Partnership

$${\text{Let C}} =× $$
$$ {\text{Then, B}} = x + 5000{\text{ and}}$$
$${\text{A}} = x + 5000 + 4000 = x + 9000$$
$${\text{So, }}$$
$$ ⇒ x = x + 5000 = x + 9000 = 50000$$
$$ ⇒ 3x = 36000$$
$$ ⇒ x = 12000$$
$$\therefore {\text{A}}:{\text{B}}:{\text{C}}$$
$$ = 21000:17000:12000$$
$$ = 21:17:12$$
$$\therefore {\text{A's share}}$$
$$ = {\text{₹}}{\text{.}}\left( {35000 \times \frac{{21}}{{50}}} \right)$$
$$ = {\text{₹}}{\text{14700}}$$

Ratio of investments of A, B and C = Ratio of their profit's
$$ = 337.50:1125:637.50$$
$$ = 9:30:17$$
$$\therefore {\text{A's investment}}$$
$$ = {\text{₹}}{\text{.}}\left( {14000 \times \frac{9}{{56}}} \right)$$
$$ = {\text{₹ }}{\text{2250}}$$
$${\text{B's investment}}$$
$$ = {\text{₹}}{\text{.}}\left( {14000 \times \frac{{30}}{{56}}} \right)$$
$$ = {\text{₹ }}{\text{7500}}$$
$${\text{Hence, required difference}}$$
$$ = {\text{₹}}{\text{.}}\left( {7500 - 2250} \right)$$
$$ = {\text{₹ }}{\text{5250}}$$

$${\text{A}}:{\text{B}}$$
$$4:{\text{ }}5$$
$${\text{we know that,}}$$
$$\left( {4 + 5} \right){\text{unit's}} = {\text{₹}}{\text{. 14130}}$$
$${\text{1 unit}} = \frac{{14130}}{9} = {\text{₹}}{\text{. }}1570$$
$$5{\text{ unit's}} = 5 \times 1570 = {\text{₹}}{\text{. 7850}}$$
$$\therefore {\text{Hence share of B}} = {\text{₹}}{\text{. 7850}}$$

Initial ratio of investments by A and B = 2 : 3
Let their respective investments are 2x and 3x
we know that,
If A added ₹ 10000 to his investment
Then the new ratio = 3 : 2
$$ ⇒ \frac{{2x + 10000}}{{3x}} = \frac{3}{2}$$
$$ ⇒ 4x + 20000 = 9x$$
$$ ⇒ 5x = 20000$$
$$ ⇒ x = 4000$$
Original investment by A = 2 × 4000 = ₹ 8000

Let the initial investments of A, B and C be ₹ 5x, ₹ 6x and ₹ 8x respectively
Then, A : B : C
[5x × 12 + (160% of 5x) × 12] : (6x × 24) : (8x × 12 + 4x × 12)
= 156x : 144x : 144x
= 13 : 12 : 12

$${\text{Let Ninad's investment be ₹}}{\text{.}}x$$
$${\text{Then, ratio of capitals}}$$
$$ = \left( {x \times 12} \right):\left( {2x \times 6} \right):\left( {3x \times 4} \right)$$
$$ = 12x:12x:12x$$
$$ = 1:1:1$$
$$\therefore {\text{Himesh's share}}$$
$$ = {\text{₹}}{\text{.}}\left( {45000 \times \frac{1}{3}} \right)$$
$$ = {\text{₹}}{\text{.15000}}$$

$$ = {\text{A}}:{\text{B}}:{\text{C}}:{\text{D}}$$
$$ = 15 \times 4:12 \times 2:18 \times 6:16 \times 5$$
$$ = 60:24:108:80$$
$$ = 15:6:27:20$$
$${\text{Let the total rent be ₹}}{\text{.}}x$$
$${\text{Then,}}$$
$${\text{A's share}} = {\text{₹}}{\text{.}}\left( {\frac{{15x}}{{68}}} \right)$$
$$\therefore \,\frac{{15x}}{{68}} = 1020$$
$$ ⇒ x = \left( {\frac{{1020 \times 68}}{{15}}} \right)$$
$$ ⇒ x = 4624$$
$${\text{Hence, C's share}}$$
$$ = {\text{₹}}{\text{.}}\left( {4624 \times \frac{{27}}{{68}}} \right)$$
$$ = {\text{₹}}{\text{.}}\,{\text{1836}}$$

$${\text{Let total profit}} = {\text{16 unit's }}$$
$${\text{we know that,}}$$
$${\text{Profit share of A}}$$
$$ = \frac{3}{{16}} \times 16{\text{ unit's}}$$
$$ = 3{\text{ unit's}}$$
$${\text{profit share of B }}$$
$${\text{ = }}\frac{1}{4} \times 16 = 4{\text{ unit's}}$$
$${\text{then profit share of C}}$$
$$ = \left[ {16 - \left( {4 + 3} \right)} \right] = 9{\text{ unit's}}$$
$${\text{But profit of C}} = {\text{₹}}{\text{. 243 (given)}}$$
$${\text{9 unit's}} = {\text{₹}}{\text{. 243}}$$
$${\text{1 unit}} = {\text{₹}}{\text{. 27}}$$
$${\text{Profit share of B}}$$
$$ = 4{\text{ unit's}}$$
$$ = 27 \times 4$$
$$ = {\text{₹}}{\text{. 108}}$$

Total profit = ₹ 880
Since A gets 15% of total profit for management
∴ Remaining profit
$$ = 880 - \frac{{880 \times 15}}{{100}}$$
$$ = {\text{₹}}.\,748$$

Ratio of initial investments
$$ = \frac{1}{3}:\frac{1}{4}:\frac{1}{5}$$
$$ = 20:15:12$$
Let their initial investments be 20x, 15x and 12x respectively
= A : B : C
= (20x × 15 + 10x × 15) : (15x × 30) : (12x × 30)
= 450x : 450x : 360x
= 5 : 5 : 4
$$\therefore {\text{C's share}}$$
$$ = {\text{₹}}{\text{.}}\left( {4340 \times \frac{4}{{14}}} \right)$$
$$ = {\text{₹}}{\text{.1240}}$$