Answer: Option D. -> Rs. 15000, Rs. 20000
Let amount invested by A = Rs. x
 
  A  
  :  
  B  
Capital →  
x
  :  
(x + 5000)
$$\eqalign{
& {\text{According to the question,}} \cr
& {\text{Share of A in profit}} \cr
& = \frac{{\left( {26000 - 6000} \right)}}{2} \cr
& = {\text{Rs}}{\text{. 10000}} \cr
& {\text{Share of B in profit}} \cr
& = \left( {26000 - 10000} \right) \cr
& = {\text{Rs}}{\text{. 16000}} \cr
& {\text{By using formulas:}} \cr
& \boxed{\frac{{{{\text{C}}_{\text{1}}} \times {{\text{T}}_1}}}{{{{\text{C}}_{\text{2}}} \times {{\text{T}}_{\text{2}}}}} = \frac{{{{\text{P}}_{\text{1}}}}}{{{{\text{P}}_2}}}} \cr
& \Leftrightarrow \frac{{x \times 5}}{{\left( {x + 5000} \right) \times 6}} = \frac{{10000}}{{16000}} \cr
& \Leftrightarrow 4x = 3x + 15000 \cr
& \Leftrightarrow x = {\text{Rs}}.15000 \cr
& {\text{Required capital of A}} \cr
& = {\text{Rs}}{\text{. 15000}} \cr
& {\text{Required capital of B}} \cr
& = \left( {15000 + 5000} \right) \cr
& = {\text{Rs}}{\text{. 20000}} \cr} $$

Answer: Option A. -> Rs. 8712
$$\eqalign{
& {\text{Investment of A for 8 months}} \cr
& = {\text{Rs}}{\text{.15400}} \cr
& {\text{Investment of B for 6 months}} \cr
& = {\text{Rs}}{\text{.18200}} \cr
& {\text{Investment of C for 8 months}} \cr
& = {\text{Rs}}{\text{.12600}} \cr
& {\text{Ratio of the share of A, B and C}} \cr
& = 15400 \times 8:18200 \times 6:12600 \times 8 \cr
& = 154 \times 8:182 \times 6:126 \times 8 \cr
& = 44:39:36 \cr
& {\text{Sum of the terms of ratio}} \cr
& = 44 + 39 + 36 \cr
& = 119 \cr
& \therefore {\text{ Share of C}} \cr
& = {\text{Rs}}{\text{.}}\left( {\frac{{36}}{{119}} \times 28790} \right) \cr
& = {\text{Rs}}{\text{.8710}} \approx {\text{Rs}}{\text{.8712}} \cr} $$

Answer: Option C. -> Rs. 9020
Let B's investment be Rs. x
∴ A's investment be Rs.2x
∴ C's investment be Rs.2x
A invests money for 12 months
B invests money for 8 months
C invests money for 6 months
Ratio of the equivalent capitals of A, B and C for 1 month
$$\eqalign{
& = 2x \times 12:x \times 8:2x \times 6 \cr
& = 6:2:3 \cr} $$
Sum of the terms of ratio
$$6 + 2 + 3 = 11$$
If the total profit at the end of the year be Rs. a
Then share of C
$$\eqalign{
& \Rightarrow \frac{{3a}}{{11}} = 2460 \cr
& \Rightarrow 3a = 2460 \times 11 \cr
& \Rightarrow a = \frac{{2460 \times 11}}{3} \cr
& \Rightarrow a = {\text{Rs}}{\text{.}}\,9020 \cr} $$

Answer: Option C. -> Rs. 1125, Rs. 1875, Rs. 2250
 
  A  
  :  
  B  
  :  
  C  
Capital →  
48000
:
48000
:
48000
Time(year) →
6
 
10
 
12
Profit →
6
:
10
:
12
 
3
:
5
:
6
Note: The capital of the partners are equal so the profit would be divided in the ratio of their time
According to the question,
$$\eqalign{
& \left( {3 + 5 + 6} \right){\text{units}} = {\text{Rs}}{\text{. 5250}} \cr
& {\text{14 units}} = {\text{Rs}}{\text{. 5250}} \cr
& {\text{1 unit}} = {\text{Rs}}{\text{. 375}} \cr
& \therefore {\text{Share of A}} \cr
& = 375 \times 3 \cr
& = {\text{Rs}}{\text{. 1125}} \cr
& {\text{Share of B}} \cr
& = 375 \times 5 \cr
& = {\text{Rs}}{\text{. 1875}} \cr
& {\text{Share of C}} \cr
& = 375 \times 6 \cr
& = {\text{Rs}}{\text{. 2250}} \cr} $$

Answer: Option C. -> Rs. 22500, Rs. 6750, Rs. 5400
Ratio of capital invested by
$${\text{A, B and C}} = 15:10:6$$
Total capital invested by A in 1 year
$$\eqalign{
& = 15x \times 4 + 30x \times 8 \cr
& = 300x \cr} $$
Total capital invested by B in 1 year
$$\eqalign{
& = 10x \times 6 + 5x \times 6 \cr
& = 90x \cr} $$
Total capital invested by C in 1 year
$$\eqalign{
& = 6x \times 12 \cr
& = 72x \cr} $$
Ratio of profits:
A
  :  
B
  :  
C
300x
:
90x
:
72x
50x
:
15x
:
12x
According to the question,
$$ \Leftrightarrow \left( {50x + 15x + 12x} \right)$$     = $${\text{Rs}}{\text{. 34650}}$$
$$\eqalign{
& \Leftrightarrow 77x = {\text{Rs}}.{\text{ }}34650 \cr
& \Leftrightarrow x = {\text{Rs}}{\text{. }}\frac{{34650}}{{77}} \cr
& \Leftrightarrow x = {\text{Rs}}{\text{. }}450 \cr
& {\text{Profit of A}} = {\text{Rs}}{\text{. }}450 \times 50 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 22500}} \cr
& {\text{Profit of B}} = {\text{Rs}}{\text{. }}450 \times 15 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 6750}} \cr
& {\text{Profit of C}} = {\text{Rs}}{\text{. }}450 \times 12 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 5400}} \cr} $$

Answer: Option D. -> Rs. 13560
Total capital invested by A in 1 year
$$\eqalign{
& = 36000 \times 12 \cr
& = {\text{Rs}}{\text{. 432000}} \cr} $$
Total capital invested by B in 1 year
$$ = 45000 \times 4$$   + $$\left( {45000 - 20000} \right) \times 5$$     + $$\left( {55000 + 25000} \right) \times 3$$
$$ = 180000 + 125000 + 240000$$
$$ = {\text{Rs}}{\text{.}}\,{\text{545000}}$$
 
A
  :  
B
Ratio of Capital →  
432000
:
545000
Ratio of Profit →
432
:
545
$$\eqalign{
& {\text{According to the question,}} \cr
& \left( {432 + 545} \right){\text{units}} = {\text{Rs}}{\text{. 117240}} \cr
& {\text{977 units}} = {\text{Rs}}{\text{. 117240}} \cr
& {\text{1 unit}} = {\text{Rs}}{\text{. }}\frac{{117240}}{{977}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{. 120}} \cr
& {\text{Difference in profit}} \cr
& = \left( {545 - 432} \right) \times 120 \cr
& = {\text{ 13560}} \cr} $$
It means B will get Rs. 13560 more than A

Answer: Option A. -> Rs. 180, Rs. 120, Rs. 144
 
  A  
  :  
  B  
  :  
  C  
Capital  
500
:
400
:
800
 
×
:
×
:
×
Time
12
:
10
:
6
Profit  
6000
:
4000
 
4800
 
15
:
10
:
12
$$\eqalign{
& {\text{According to the question,}} \cr
& \left( {15 + 10 + 12} \right){\text{units}} = {\text{Rs}}{\text{.}}\,{\text{444}} \cr
& {\text{37 units}} = {\text{Rs}}{\text{. 444}} \cr
& 1{\text{ unit}} = \frac{{444}}{{37}}{\text{ = Rs}}{\text{. 12}} \cr
& {\text{Profit of A}} = 12 \times 15 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{180}} \cr
& {\text{Profit of B}} = 12 \times 10 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{120}} \cr
& {\text{Profit of C}} = 12 \times 12 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{144}} \cr} $$

Answer: Option B. -> Rs. 370000
 
  A  
  :  
  B  
  :  
  C  
Capital →  
24000
:
32000
:
18000
 
24
:
32
:
18
 
12
:
16
:
9
$$\eqalign{
& {\text{Let the total profit}} = 100x \cr
& {\text{Extra share of A}} \cr
& = 100x \times \frac{{15}}{{100}} \cr
& = 15x \cr
& {\text{Extra share of B}} \cr
& = 100x \times \frac{{12}}{{100}} \cr
& = 12x \cr
& {\text{Remaining profit}} \cr
& = \left[ {100x - \left( {15x + 12x} \right)} \right] \cr
& = 73x \cr} $$
According to the question,
Note: Remaining profit will be distributed in the ratio of their capitals.
∴ Share of C
$$\eqalign{
& \Leftrightarrow \frac{{73x}}{{\left( {12 + 16 + 9} \right)}} \times 9 = {\text{Rs}}{\text{. }}65700 \cr
& \Leftrightarrow \frac{{657x}}{{37}} = {\text{Rs}}{\text{. }}65700 \cr
& \Leftrightarrow x = {\text{Rs}}{\text{. }}\frac{{65700 \times 37}}{{657}} \cr
& \Leftrightarrow x = {\text{Rs}}{\text{. 3}}700 \cr
& {\text{Hence, required profit}} \cr
& = 100x \cr
& = 100 \times 3700 \cr
& = {\text{Rs}}{\text{. 3}}70000 \cr} $$

Answer: Option D. -> Rs. 4800
A invests Rs. 2400 for 12 months
B invests Rs. 3600 for 8 months
And Rs. 3000 for 4 months
C invests Rs. X for 8 months
Ratio of profit of A, B and C
$$ \Rightarrow {\text{Profit of A}}$$   : $${\text{Profit of B}}$$   : $${\text{Profit of C}}$$
$$ \Rightarrow {\text{2400}} \times {\text{12}}$$   : $$\left( {3600 \times 8} \right)$$   + $$\left( {3000 \times 4} \right)$$   : $${\text{X}} \times 8$$
$$\eqalign{
& \Rightarrow 28800:40800:8{\text{X}} \cr
& \Rightarrow 3600:5100:{\text{X}} \cr} $$
Given profit of C = Rs. 8000
And total profit of A, B and C = Rs. 22500
$$\eqalign{
& \therefore \frac{{{\text{X}} \times 22500}}{{3600 + 5100 + {\text{X}}}} = 8000 \cr
& \Rightarrow \frac{{{\text{X}} \times 22500}}{{8700 + {\text{X}}}} = 8000 \cr} $$
$$ \Rightarrow 22500{\text{X}}$$   = $$69600000$$   + $$8000{\text{X}}$$
$$\eqalign{
& \Rightarrow 14500{\text{X}} = 69600000 \cr
& \Rightarrow {\text{X}} = {\text{Rs}}.\,4800 \cr} $$

Answer: Option A. -> Rs. 3600, Rs. 6600, Rs. 6750
Total capital invested by A in 1 year
$$\eqalign{
& = 12 \times 4000 \cr
& = {\text{Rs}}{\text{. 48000}} \cr} $$
Total capital invested by B in 1 year
$$\eqalign{
& = 4 \times 6000 + 8 \times 8000 \cr
& = 24000 + 64000 \cr
& = {\text{Rs}}{\text{.}}\,{\text{88000}} \cr} $$
Total capital invested by C in 1 year
$$\eqalign{
& = 9 \times 8000 + 3 \times 6000 \cr
& = 72000 + 18000 \cr
& = {\text{Rs}}{\text{. 90000}} \cr} $$
 
  A  
  :  
  B  
  :  
  C  
Capital  
48000
:
88000
:
90000
 
24
:
44
:
45
According to the question,
$$\left( {24 + 44 + 45} \right){\text{units}}$$     = $${\text{Rs}}{\text{.}}\,{\text{16950}}$$
$$\eqalign{
& {\text{113 units}} = {\text{Rs}}{\text{. 16950}} \cr
& 1{\text{ unit}} = \frac{{16950}}{{113}}{\text{ = Rs}}{\text{. 150}} \cr
& {\text{Hence,}} \cr
& {\text{Profit of A}} = 150 \times 24 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{3600}} \cr
& {\text{Profit of B}} = 150 \times 44 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{6600}} \cr
& {\text{Profit of C}} = 150 \times 45 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,{\text{6750}} \cr} $$