Answer: Option C. -> Rs. 15000
Total profit = Rs. 4000
According to the question,
$$\eqalign{
& {\text{20% of B's capital}} = {\text{Rs}}{\text{.}}\,{\text{4000}} \cr
& {\text{1% of B's capital}} = {\text{Rs}}{\text{.}}\,\frac{{{\text{4000}}}}{{20}} \cr
& {\text{B's total capital}} \cr
& = {\text{Rs}}{\text{. }}\frac{{{\text{4000}}}}{{20}} \times 100 \cr
& = {\text{Rs}}{\text{. }}20000 \cr} $$
Let total capital required for business = 100 units.
 
  A  
  :  
  B  
  :  
  C  
Capital  
30
:
40
:
30
 
× 500
:
× 500
:
× 500
 
15000
:
20000
:
$$\boxed{15000}$$
Hence, Required capital for C = Rs. 15000

Answer: Option B. -> Rs. 450
$$\eqalign{
& {\text{Given share of }}{{\text{2}}^{{\text{nd}}}} \cr
& = \frac{5}{{13}}{\text{of}}\left( {{{\text{1}}^{{\text{st}}}} + {{\text{3}}^{{\text{rd}}}}} \right) \cr
& {\text{or, }}\frac{{{{\text{2}}^{{\text{nd}}}}}}{{{{\text{1}}^{{\text{st}}}}{\text{ + }}{{\text{3}}^{{\text{rd}}}}}} = \frac{5}{{13}} \cr
& \therefore {1^{{\text{st}}}} + {{\text{2}}^{{\text{nd}}}} + {{\text{3}}^{{\text{rd}}}} = 13 + 5 = 18 \cr
& \because 18{\text{units}} = 1620 \cr
& \therefore {\text{1 unit}} = \frac{{1620}}{{18}} \cr
& \therefore 5{\text{ units}} = \frac{{1620}}{{18}} \times {\text{5}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4{\text{50}} \cr
& {\text{Hence share of }}{{\text{2}}^{{\text{nd}}}} = {\text{Rs}}{\text{.}}\,{\text{450}} \cr} $$

Answer: Option A. -> Rs. 3535, Rs. 6666
Note : In such type of question we can assume ratio as per our need to avoid fraction
Capital →  
A7 × 3
  :  
B9 × 3
New Ratio, →
A21x
:
B27x
Total capital invested by A in 9 months
$$\eqalign{
& = 21x \times 3 + 7x \times 6 \cr
& = 105x \cr} $$
Total capital of B invested in 9 months
$$\eqalign{
& = 27x \times 4 + 18x \times 5 \cr
& = 198x \cr} $$
$$\eqalign{
& {\text{According to the question,}} \cr
& \left( {105x + 198x} \right) = {\text{Rs}}{\text{. 10201}} \cr
& 303x = {\text{Rs}}{\text{. 10201}} \cr
& x = \frac{{10201}}{{303}} \cr
& {\text{Hence,}} \cr
& {\text{Share of A}} = 105 \times \frac{{10201}}{{303}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,3535 \cr
& {\text{Share of B}} = 198 \times \frac{{10201}}{{303}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{6666}} \cr} $$

Answer: Option B. -> Rs. 1200
$$\eqalign{
& {\text{investment of A}} = 1000 \cr
& {\text{So, investment of B}} + {\text{C}} \cr
& = {\text{6000}} - 1000 \cr
& = 5000 \cr
& {\text{B}}:{\text{C}} = 5000 \cr
& 2:3 = 2000:3000 \cr} $$
So,
A
  :  
B
  :  
C
1000
:
2000
:
3000
1
:
2
:
3 = 6
$$\eqalign{
& \therefore {\text{Profit of C}} = \frac{3}{6} \times 2400 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1200 \cr} $$

Answer: Option B. -> 5 : 6 : 10
Let their initial investments be x, 3x and 5x respectively
Then,
$$ = {\text{A}}:{\text{B}}:{\text{C}}$$
$$ = \left( {x \times 4 + 2x \times 8} \right)$$     : $$\left( {3x \times 4 + \frac{{3x}}{2} \times 8} \right)$$     : $$\left( {5x \times 4 + \frac{{5x}}{2} \times 8} \right)$$
$$\eqalign{
& = 20x:24x:40x \cr
& = 5:6:10 \cr} $$

Answer: Option B. -> Rs. 800
$$\eqalign{
& {\text{Suppose, }} \cr
& {\text{A invests Rs}}{\text{.}}\frac{x}{6}{\text{ for }}\frac{y}{6}{\text{ months}} \cr
& {\text{Then,}} \cr
& {\text{ B invests Rs}}{\text{.}}\frac{x}{3}{\text{ for }}\frac{y}{3}{\text{ months}} \cr
& {\text{C invests}}\left[ {x - \left( {\frac{x}{6} + \frac{x}{3}} \right)} \right]i.e.,{\text{ Rs}}{\text{.}}\frac{x}{2}{\text{ for }}y{\text{ months}} \cr
& \therefore {\text{A}}:{\text{B}}:{\text{C}} \cr
& {\text{ = }}\left( {\frac{x}{6} \times \frac{y}{6}} \right):\left( {\frac{x}{3} \times \frac{y}{3}} \right):\left( {\frac{x}{2} \times y} \right) \cr
& = \frac{1}{{36}}:\frac{1}{9}:\frac{1}{2} \cr
& = 1:4:18 \cr
& {\text{Hence, B's share}} \cr
& = {\text{Rs}}{\text{. }}\left( {4600 \times \frac{4}{{23}}} \right) \cr
& = {\text{Rs}}{\text{. 800}} \cr} $$

Answer: Option B. -> Rs. 16800
 
A
  B  
  C
Amounts invested  
14000  
 
 
Time (in months)
12
  7
  5
 
168000  
 
 
$$\eqalign{
& {\text{Ratio of profits }}4:3:2 \cr
& {\text{Let their profits }}4x:3x:2x \cr
& \Leftrightarrow 4x = 168000 \cr
& \Leftrightarrow x = 42000 \cr
& \Rightarrow {\text{Profit share of C}} \cr
& = 2x \cr
& = 2 \times 42000 \cr
& = {\text{Rs}}{\text{. 84000}} \cr
& \Rightarrow {\text{Capital invested by C}} \cr
& = \frac{{84000}}{5} \cr
& = {\text{Rs. }}16800 \cr} $$

Answer: Option C. -> Rs. 4280
Let C's capital be Rs. x
Then,
A : B : C = 2560 : 2000 : x
$$\eqalign{
& {\text{A's share}} \cr
& = {\text{Rs}}{\text{.}}\left( {1105 \times \frac{{2560}}{{4560 + x}}} \right) \cr
& \therefore 1105 \times \frac{{2560}}{{4560 + x}} = 320 \cr
& \Rightarrow 320x + 1459200 = 2828800 \cr
& \Rightarrow 320x = 1369600 \cr
& \Rightarrow x = 4280 \cr} $$

Answer: Option B. -> Rs. 260
Let the total capital of A, B and C = 15 units
Let total time for investment = 12 units
Now, According to the question,
Capital Investment of A = $$\frac{1}{3} \times 15$$   = 5units
Capital Investment of B = $$\frac{1}{5} \times 15$$   = 3units
Capital Investment of C = 15 - (5 + 3) = 7
A's Capital Invested for time = $$\frac{1}{4} \times 12$$   = 3units
B's Capital Invested for time = $$\frac{1}{6} \times 12$$   = 2units
C's Capital Invested all the time. i.e = 12units
∴ Profit ratio of A : B : C = (5 × 3) : (3 × 2) : (7 × 12)
= 15 : 6 : 84
= 5 : 2 : 28
Total profit = 5 + 2 + 28 = 35 units
Also, total profit = Rs. 1820 (given)
$$\eqalign{
& {\text{35 units}} = {\text{Rs}}{\text{. 1820}} \cr
& {\text{1 unit}} = \frac{{1820}}{{35}} = {\text{Rs}}.52 \cr
& {\text{Hence A's share in profit}} \cr
& = 5{\text{ units}} \cr
& = 52 \times 5 \cr
& = {\text{Rs}}{\text{. 260}} \cr} $$

Answer: Option A. -> 3
Clearly, A invested his capital for 12 months
while each one of B and C invested his capital for (12 - x) months
Ratio of profits os A, B, C
$$ = \left( {50000 \times 12} \right)$$   : $$\left[ {60000 \times \left( {12 - x} \right)} \right]$$     : $$\left[ {70000 \times \left( {12 - x} \right)} \right]$$
$$\eqalign{
& = 60:6\left( {12 - x} \right):7\left( {12 - x} \right) \cr
& {\text{But ratio of profits}} \cr
& = 20:18:21 \cr
& = 60:54:63 \cr} $$
$$\therefore 60:\left( {72 - 6x} \right):\left( {84 - 7x} \right)$$     = $$60$$ : $$54$$ : $$63$$
$$\eqalign{
& {\text{So, }}72 - 6x = 54 \cr
& \Rightarrow 6x = 18 \cr
& \Rightarrow x = 3 \cr} $$