1. A, B and C invested money in the ratio of $$\frac{1}{2}:\frac{1}{3}:\frac{1}{5}$$ in a business. After 4 months A doubled his investment and after 6 months B halves his investment. If the total profit at the end of the year be Rs. 34650 then find the share of each in profit ?
a) Rs. 20000, Rs. 25000, Rs. 18000
b) Rs. 15500, Rs. 27200, Rs. 20450
c) Rs. 22500, Rs. 6750, Rs. 5400
d) Rs. 10350, Rs. 21540, Rs. 12050
Explanation: 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 |
$$ \left( {50x + 15x + 12x} \right)$$ = $${\text{Rs}}{\text{. 34650}}$$
$$\eqalign{ & 77x = {\text{Rs}}.{\text{ }}34650 \cr & x = {\text{Rs}}{\text{. }}\frac{{34650}}{{77}} \cr & x = {\text{Rs}}{\text{. }}450 \cr & {\text{Profit of A}} = {\text{Rs}}{\text{. }}450 \times 50 = {\text{Rs}}{\text{. 22500}} \cr & {\text{Profit of B}} = {\text{Rs}}{\text{. }}450 \times 15 = {\text{Rs}}{\text{. 6750}} \cr & {\text{Profit of C}} = {\text{Rs}}{\text{. }}450 \times 12 = {\text{Rs}}{\text{. 5400}} \cr} $$
2. A and B started a business by investing Rs. 36000 and Rs. 45000 respectively. After 4 months B withdraws $$\frac{4}{9}$$ of his investment. Its 5 months after she again invested $$\frac{{11}}{9}$$ of its original investment. If the total earned profit at the end of the year, is Rs. 117240, then who will get more money as a share of profit and how much ?
a) Rs. 15500
b) Rs. 12450
c) Rs. 14245
d) Rs. 13560
Explanation: 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 |
$$\eqalign{ & \left( {432 + 545} \right){\text{units}} = {\text{Rs}}{\text{. 117240}} \cr & {\text{977 units}} = {\text{Rs}}{\text{. 117240}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{. }}\frac{{117240}}{{977}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{. 120}} \cr & {\text{Difference in profit}} = \left( {545 - 432} \right) \times 120 \cr & = {\text{ 13560}} \cr} $$
It means B will get Rs. 13560 more than A
3. A, B and C started a business by investing Rs. 24000, Rs. 32000 and Rs. 18000 respectively. A and B are active partners and get 15% and 12% of total profit and remaining profit is to be distributed among them in the ratio of their investment. If C got total Rs. 65700 as a profit, what was the total amount of profit ?
a) Rs. 470000
b) Rs. 370000
c) Rs. 345000
d) Rs. 157000
Explanation:
| A | : | B | : | C | |
| Capital → | 24000 | : | 32000 | : | 18000 |
| 24 | : | 32 | : | 18 | |
| 12 | : | 16 | : | 9 |
$$\eqalign{ & {\text{Let the total profit}} = 100x \cr & {\text{Extra share of A}} = 100x \times \frac{{15}}{{100}} \cr & = 15x \cr & {\text{Extra share of B}} = 100x \times \frac{{12}}{{100}} \cr & = 12x \cr & {\text{Remaining profit}} = \left[ {100x - \left( {15x + 12x} \right)} \right] \cr & = 73x \cr} $$
Note: Remaining profit will be distributed in the ratio of their capitals.
∴ Share of C
$$\eqalign{ & \frac{{73x}}{{\left( {12 + 16 + 9} \right)}} \times 9 = {\text{Rs}}{\text{. }}65700 \cr & \frac{{657x}}{{37}} = {\text{Rs}}{\text{. }}65700 \cr & x = {\text{Rs}}{\text{. }}\frac{{65700 \times 37}}{{657}} \cr & x = {\text{Rs}}{\text{. 3}}700 \cr & {\text{Required profit}} = 100x \cr & = 100 \times 3700 \cr & = {\text{Rs}}{\text{. 3}}70000 \cr} $$
4. A started a business with the capital of Rs. 500. After 2 months B joined A with Rs. 400. 6 months after the business started C joined with Rs. 800. If the total profit earned at the end of the year is Rs. 444 find the share of their profit ?
a) Rs. 180, Rs. 120, Rs. 144
b) Rs. 150, Rs. 130, Rs. 123
c) Rs. 160, Rs. 141, Rs. 125
d) Rs. 141, Rs. 110, Rs. 140
Explanation:
| A | : | B | : | C | |
| Capital | 500 | : | 400 | : | 800 |
| × | : | × | : | × | |
| Time | 12 | : | 10 | : | 6 |
| Profit | 6000 | : | 4000 | 4800 | |
| 15 | : | 10 | : | 12 |
$$\eqalign{ & \left( {15 + 10 + 12} \right){\text{units}} = {\text{Rs}}{\text{.}}\,{\text{444}} \cr & {\text{37 units}} = {\text{Rs}}{\text{. 444}} \cr & 1{\text{ unit}} = \frac{{444}}{{37}}{\text{ = Rs}}{\text{. 12}} \cr & {\text{Profit of A}} = 12 \times 15 = {\text{Rs}}{\text{.}}\,{\text{180}} \cr & {\text{Profit of B}} = 12 \times 10 = {\text{Rs}}{\text{.}}\,{\text{120}} \cr & {\text{Profit of C}} = 12 \times 12 = {\text{Rs}}{\text{.}}\,{\text{144}} \cr} $$
5. A and B started a business by investing Rs. 2400 and Rs. 3600 respectively. At the end 4th months from the stat of the business, C joined with Rs. X. After 8 months from the start of the business, B withdrew Rs. 600. If C's share is Rs. 8000 in the annual profit of Rs. 22500, what was the amount C invested in the business ?
a) Rs. 7200
b) Rs. 5800
c) Rs. 4000
d) Rs. 4800
Explanation: 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
$$\eqalign{ & \frac{{{\text{X}} \times 22500}}{{3600 + 5100 + {\text{X}}}} = 8000 \cr & \frac{{{\text{X}} \times 22500}}{{8700 + {\text{X}}}} = 8000 \cr} $$
$$ 22500{\text{X}}$$ = $$69600000$$ + $$8000{\text{X}}$$
$$\eqalign{ & 14500{\text{X}} = 69600000 \cr & {\text{X}} = {\text{Rs}}.\,4800 \cr} $$
6. M, P and Q together started a business. M invested Rs. 6500 for 6 months, P invested Rs. 8400 for 5 months and Q invested Rs. 10000 for 3 months. M is working member for which he gets 5% of total profit extra. If the total gain is Rs. 7400, then Q's share is ?
a) Rs. 1900
b) Rs. 2100
c) Rs. 3200
d) Data are incomplete
Explanation:
| 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
$$\eqalign{ & = {\text{Rs}}.7400 \times \frac{5}{{100}} \cr & = {\text{Rs}}{\text{.370 }} \cr & {\text{Remaining profit}} = {\text{Rs}}{\text{.}}\left( {{\text{7400}} - {\text{370}}} \right) \cr & = {\text{Rs}}.\,7030 \cr} $$
(13 + 14 + 10) units = Rs. 7030
$$\eqalign{ & {\text{37 units}} = {\text{Rs}}{\text{. 7030}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{.}}\frac{{7030}}{{37}} \cr & {\text{Profit of Q}} = {\text{10 units}} \cr & = {\text{Rs}}.\frac{{7030}}{{37}} \times 10 \cr & = {\text{Rs}}.\,1900 \cr} $$
7. A started a business by investing Rs. 50000. After 6 months B joined her by investing Rs. 75000. After its 6 months C joined with Rs. 125000. What is the ratio of profit share after 2 year among A, B and C ?
a) 4 : 5 : 6
b) 8 : 9 : 10
c) 8 : 9 : 12
d) 4 : 5 : 8
Explanation:
| 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
8. A and B started a business with initial investments in the respective ratio of 18 : 7. After 4 months from the start of the business, A invested Rs. 2000 more and B invested Rs. 7000 more. At the end of one year, if the profit was distributed among them in the ratio of 2 : 1 respectively, what was the total initial investment with which A and B started the business ?
a) Rs. 50000
b) Rs. 25000
c) Rs. 150000
d) Rs. 75000
Explanation: 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
$$\eqalign{ & = 7x \times 4 + \left( {7x + 7000} \right) \times 8 \cr & = 28x + 56x + 56000 \cr & = 84x + 56000 \cr} $$
$$ \Rightarrow \frac{{216x + 16000}}{{84x + 56000}} = \frac{2}{1}$$
216x + 16000 = 168x + 112000
216x - 168x = 112000 - 16000
48x = 96000
x = 2000
Total initial investment of A and B
= (18 + 7) × 2000
= Rs. 50000
9. Anil, Kamal and Vini invested Rs. 8000, Rs. 4000 and Rs. 8000 respectively in a business. Anil left after 6 months. If after 8 months, there was a gain of Rs. 4005, then what will be the share of Kamal ?
a) Rs. 800
b) Rs. 890
c) Rs. 500
d) Rs. 900
Explanation: Ratio of profit of Anil : Kamal : Vini
(8000 × 6) : (4000 × 8) : (8000 × 8)
= 48000 : 32000 : 64000
= 48 : 32 : 64
= 3 : 2 : 4
$$\eqalign{ & {\text{Kamal's share}} = {\text{Rs}}{\text{.}}\left( {4005 \times \frac{2}{9}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{890}} \cr} $$
10. A starts a business by investing Rs. 28000. After 2 months, B joins with Rs. 20000 and after another 2 months C joins with Rs. 18000. At the end of 10 months from the start of the business, if B withdraws Rs. 2000 and C withdraws Rs. 2000, in what ratio should the profit be distributed among A, B and C at the end of the year ?
a) 12 : 7 : 5
b) 12 : 9 : 5
c) 12 : 6 : 3
d) 14 : 7 : 5
Explanation: 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