Partnership

Let the money invested by A, B and C be ₹ x, ₹ 2x and ₹ 3x respectively
Then, A : B : C
$$ = \left( {x \times 12} \right):\left( {2x \times 6} \right):\left( {3x \times 4} \right)$$
$$ = 12x:12x:12x$$
$$ = 1:1:1$$
$$\therefore {\text{C's share}}$$
$$ = {\text{₹}}{\text{.}}\left( {27000 \times \frac{1}{3}} \right)$$
$$ = {\text{₹ }}{\text{9000}}$$

Let the capitals of Y was used for T months
we know that,
$$\frac{{5 \times 8}}{{6 \times {\text{T}}}}{\text{ = }}\frac{5}{9}$$
$$ ⇒ {\text{T}} = {\text{12 months}}$$
Hence capital of Y was used for = 12 months

30% of the total profit is divided in the ratio 50000 : 70000 = 5 : 7 between the two partners.
Suppose×= 30% of the total profit. Then
$$\eqalign{ &×\times \frac{{7 - 5}}{{7 + 5}} = 90 \cr &×\times \frac{1}{6} = 90 \cr &×= 540$$
$$30\% \,{\text{of}}\,{\text{the}}\,{\text{total}}\,{\text{profit}} = 540$$
$${\text{Total Profit}}$$
$$ = \frac{{540 \times 100}}{{30}}$$
$$ = 1800$$
Alternative
From the 30% of the profit, let first partner gets×and second partner gets (x + 90)
$$x:(x + 90) = 50000:70000$$
$$x:(x + 90) = 5:7$$
$$7x = 5x + 450$$
$$2x = 450 \cr &×= 225$$
$$\therefore 30\% \,{\text{of}}\,{\text{the}}\,{\text{profit}}$$
$$ = x + {\text{x}} + 90$$
$$ = 225 + 225 + 90$$
$$ = 540$$
$${\text{Total Profit}} = \frac{{540 \times 100}}{{30}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1800$$

A : B : C
(40000 × 36) :(80000 × 12 + 40000 × 24) : (120000 × 24 + 40000 × 12)
= 144 : 192 : 336
= 3 : 4 : 7

$${\text{Abdul}}:{\text{Madhu}}$$
$$ = \left( {38000 \times 12} \right):\left( {55000 \times 7} \right)$$
$$ = 456000:385000$$
$$ = 456:385$$
$$\therefore {\text{Required difference}}$$
$$ = {\text{₹}}{\text{.}}\,\left[ {22000 \times \left( {\frac{{456}}{{841}} - \frac{{385}}{{841}}} \right)} \right]$$
$$ = {\text{₹}}{\text{.}}\left( {22000 \times \frac{{71}}{{841}}} \right)$$
$$ = {\text{₹ }}{\text{1857}}{\text{.31}}$$
$$ = {\text{₹ }}{\text{1857 }}$$

$${\text{Gautam}}:{\text{Jatin}}$$
$$ = \left( {6000 \times 12} \right):\left( {35000 \times 8} \right)$$
$$ = 720000:280000$$
$$ = 18:7$$

$${\text{Fatima}}:{\text{Arjun}}$$
$$ = \left( {50000 \times 36} \right):\left( {80000 \times 30} \right)$$
$$ = 1800000:2400000$$
$$ = 3:4$$
$$\therefore \,{\text{Fatima's share}}$$
$$ = {\text{₹}}{\text{.}}\left( {24500 \times \frac{3}{7}} \right)$$
$$ = {\text{₹ }}{\text{10500 }}$$

$${\text{Pratap}}:{\text{Ram}}:{\text{Harshad}}$$
$$ = 2700:8100:7200$$
$$ = 3:9:8$$
$${\text{Let the total profit be ₹}}.\,x$$
$${\text{Then, Ram's share}}$$
$$ = {\text{₹}}{\text{.}}\,\left( {\frac{9}{{20}}x} \right)$$
$$\therefore \frac{9}{{20}}x = 3600$$
$$ ⇒ x = \left( {\frac{{3600 \times 20}}{9}} \right)$$
$$ ⇒ x = 8000$$
$${\text{Hence,}}$$
$${\text{Total profit}} = {\text{₹}}{\text{. 8000}}$$

Chandan : Rani : Joydeep
= (70000 × 36) : (105000 × 30) : (140000 × 24)
= 12 : 15 : 16