Partnership

$${\text{Let}}\,{\text{the}}\,{\text{total}}\,{\text{profit}}\,{\text{be}}\,{\text{₹}}{\text{.}}\,{\text{100}}$$
$${\text{After}}\,{\text{paying}}\,{\text{to}}\,{\text{trust,}}\,{\text{A's}}\,{\text{share}}$$
$$ = ₹\,\left( {95 \times \frac{3}{5}} \right) = ₹\,57$$
$${\text{If}}\,{\text{A's}}\,{\text{share}}\,{\text{is}}\,₹\,57,$$
$${\text{Total}}\,{\text{profit}} = ₹\,100$$
$${\text{If}}\,{\text{A's}}\,{\text{share}}\,₹\,855,$$
$${\text{Total}}\,{\text{profit}}$$
$$ = {\frac{{100}}{{57}} \times 855} $$
$$ = 1500$$

For managing, A received = 5% of ₹ 7400 = ₹ 370.
Balance = ₹ (7400 - 370) = ₹ 7030.
Ratio of their investments = (6500×6) : (8400×5) : (10000×3)
= 39000 : 42000 : 30000
= 13 : 14 : 10
$$\therefore {\text{ B's Share}}$$
$$ = {\text{₹}}{\text{. 7030}} \times \frac{{14}}{{37}}$$
$$ = {\text{₹}}{\text{. 2660}}$$

Ratio of initial investments
$$ = {\frac{7}{2}:\frac{4}{3}:\frac{6}{5}} $$
$$ = 105:40:36$$
$$Let the initial investments be 105x, 40x and 36x$$
$$\left( {105x \times 4 + \frac{{150}}{{100}} \times 105x \times 8} \right)$$      $$:$$ $$\left( {40x \times 12} \right)$$   $$:$$ $$\left( {36x \times 12} \right)$$
$$ = 1680x:480x:432x$$
$$ = 35:10:9$$
$${\text{Hence,}}\ {\text{B's}}\  {\text{share}}$$
$$ = {\text{₹}}{\text{.}}\left( {21600 \times \frac{{10}}{{54}}} \right)$$
$$ = {\text{₹}}{\text{.}}\,4000$$

$${\text{Let}}\,C = x $$
$$ {\text{Then,}}\,{\text{B}} = x + 50000\,{\text{and}}$$
$${\text{A}} = x + 50000 + 40000$$
$$\,\,\,\,\,\,\,\, = x + 90000$$
$$So,\,x = x + 50000 = x + 90000 = 500000$$
$$ ⇒3x = 360000$$
$$ ⇒ x = 120000 $$
$$ {\text{A:B:C}} = 210000:170000:120000$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 21:17:12$$
$$\therefore {\text{A's}}\,{\text{share}}$$
$$ = ₹\,\left( {350000 \times \frac{{21}}{{50}}} \right)$$
$$ = ₹\,1,47,000$$

$${\text{Let their investments be}}\,$$
$$₹\,x\,{\text{for}}\,{\text{14}}\,{\text{months}},\,$$
$$₹\,y\,{\text{for}}\,{\text{8}}\,{\text{months}}\,{\text{and}}\,$$
$$₹\,z\,{\text{for}}\,{\text{7}}\,{\text{months}}\,{\text{respectively}}.$$
$${\text{Then}},\,14x:8y:7z = 5:7:8$$
$${\text{Now}},\frac{{14x}}{{8y}} = \frac{5}{7}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,98x = 40y$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y = \frac{{49}}{{20}}x$$
$${\text{And}},\frac{{14x}}{{7z}} = \frac{5}{8}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,112x = 35z$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,z = \frac{{112}}{{35}}x = \frac{{16}}{5}x$$
$$\therefore x:y:z$$
$$ = x:\frac{{49}}{{20}}x:\frac{{16}}{5}x$$
$$ = 20:49:64$$

$${\text{Suppose}}\,{\text{B}}\,{\text{joined}}\,{\text{for}}\,x\,{\text{months}}.$$
$${\text{Then}},$$
$$ {\frac{{85000 \times 12}}{{42500 \times x}} = \frac{3}{1}} $$
$$ ⇒ x = {\frac{{85000 \times 12}}{{42500 \times 3}}} = 8$$
$${\text{So,}}\,{\text{B}}\,{\text{joined}}\,{\text{for}}\,{\text{8}}\,{\text{months}}$$

Chandan : Rani : Joydeep
= (70,000×36) : (1,05,000×30) : (1,40,000×24)
= 12 : 15 : 16

$${\text{Abdul}}\,{\text{:}}\,{\text{Sekh}}\,{\text{:}}\,{\text{Gopal}}$$
$$ = \left( {8000 \times 6} \right):\left( {4000 \times 8} \right):\left( {8000 \times 8} \right)$$
$$ = 48:32:64$$
$$ = 3:2:4$$
$$\therefore {\text{Sekh's}}\,{\text{share}}$$
$$ = ₹\,\left( {4005 \times \frac{2}{9}} \right)$$
$$ = ₹\,890$$

$${\text{Fatima}}\,{\text{:}}\,{\text{Arjun}}$$
$$ = \left( {50000 \times 36} \right):\left( {80000 \times 30} \right)$$
$$ = 3:4$$
$$\therefore {\text{Fatima's}}\,{\text{share}}$$
$$ = ₹\,\left( {24500 \times \frac{3}{7}} \right)$$
$$ = ₹\,10,500$$

Note: If you have clear concept of ratio and proportion chapter then it will really easy for you to solve partnership problems.
P : Q = 85000 : 15000 = 17 : 3
Important to note there that if both have invested for different period of times then we had to multiply with number of months to get the desired ratio.