Answer: Option A. -> 4 : 3 : 2
Let their investments be Rs. 3x for p months
Rs. 4x for q months and Rs. 6x for r months respectively
Then,
$$\eqalign{
& 3xp:4xq:6xr = 1:1:1 \cr
& \Rightarrow 3p:4q:6r = 1:1:1 \cr
& {\text{So,}}3p = 4q \cr
& \Leftrightarrow q = \frac{{3p}}{4} \cr
& {\text{And, }}4q = 6r \cr
& \Leftrightarrow r = \frac{{2q}}{3} = \left( {\frac{2}{3} \times \frac{3}{4}p} \right) = \frac{p}{2} \cr
& \therefore p:q:r \cr
& = p:\frac{{3p}}{4}:\frac{p}{2} \cr
& = 4:3:2 \cr} $$

Answer: Option D. -> None of these
Suppose,
Swati invested Rs. 5x for 7 months
And
Rajni invested Rs. 6x for y months
Then,
$$\eqalign{
& \Rightarrow \frac{{5x \times 7}}{{6x \times y}} = \frac{5}{9} \cr
& \Rightarrow 30y = 315 \cr
& \Rightarrow y = 10\frac{1}{2} \cr} $$
Hence, Rajni's capital was used for $${\text{10}}\frac{1}{2}$$ months

Answer: Option C. -> Rs. 210000
Let the total investment be Rs. z
Then,
$$\eqalign{
& 20\% {\text{ of }}z = 98000 \cr
& \Leftrightarrow z = \left( {\frac{{98000 \times 100}}{{20}}} \right) \cr
& \Leftrightarrow z = 490000 \cr} $$
Let the capital of P, Q and R be
Rs. 5x, Rs. 6x and Rs. 6x respectively
Then,
$$ \Leftrightarrow \left( {5x \times 12} \right)$$   + $$\left( {6x \times 12} \right)$$   + $$\left( {6x \times 6} \right)$$   = $$490000 \times 12$$
$$\eqalign{
& \Leftrightarrow 168x = 490000 \times 12 \cr
& \Leftrightarrow x = \left( {\frac{{490000 \times 12}}{{168}}} \right) \cr
& \Leftrightarrow x = 35000 \cr
& \therefore {\text{R's investment}} \cr
& = {\text{Rs}}{\text{. }}6x \cr
& = {\text{Rs}}{\text{.}}\left( {6 \times 35000} \right) \cr
& = {\text{Rs}}{\text{. 210000}} \cr} $$

Answer: Option B. -> 3 months
$$\eqalign{
& {\text{Let the total profit be Rs}}{\text{.}}\,z \cr
& {\text{Then,}} \cr
& {\text{Y's share}} = {\text{Rs}}{\text{.}}\,\frac{{2z}}{5} \cr
& {\text{X's share}} = {\text{Rs}}{\text{.}}\left( {z - \frac{{2z}}{5}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}.\,\frac{{3z}}{5} \cr
& \therefore {\text{X}}:{\text{Y}} \cr
& = \frac{{3z}}{5}:\frac{{2z}}{5} \cr
& = 3:2 \cr} $$
Let the total capital be Rs. x and
Suppose Y's money was used for y months
Then,
$$\eqalign{
& \Rightarrow \frac{{\frac{1}{3}x \times 9}}{{\frac{2}{3}x \times y}} = \frac{3}{2} \cr
& \Rightarrow 18x = 6xy \cr
& \Rightarrow y = 3 \cr} $$
Hence, Y's money was used for 3 months.

Answer: Option A. -> Rs. 1900
 
  M  
  :  
  P  
  :  
  Q  
Capital →  
6500
:
8400
:
10000
 
65
:
84
:
100
Time →
×6
:
×5
:
×3
 
390
:
420
:
300
Profit →
13
:
14
:
10
M's extra share on work in partner
$$\eqalign{
& = {\text{Rs}}.7400 \times \frac{5}{{100}} \cr
& = {\text{Rs}}{\text{.370 }} \cr
& {\text{Remaining profit}} \cr
& = {\text{Rs}}{\text{.}}\left( {{\text{7400}} - {\text{370}}} \right) \cr
& = {\text{Rs}}.\,7030 \cr} $$
According to the question,
(13 + 14 + 10) units = Rs. 7030
$$\eqalign{
& {\text{37 units}} = {\text{Rs}}{\text{. 7030}} \cr
& {\text{1 unit}} = {\text{Rs}}{\text{.}}\frac{{7030}}{{37}} \cr
& {\text{Profit of Q}} = {\text{10 units}} \cr
& = {\text{Rs}}.\frac{{7030}}{{37}} \times 10 \cr
& = {\text{Rs}}.\,1900 \cr} $$

Answer: Option A. -> 12 : 7 : 5
A invests money for 12 months
B invests money for 10 months
C invests money for 8 months
Ratio of profit of A to B to C
= 28000 × 12 : 20000 × 8 + 18000 × 2 : 18000 × 6 + 16000 × 2
= 28 × 12 × 1000 : (160 + 36) × 1000 : (108 + 32) × 1000
= 28 × 12 : 160 + 36 : 108 + 32
= 336 : 196 : 140
= 12 : 7 : 5

Answer: Option B. -> 8 : 9 : 10
 
  A  
  :  
  B  
  :  
  C  
Capital →  
50000
:
75000
:
125000
Time(year) →
2
 
$$\frac{3}{2}$$
 
1
Profit →
100
:
$${\frac{{75 \times 3}}{2}}$$
:
125
 
8
:
9
:
10
∴ Required ratio of profit = 8 : 9 : 10

Answer: Option B. -> Rs. 890
Ratio of profit of Anil : Kamal : Vini
(8000 × 6) : (4000 × 8) : (8000 × 8)
= 48000 : 32000 : 64000
= 48 : 32 : 64
= 3 : 2 : 4
$$\eqalign{
& \therefore {\text{Kamal's share}} \cr
& = {\text{Rs}}{\text{.}}\left( {4005 \times \frac{2}{9}} \right) \cr
& = {\text{Rs}}{\text{.}}\,{\text{890}} \cr} $$

Answer: Option A. -> Rs. 50000
Let the initial investment of A and B is 18x and 7x
After 4 months from the start of business,
A invest Rs. 2000 more for each eight months.
Then total investment of A
$$\eqalign{
& = 18x \times 4 + \left( {18x + 2000} \right) \times 8 \cr
& = 72x + 144x + 16000 \cr
& = 216x + 16000 \cr} $$
After 4 months, from the start of business,
B invest Rs. 7000 more for each eight months.
Total investment by B
$$\eqalign{
& = 7x \times 4 + \left( {7x + 7000} \right) \times 8 \cr
& = 28x + 56x + 56000 \cr
& = 84x + 56000 \cr} $$
According to the question,
$$ \Rightarrow \frac{{216x + 16000}}{{84x + 56000}} = \frac{2}{1}$$
⇒ 216x + 16000 = 168x + 112000
⇒ 216x - 168x = 112000 - 16000
⇒ 48x = 96000
⇒ x = 2000
Total initial investment of A and B
= (18 + 7) × 2000
= Rs. 50000