
$$ = {\text{₹}}{\text{.}}\left[ {\frac{{400}}{2} + \left( {400 \times \frac{5}{8}} \right)} \right]$$
$$ = {\text{₹}}.\left( {200 + 250} \right)$$
$$ = {\text{₹}}{\text{. 450}}$$

Let A's capital = ₹ x
Then, B's capital = ₹ (x - 600)
C's capital = ₹ (x - 600) - 300 = ₹ (x - 900)
∴×+ (x - 600) + (x - 900) = 3000
⇒ 3x = 4500
⇒×= 1500
$${\text{So, A}}:{\text{B}}:{\text{C}}$$
$$ = 1500:900:600$$
$$ = 5:3:2$$
$${\text{Hence, B's share}}$$
$$ = {\text{₹}}{\text{.}}\left( {886 \times \frac{3}{{10}}} \right)$$
$$ = {\text{₹}}{\text{. 265}}{\text{.80}}$$

Let B's investment is used for x months → Now by using formula
$$\frac{{5 \times 8}}{{6 \times {{\text{x}}_2}}} = \frac{5}{9}$$
$${\text{x}} = 12{\text{ months}}$$

A graze 16 cows for 3 months.
B graze 20 cows for 4 months.
C graze 18 cows for 6 months.
D graze 42 cows for 2 months.
So, Ration of Rent
= A : B : C : D
= (16 × 3) : (20 × 4) : (18 × 6) : (42 × 2)
= 48 : 80 : 108 : 84
= 12 : 20 : 27 : 21
we know that,
$${\text{12 unit's}} = {\text{₹}}{\text{. 2400}}$$
$${\text{1 unit}} = \frac{{2400}}{{12}}$$
$$27{\text{ unit's}} = \frac{{2400 \times 27}}{{12}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{₹}}{\text{. }}5400$$

$${\text{Pranit}}:{\text{Rishit}}$$
$$ = \left( {120000 \times 12} \right):\left( {190000 \times 9} \right)$$
$$ = 1440000:1710000$$
$$ = 16:19$$
$$\therefore {\text{Rishit's share}}$$
$$ = {\text{₹}}{\text{.}}\left( {1750000 \times \frac{{19}}{{35}}} \right)$$
$$ = {\text{₹ }}{\text{950000 }}$$

Let C = x
Then B = $$\frac{{\text{x}}}{2}$$
and A = $$\frac{{\text{x}}}{4}$$
A : B : C = 1 : 2 : 4
C's share = $${\text{₹}}{\text{.}}\left( {\frac{4}{7} \times 700} \right) = 400$$

Ratio of their shares
= 22500 : 35000
= 9 : 14
Dhruv's share
$$ = {\text{₹}}{\text{.}}\left( {13800 \times \frac{{14}}{{23}}} \right)$$
$$ = {\text{₹}}{\text{. 8}}400$$

Suppose B joined after×months
then,
21000 × 12 = 36000 × (12 - x)
⇒ 36x = 180
⇒×= 5

Amar : Varun
= 9000 × 12 : (12000 × 6 + 6000 × 6)
= 1 : 1
Varuns share
$$ = {\text{₹}}{\text{.}}\left( {4600 \times \frac{1}{2}} \right)$$
$$ = {\text{₹}}{\text{. 2300}}$$

Suppose Pratap invested ₹ x. Then,
Alok : Pratap = 120000 × 6 :×× 12.
$$\frac{{120000}}{{12{\text{x}}}}:\frac{{6000}}{{3000}}$$
$$ ⇒ {\text{x}} = \frac{{120000}}{{24}}$$
$$ ⇒ {\text{x}} = 5000$$