1. In a business, A and C invested amounts in the ratio 2 : 1 , whereas the ratio between amounts invested by A and B was 3 : 2 . If Rs. 157300 was their profit, how much amount did B receive?
a) Rs. 48400
b) Rs. 48600
c) Rs. 48000
d) Rs. 48200
Discussion
Explanation:
Assume that investment of C = x
Then, investment of A = 2x
Investment of B = $$\frac{{2{\text{A}}}}{3}$$ = $$\frac{{4{\text{x}}}}{3}$$ (since ratio of investment of A : B = 2 : 3 i.e B = $$\frac{{2{\text{A}}}}{3}$$)
A : B : C
= 2x : $$\frac{{4{\text{x}}}}{3}$$ : x
= 2 : $$\frac{4}{3}$$ : 1
= 6 : 4 : 3
$$\eqalign{ & {\text{B's Share}} \cr & = 157300 \times \frac{4}{{6 + 4 + 3}} \cr & = 157300 \times \frac{4}{{13}} \cr & = 12100 \times 4 \cr & = {\text{Rs}}{\text{. 48400}} \cr} $$
2. A starts business with Rs. 3500 and after 5 months, B joins with A as his partner. After a year, the profit is divided in the ratio 2 : 3. What is B's contribution in the capital?
a) Rs. 8000
b) Rs. 7500
c) Rs. 8500
d) Rs. 9000
Discussion
Explanation:
Let B's capital be Rs. x
∴ A's share in 12 months = 3500 × 12
And, B's share in 7 months = 7x
$$\eqalign{ & {\text{Then}},\, {\frac{{3500 \times 12}}{{7x}} = \frac{2}{3}} \cr & \Rightarrow 14x = 126000 \cr & \Rightarrow x = 9000 \cr} $$
3. A, B, C subscribe Rs. 5,00,000 for a business. A subscribes Rs. 40,000 more than B and B Rs. 50,000 more than C. Out of a total profit of Rs. 3,50,000 A receives:
a) Rs. 84,000
b) Rs. 1,47,000
c) Rs. 1,36,000
d) Rs. 1,19,000
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,C = x \cr & {\text{Then,}}\,{\text{B}} = x + 50000\,{\text{and}} \cr & {\text{A}} = x + 50000 + 40000 \cr & \,\,\,\,\,\,\,\, = x + 90000 \cr & So,\,x + x + 50000 + x + 90000 = 500000 \cr & \Rightarrow 3x = 360000 \cr & \Rightarrow x = 120000 \cr & {\text{A:B:C}} = 210000:170000:120000 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 21:17:12 \cr & \therefore {\text{A's}}\,{\text{share}} \cr & = Rs.\,\left( {350000 \times \frac{{21}}{{50}}} \right) \cr & = Rs.\,1,47,000 \cr} $$
4. Arun, Kamal and Vinay invested Rs. 8000, Rs. 4000 and Rs. 8000 respectively in a business. Arun left after six months. If after eight months, there was a gain of Rs. 4005, then what will be the share of Kamal?
a) Rs. 890
b) Rs. 1780
c) Rs. 1335
d) Rs. 1602
Discussion
Explanation:
$$\eqalign{ & {\text{Arun}}\,{\text{:}}\,{\text{Kamal}}\,{\text{:}}\,{\text{Vinay}} \cr & = \left( {8000 \times 6} \right):\left( {4000 \times 8} \right):\left( {8000 \times 8} \right) \cr & = 48:32:64 \cr & = 3:2:4 \cr & \therefore {\text{Kamal's}}\,{\text{share}} \cr & = Rs.\,\left( {4005 \times \frac{2}{9}} \right) \cr & = Rs.\,890 \cr} $$
5. Three partners shared the profit in a business in the ratio 5 : 7 : 8. They had partnered for 14 months, 8 months and 7 months respectively. What was the ratio of their investments?
a) 5 : 7 : 8
b) 38 : 28 : 21
c) 20 : 49 : 64
d) None of these
Discussion
Explanation:
$$\eqalign{ & {\text{Let their investments be}}\, \cr & Rs.\,x\,{\text{for}}\,{\text{14}}\,{\text{months}},\, \cr & Rs.\,y\,{\text{for}}\,{\text{8}}\,{\text{months}}\,{\text{and}}\, \cr & Rs.\,z\,{\text{for}}\,{\text{7}}\,{\text{months}}\,{\text{respectively}}. \cr & {\text{Then}},\,14x:8y:7z = 5:7:8 \cr & {\text{Now}},\frac{{14x}}{{8y}} = \frac{5}{7} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,98x = 40y \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y = \frac{{49}}{{20}}x \cr & {\text{And}},\frac{{14x}}{{7z}} = \frac{5}{8} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,112x = 35z \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,z = \frac{{112}}{{35}}x = \frac{{16}}{5}x \cr & \therefore x:y:z \cr & = x:\frac{{49}}{{20}}x:\frac{{16}}{5}x \cr & = 20:49:64 \cr} $$
6. A, B, C rent a pasture. A puts 10 oxen for 7 months, B puts 12 oxen for 5 months and C puts 15 oxen for 3 months for grazing. If the rent of the pasture is Rs. 175, how much must C pay as his share of rent?
a) Rs. 30
b) Rs. 50
c) Rs. 40
d) Rs. 45
Discussion
Explanation:
$$\eqalign{ & A:B:C \cr & = \left( {10 \times 7} \right):\left( {12 \times 5} \right):\left( {15 \times 3} \right) \cr & = 70:60:45 \cr & = 14:12:9 \cr & \therefore {\text{C's rent}} = {\text{Rs}}{\text{.}}\left( {175 \times \frac{9}{{35}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}45 \cr} $$
7. A and B started a business in partnership investing Rs. 20,000 and Rs. 15,000 respectively. After six months, C joined them with Rs. 20,000. What will be B's share in total profit of Rs. 25,000 earned at the end of 2 years from the starting of the business?
a) Rs. 10,000
b) Rs. 7500
c) Rs. 9000
d) Rs. 9500
Discussion
Explanation:
$$\eqalign{ & A:B:C \cr & = \left( {20000 \times 24} \right):\left( {15000 \times 24} \right):\left( {20000 \times 18} \right) \cr & = 4:3:3 \cr & \therefore {\text{B's}}\,{\text{share}} = Rs.\,\left( {25000 \times \frac{3}{{10}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,7500 \cr} $$
8. Sumit and Ravi started a business by investing Rs. 85000 and Rs. 15000 respectively. In what ratio the profit earned after 2 years be divided between Sumit and Ravi respectively.
a) 17 : 2
b) 17 : 4
c) 17 : 3
d) 17 : 1
Discussion
Explanation:
Sumit : Ravi = 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.
9. A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and A's share is Rs. 855, the total profit is:
a) Rs. 1537.50
b) Rs. 1500
c) Rs. 1425
d) Rs. 1576
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{total}}\,{\text{profit}}\,{\text{be}}\,{\text{Rs}}{\text{.}}\,{\text{100}} \cr & {\text{After}}\,{\text{paying}}\,{\text{to}}\,{\text{charity,}}\,{\text{A's}}\,{\text{share}} \cr & = Rs.\,\left( {95 \times \frac{3}{5}} \right) = Rs.\,57 \cr & {\text{If}}\,{\text{A's}}\,{\text{share}}\,{\text{is}}\,Rs.\,57, \cr & {\text{Total}}\,{\text{profit}} = Rs.\,100 \cr & {\text{If}}\,{\text{A's}}\,{\text{share}}\,Rs.\,855, \cr & {\text{Total}}\,{\text{profit}} \cr & = {\frac{{100}}{{57}} \times 855} \cr & = 1500 \cr} $$
10. A and B started a partnership business investing some amount in the ratio of 3 : 5. C joined then after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed among A, B and C?
a) 6 : 10 : 5
b) 3 : 5 : 2
c) 3 : 5 : 5
d) Data inadequate
Discussion
Explanation:
Let initial investment of A is 3x and B is 5x, then C investment is also 5x
A invested for 12 months, B invested for 12 months and C invested for 6 months.
A : B : C = (3x x 12) : (5x x 12) : (5x x 6)
= 36 : 60 : 30
= 6 : 10 : 5
11. A, B, C subscribe Rs. 50,000 for a business. A subscribes Rs. 4000 more than B and B Rs. 5000 more than C. Out of a total profit of Rs. 35,000, A receives
a) Rs. 8400
b) Rs. 13,600
c) Rs. 11,900
d) Rs. 14,700
Discussion
Explanation:
Let C = x.
Then, B = x + 5000 and A = x + 5000 + 4000 = x + 9000
So, x + x + 5000 + x + 9000 = 50000
⇒ 3x = 36000
⇒ x = 12000
A : B : C = 21000 : 17000 : 12000 = 21 : 17 : 12
$$\eqalign{ & {\text{So A's Share}} \cr & = {\text{Rs}}{\text{. 3}}5000 \times \frac{{21}}{{50}} \cr & = {\text{Rs}}{\text{. 14700}} \cr} $$
12. A and B entered into partnership with capitals in the ratio 4 : 5. After 3 months, A withdrew $$\frac{1}{4}$$ of his capital and B withdrew $$\frac{1}{5}$$ of his capital. The gain at the end of 10 months was Rs. 760. A's share in this profit is:
a) Rs. 360
b) Rs. 330
c) Rs. 430
d) Rs. 380
Discussion
Explanation:
$$A:B$$
$$ = \left[ {4x \times 3 + \left( {4x - \frac{1}{4} \times 4x} \right) \times 7} \right]$$ $$:$$ $$\left[ {5x \times 3 + \left( {5x - \frac{1}{5} \times 5x} \right) \times 7} \right]$$
$$\eqalign{ & = \left( {12x + 21x} \right):\left( {15x + 28x} \right) \cr & = 33x:43x = 33:43 \cr & \therefore {\text{A's share}} = {\text{Rs}}{\text{.}}\left( {760 \times \frac{{33}}{{76}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,330 \cr} $$
13. A, B and C enter into a partnership in the ratio $$\frac{7}{2}$$ : $$\frac{4}{3}$$ : $$\frac{6}{5}$$. After 4 months, A increases his share 50%. If the total profit at the end of one year be Rs. 21,600, then B's share in the profit is:
a) Rs. 2100
b) Rs. 3600
c) Rs. 4000
d) Rs. 2400
Discussion
Explanation:
Ratio of initial investments
$$\eqalign{ & = {\frac{7}{2}:\frac{4}{3}:\frac{6}{5}} \cr & = 105:40:36 \cr} $$
Let the initial investments be 105x, 40x and 36x
$$\left( {105x \times 4 + \frac{{150}}{{100}} \times 105x \times 8} \right)$$ $$:$$ $$\left( {40x \times 12} \right)$$ $$:$$ $$\left( {36x \times 12} \right)$$
$$\eqalign{ & = 1680x:480x:432x \cr & = 35:10:9 \cr & {\text{Hence,}}{\kern 1pt} {\text{B's}}{\kern 1pt} {\text{share}} \cr & = {\text{Rs}}{\text{.}}\left( {21600 \times \frac{{10}}{{54}}} \right) \cr & = {\text{Rs}}{\text{.}}\,4000 \cr} $$
14. Aman started a business investing Rs. 70,000. Rakhi joined him after six months with an amount of Rs. 1,05,000 and Sagar joined them with Rs. 1.4 lakhs after another six months. The amount of profit earned should be distributed in what ratio among Aman, Rakhi and Sagar respectively, 3 years after Aman started the business?
a) 12 : 15 : 16
b) 42 : 45 : 56
c) 7 : 6 : 10
d) Cannot be determined
Discussion
Explanation:
Aman : Rakhi : Sagar
= (70,000 x 36) : (1,05,000 x 30) : (1,40,000 x 24)
= 12 : 15 : 16
15. A, B, C rent a pasture. A puts 10 oxen for 7 months, B puts 12 oxen for 5 months and C puts 15 oxen for 3 months for grazing. If the rent of the pasture is Rs. 175, how much must C pay as his share of rent?
a) Rs. 45
b) Rs. 55
c) Rs. 50
d) Rs. 60
Discussion
Explanation:
$$\eqalign{ & A:B:C \cr & = \left( {10 \times 7} \right):\left( {12 \times 5} \right):\left( {15 \times 3} \right) \cr & = 70:60:45 \cr & = 14:12:9 \cr & \therefore {\text{C's}}\,{\text{rent}} = Rs.\,\left( {175 \times \frac{9}{{35}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,45 \cr} $$
16. P, Q, R enter into a partnership. P initially invests 25 lakh & adds another 10 lakhs after one year. Q initially invests 35 lakh & withdrawal 10 lakh after 2 years and R invests Rs 30 Lakhs. In what ratio should the profit be divided at the end of 3 years?
a) 18 : 18 : 19
b) 19 : 19 : 18
c) 18 : 19 : 19
d) 18 : 19 : 18
Discussion
Explanation:
P : Q : R
= (25 × 1 + 35 × 2) : (35 × 2 : 25 × 1) : (30 × 3)
= 95 : 95 : 90
= 19 : 19: 18
17. A began a business with Rs. 85,000. He was joined afterwards by B with Rs. 42,500. For how much period does B join, if the profits at the end of the year are divided in the ratio of 3 : 1?
a) 6 months
b) 5 months
c) 4 months
d) 8 months
Discussion
Explanation:
$$\eqalign{ & {\text{Suppose}}\,{\text{B}}\,{\text{joined}}\,{\text{for}}\,x\,{\text{months}}. \cr & {\text{Then}}, \cr & {\frac{{85000 \times 12}}{{42500 \times x}} = \frac{3}{1}} \cr & \Rightarrow x = {\frac{{85000 \times 12}}{{42500 \times 3}}} = 8 \cr & {\text{So,}}\,{\text{B}}\,{\text{joined}}\,{\text{for}}\,{\text{8}}\,{\text{months}} \cr} $$
18. A, B and C jointly thought of engaging themselves in a business venture. It was agreed that A would invest Rs. 6500 for 6 months, B, Rs. 8400 for 5 months and C, Rs. 10,000 for 3 months. A wants to be the working member for which, he was to receive 5% of the profits. The profit earned was Rs. 7400. Calculate the share of B in the profit.
a) Rs. 2840
b) Rs. 1900
c) Rs. 2660
d) Rs. 2800
Discussion
Explanation:
For managing, A received = 5% of Rs. 7400 = Rs. 370.
Balance = Rs. (7400 - 370) = Rs. 7030.
Ratio of their investments = (6500 x 6) : (8400 x 5) : (10000 x 3)
= 39000 : 42000 : 30000
= 13 : 14 : 10
$$\eqalign{ & \therefore {\text{ B's Share}} \cr & = {\text{Rs}}{\text{. 7030}} \times \frac{{14}}{{37}} \cr & = {\text{Rs}}{\text{. 2660}} \cr} $$
19. A and B started a business in partnership investing Rs. 20,000 and Rs. 15,000 respectively. After six months, C joined them with Rs. 20,000. What will be B's share in total profit of Rs. 25,000 earned at the end of 2 years from the starting of the business?
a) Rs. 7500
b) Rs. 10,000
c) Rs. 9500
d) Rs. 9000
Discussion
Explanation:
A : B : C
= (20,000 × 24) : (15,000 × 24) : (20,000 × 18)
= 4 : 3 : 3.
$$\eqalign{ & {\text{So B's Share}} \cr & = {\text{Rs}}{\text{. }}25000 \times \frac{3}{{10}} \cr & = {\text{Rs}}{\text{. 7500}} \cr} $$
20. Simran started a software business by investing Rs. 50,000. After six months, Nanda joined her with a capital of Rs. 80,000. After 3 years, they earned a profit of Rs. 24,500. What was Simran's share in the profit?
a) Rs. 9,423
b) Rs. 10,250
c) Rs. 10,500
d) Rs. 12,500
Discussion
Explanation:
$$\eqalign{ & {\text{Simran}}\,{\text{:}}\,{\text{Nanda}} \cr & = \left( {50000 \times 36} \right):\left( {80000 \times 30} \right) \cr & = 3:4 \cr & \therefore {\text{Simran's}}\,{\text{share}} \cr & = Rs.\,\left( {24500 \times \frac{3}{7}} \right) \cr & = Rs.\,10,500 \cr} $$
21. A start a business with Rs. 45000. After 6 months B enters in his business with Rs. 80000. After one year C invests Rs. 120000. In what ratio the profit will be divided among A, B and C ?
a) 9 : 16 :24
b) 3 : 4 : 4
c) 3 : 4 : 8
d) 3 : 3 : 8
Discussion
Explanation:
| A | : | B | : | C | |
| Capital → | 45000 | : | 80000 | : | 120000 |
| Time(year) → | 2 | $$\frac{3}{2}$$ | 1 | ||
| Profit → | 90 | : | 120 | : | 120 |
| 3 | : | 4 | : | 4 |
Required ratio of profit = 3 : 4 : 4
22. Three partner A, B and C started a business by investing Rs. 48000 each. After 6 months A left the business after 10 months B left the business and after 12 months C left the business. If total earned profit is Rs. 5250, then find the share of A, B and C ?
a) Rs. 1125, Rs. 1825, Rs. 2250
b) Rs. 1125, Rs. 1800, Rs. 2200
c) Rs. 1125, Rs. 1875, Rs. 2250
d) Rs. 1175, Rs. 1256, Rs. 2350
Discussion
Explanation:
| 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
$$\eqalign{ & \left( {3 + 5 + 6} \right){\text{units}} = {\text{Rs}}{\text{. 5250}} \cr & {\text{14 units}} = {\text{Rs}}{\text{. 5250}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{. 375}} \cr & {\text{Share of A}} = 375 \times 3 \cr & = {\text{Rs}}{\text{. 1125}} \cr & {\text{Share of B}} = 375 \times 5 \cr & = {\text{Rs}}{\text{. 1875}} \cr & {\text{Share of C}} = 375 \times 6 \cr & = {\text{Rs}}{\text{. 2250}} \cr} $$
23. A started a business by investing some money and B invested Rs. 5000 each more than that of A. A remained in business for 5 months and B remained in business 1 month more than A. Out of the total profit of Rs. 26000, B got Rs. 6000 more than A. Find the capitals invested A and B ?
a) Rs. 29000, Rs. 18000
b) Rs. 25000, Rs. 3000
c) Rs. 15000, Rs. 10000
d) Rs. 15000, Rs. 20000
Discussion
Explanation: Let amount invested by A = Rs. x
| A | : | B | |
| Capital → | x | : | (x + 5000) |
$$\eqalign{ & {\text{Share of A in profit}} = \frac{{\left( {26000 - 6000} \right)}}{2} \cr & = {\text{Rs}}{\text{. 10000}} \cr & {\text{Share of B in profit}} = \left( {26000 - 10000} \right) \cr & = {\text{Rs}}{\text{. 16000}} \cr & {\text{By using formulas:}} \cr & {\frac{{{{\text{C}}_{\text{1}}} \times {{\text{T}}_1}}}{{{{\text{C}}_{\text{2}}} \times {{\text{T}}_{\text{2}}}}} = \frac{{{{\text{P}}_{\text{1}}}}}{{{{\text{P}}_2}}}} \cr & \frac{{x \times 5}}{{\left( {x + 5000} \right) \times 6}} = \frac{{10000}}{{16000}} \cr & 4x = 3x + 15000 \cr & x = {\text{Rs}}.15000 \cr & {\text{Required capital of A}} = {\text{Rs}}{\text{. 15000}} \cr & {\text{Required capital of B}} = \left( {15000 + 5000} \right) \cr & = {\text{Rs}}{\text{. 20000}} \cr} $$
24. In a business, B invests half the amount invested by A. After 6 months from the start of the business, C joins the business with an amount equal to twice of B's investment. After 8 months from the start of the business B withdraws completely from the business. If at the end of the year, C's share in the profit was Rs. 2460, what was the total profit received that year ?
a) Rs. 11200
b) Rs. 9600
c) Rs. 9020
d) Rs. 12000
Discussion
Explanation: 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
$$\eqalign{ & \frac{{3a}}{{11}} = 2460 \cr & 3a = 2460 \times 11 \cr & a = \frac{{2460 \times 11}}{3} \cr & a = {\text{Rs}}{\text{.}}\,9020 \cr} $$
25. A, B and C entered in to a partnership by investing Rs. 15400, Rs.18200 and Rs. 12600 respectively. B left after 6 months. If after 8 months, there was a profit of Rs. 28790, then what is the share of C in the profit ?
a) Rs. 8712
b) Rs. 9432
c) Rs. 8352
d) Rs. 8568
Discussion
Explanation:
$$\eqalign{ & {\text{Investment of A for 8 months}} = {\text{Rs}}{\text{.15400}} \cr & {\text{Investment of B for 6 months}} = {\text{Rs}}{\text{.18200}} \cr & {\text{Investment of C for 8 months}} = {\text{Rs}}{\text{.12600}} \cr & {\text{Ratio of the share of A, B and C}} \cr & = 15400 \times 8:18200 \times 6:12600 \times 8 \cr & = 154 \times 8:182 \times 6:126 \times 8 \cr & = 44:39:36 \cr & {\text{Sum of the terms of ratio}} \cr & = 44 + 39 + 36 \cr & = 119 \cr & {\text{ Share of C}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{36}}{{119}} \times 28790} \right) \cr & = {\text{Rs}}{\text{.8710}} \approx {\text{Rs}}{\text{.8712}} \cr} $$
26. A, B, C started a business with their investments in the ratio 1 : 3 : 5. After 4 months, A invested the same amount as before and B as well as C withdrew half of their investments. The ratio of their profits at the end of the year is ?
a) 4 : 3 : 5
b) 5 : 6 : 10
c) 6 : 5 : 10
d) 10 : 5 : 6
Discussion
Explanation: Let their initial investments be x, 3x and 5x respectively
$$ = {\text{A}}:{\text{B}}:{\text{C}}$$
$$ = \left( {x \times 4 + 2x \times 8} \right)$$ : $$\left( {3x \times 4 + \frac{{3x}}{2} \times 8} \right)$$ : $$\left( {5x \times 4 + \frac{{5x}}{2} \times 8} \right)$$
$$\eqalign{ & = 20x:24x:40x \cr & = 5:6:10 \cr} $$
27. In a partnership, A invests $$\frac{1}{6}$$ of the capital $$\frac{1}{6}$$ for of the time, B invests $$\frac{1}{3}$$
of the capital for $$\frac{1}{3}$$ of the time and C, the rest of the capital for the whole time. Out of a profit of Rs. 4600, B's share is ?
a) Rs. 650
b) Rs. 800
c) Rs. 960
d) Rs. 1000
Discussion
Explanation:
$$\eqalign{ & {\text{Suppose, }} {\text{A invests Rs}}{\text{.}}\frac{x}{6}{\text{ for }}\frac{y}{6}{\text{ months}} \cr & {\text{ B invests Rs}}{\text{.}}\frac{x}{3}{\text{ for }}\frac{y}{3}{\text{ months}} \cr & {\text{C invests}}\left[ {x - \left( {\frac{x}{6} + \frac{x}{3}} \right)} \right]i.e.,{\text{ Rs}}{\text{.}}\frac{x}{2}{\text{ for }}y{\text{ months}} \cr & {\text{A}}:{\text{B}}:{\text{C}} {\text{ = }}\left( {\frac{x}{6} \times \frac{y}{6}} \right):\left( {\frac{x}{3} \times \frac{y}{3}} \right):\left( {\frac{x}{2} \times y} \right) \cr & = \frac{1}{{36}}:\frac{1}{9}:\frac{1}{2} \cr & = 1:4:18 \cr & {\text{B's share}} \cr & = {\text{Rs}}{\text{. }}\left( {4600 \times \frac{4}{{23}}} \right) \cr & = {\text{Rs}}{\text{. 800}} \cr} $$
28. A starts business with a capital of Rs. 14000. Five months later B joins and further two months later C joins them. If the profit sharing ratio in the end of year is 4 : 3 : 2, then the money invested by C was ?
a) Rs. 18000
b) Rs. 16800
c) Rs. 18600
d) Rs. 10800
Discussion
Explanation:
| A | B | C | |
| Amounts invested | 14000 | ||
| Time (in months) | 12 | 7 | 5 |
| 168000 |
$$\eqalign{ & {\text{Ratio of profits }}4:3:2 \cr & {\text{Let their profits }}4x:3x:2x \cr & 4x = 168000 \cr & x = 42000 \cr & \Rightarrow {\text{Profit share of C}} = 2x \cr & = 2 \times 42000 \cr & = {\text{Rs}}{\text{. 84000}} \cr & \Rightarrow {\text{Capital invested by C}} = \frac{{84000}}{5} \cr & = {\text{Rs. }}16800 \cr} $$
29. A, B and C become partners in a business. A contributes $$\frac{1}{3}$$ rd of the capital for $$\frac{1}{4}$$ th of the time. B contributes $$\frac{1}{5}$$ th of the capital for $$\frac{1}{6}$$ th of the time and C the rest of the capital for the whole time. If the profit is Rs. 1820, then the A's share in profit is ?
a) Rs. 130
b) Rs. 260
c) Rs. 292
d) Rs. 304
Discussion
Explanation: Let the total capital of A, B and C = 15 units
Let total time for investment = 12 units
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)
$$\eqalign{ & {\text{35 units}} = {\text{Rs}}{\text{. 1820}} \cr & {\text{1 unit}} = \frac{{1820}}{{35}} = {\text{Rs}}.52 \cr & {\text{Hence A's share in profit}} \cr & = 5{\text{ units}} \cr & = 52 \times 5 \cr & = {\text{Rs}}{\text{. 260}} \cr} $$
30. A, B and C entered into a partnership. A invested Rs. 2560 and B invested Rs. 2000. At the end of the year, they gained Rs. 1105, out of which A got Rs. 320. C's capital was ?
a) Rs. 2840
b) Rs. 4028
c) Rs. 4280
d) Rs. 4820
Discussion
Explanation: Let C's capital be Rs. x
A : B : C = 2560 : 2000 : x
$$\eqalign{ & {\text{A's share}} \cr & = {\text{Rs}}{\text{.}}\left( {1105 \times \frac{{2560}}{{4560 + x}}} \right) \cr & \therefore 1105 \times \frac{{2560}}{{4560 + x}} = 320 \cr & 320x + 1459200 = 2828800 \cr & 320x = 1369600 \cr & x = 4280 \cr} $$
31. A and B enter into a partnership with Rs. 50000 and Rs. 60000 respectively. C joins them after x months, contributing Rs. 70000 and B leaves x months before the end of the year. If they share the profit in the ratio of 20 : 18 : 21, then the value of x is = ?
a) 3
b) 6
c) 8
d) 9
Discussion
Explanation: 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
$$ = \left( {50000 \times 12} \right)$$ : $$\left[ {60000 \times \left( {12 - x} \right)} \right]$$ : $$\left[ {70000 \times \left( {12 - x} \right)} \right]$$
$$\eqalign{ & = 60:6\left( {12 - x} \right):7\left( {12 - x} \right) \cr & {\text{But ratio of profits}} \cr & = 20:18:21 \cr & = 60:54:63 \cr} $$
$$ 60:\left( {72 - 6x} \right):\left( {84 - 7x} \right)$$ = $$60$$ : $$54$$ : $$63$$
$$\eqalign{ & {\text{So, }}72 - 6x = 54 \cr & 6x = 18 \cr & x = 3 \cr} $$
32. Two friends P and Q started a business investing in the ratio 5 : 6. R joined them after six months investing an amount equal to that of Q's. At the end of the year, 20% profit was earned which was equal to Rs. 98000. What was the amount invested by R?
a) Rs. 105000
b) Rs. 175000
c) Rs. 210000
d) Data inadequate
Discussion
Explanation: Let the total investment be Rs. z
$$\eqalign{ & 20\% {\text{ of }}z = 98000 \cr & z = \left( {\frac{{98000 \times 100}}{{20}}} \right) \cr & z = 490000 \cr} $$
Let the capital of P, Q and R be
Rs. 5x, Rs. 6x and Rs. 6x respectively
$$ \left( {5x \times 12} \right)$$ + $$\left( {6x \times 12} \right)$$ + $$\left( {6x \times 6} \right)$$ = $$490000 \times 12$$
$$\eqalign{ & 168x = 490000 \times 12 \cr & x = \left( {\frac{{490000 \times 12}}{{168}}} \right) \cr & x = 35000 \cr & {\text{R's investment}} = {\text{Rs}}{\text{. }}6x \cr & = {\text{Rs}}{\text{.}}\left( {6 \times 35000} \right) \cr & = {\text{Rs}}{\text{. 210000}} \cr} $$
33. A, B and C invested their capitals in the ratio 3 : 4 : 6. However their share of profit are equal. The duration of their investments must be in the ratio ?
a) 4 : 3 : 2
b) 6 : 4 : 3
c) 3 : 4 : 6
d) 1 : 1 : 1
Discussion
Explanation: Let their investments be Rs. 3x for p months
Rs. 4x for q months and Rs. 6x for r months respectively
$$\eqalign{ & 3xp:4xq:6xr = 1:1:1 \cr & 3p:4q:6r = 1:1:1 \cr & {\text{So, }}3p = 4q \cr & \Leftrightarrow q = \frac{{3p}}{4} \cr & 4q = 6r \cr & \Leftrightarrow r = \frac{{2q}}{3} = \left( {\frac{2}{3} \times \frac{3}{4}p} \right) = \frac{p}{2} \cr & \therefore p:q:r \cr & = p:\frac{{3p}}{4}:\frac{p}{2} \cr & = 4:3:2 \cr} $$
34. Swati and Rajni enter into a partnership with their capitals in the ratio 5 : 6. At the end of 7 months Swati withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long was Rajni's capital used ?
a) 10 months
b) 12 months
c) 14 months
d) None of these
Discussion
Explanation: Suppose, Swati invested Rs. 5x for 7 months
Rajni invested Rs. 6x for y months
$$\eqalign{ & \frac{{5x \times 7}}{{6x \times y}} = \frac{5}{9} \cr & 30y = 315 \cr & y = 10\frac{1}{2} \cr} $$
35. X and Y are partners in a business. X contributed $$\frac{1}{3}$$ of the capital for 9 months and Y received $$\frac{2}{5}$$ of the profit. For how long was Y's money used in the business ?
a) 2 months
b) 3 months
c) 4 months
d) 5 months
Discussion
Explanation:
$$\eqalign{ & {\text{Let the total profit be Rs}}{\text{.}}\,z \cr & {\text{Y's share}} = {\text{Rs}}{\text{.}}\,\frac{{2z}}{5} \cr & {\text{X's share}} = {\text{Rs}}{\text{.}}\left( {z - \frac{{2z}}{5}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}.\,\frac{{3z}}{5} \cr & \therefore {\text{X}}:{\text{Y}} = \frac{{3z}}{5}:\frac{{2z}}{5} \cr & = 3:2 \cr} $$
Let the total capital be Rs. x and suppose Y's money was used for y months
$$\eqalign{ & \frac{{\frac{1}{3}x \times 9}}{{\frac{2}{3}x \times y}} = \frac{3}{2} \cr & 18x = 6xy \cr & y = 3 \cr} $$
36. A, B and C enter into a partnership. A contributes one-third of the capital while B contributes as much as A and C together contribute. If the profit at the end of the year amounts to Rs. 900, what would C receive ?
a) Rs. 100
b) Rs. 150
c) Rs. 200
d) Rs. 300
Discussion
Explanation:
$$\eqalign{ & {\text{Let total capital}} = {\text{Rs}}{\text{.}}\,{\text{x}} \cr & {\text{Then, A's capital}} = {\text{Rs}}{\text{.}}\,\frac{x}{3} \cr} $$
B's capital = (A + C)'s capital ⇒ 2(B's capital)
B's capital = (A + B + C)'s capital = Rs. x
$$\eqalign{ & {\text{B's capital}} = {\text{Rs}}{\text{.}}\frac{x}{2} \cr & {\text{C's capital}} = {\text{Rs}}{\text{.}}\left[ {x - \left( {\frac{x}{3} + \frac{x}{2}} \right)} \right] \cr & = {\text{Rs}}{\text{.}}\,\frac{x}{6} \cr & \therefore {\text{A}}:{\text{B}}:{\text{C}} = \frac{x}{3}:\frac{x}{2}:\frac{x}{6} \cr & = 2:3:1 \cr & {\text{C's share}} = {\text{Rs}}{\text{.}}\left( {900 \times \frac{1}{6}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{150}} \cr} $$
37. A and B started a business jointly. A's investment was thrice the investment of B and the period of his investment was two times the period of investment of B. If B received Rs. 4000 as profit, then their total profit is ?
a) Rs. 16000
b) Rs. 20000
c) Rs. 24000
d) Rs. 28000
Discussion
Explanation: Suppose B invested Rs. x for y months
A invested Rs. 3x for 2y months
$$\eqalign{ & {\text{A}}:{\text{B}} = \left( {3x \times 2y} \right):\left( {x \times y} \right) \cr & = 6xy:xy \cr & = 6:1 \cr & {\text{B's profit}}:{\text{Total profit}} = 1:7 \cr & {\text{Let the total profit is Rs}}{\text{. }}x \cr & \frac{1}{7} = \frac{{4000}}{x} \cr & x = 28000 \cr} $$
38. A began a business with Rs. 85000, he was joined afterwards by B with Rs. 42500. For how much period does B join, if the profits at the end of the year are divided in the ratio of 3 : 1 ?
a) 4 months
b) 5 months
c) 6 months
d) 8 months
Discussion
Explanation:
$$\eqalign{ & {\text{Suppose B joined for }}x{\text{ months}} \cr & \frac{{85000 \times 12}}{{42500 \times x}} = \frac{3}{1} \cr & x = \frac{{85000 \times 12}}{{42500 \times 3}} \cr & x = 8 \cr} $$
39. A and B share profits and losses in a firm in the ratio of 3 : 2. And C entered in the firm as a new partner; his profit sharing ratio is $$\frac{1}{4}$$. If C has taken his share of profit from A and B in equal ratio, then the new profit shearing ratio will be ?
a) 19 : 11 : 1
b) 19 : 11 : 10
c) 10 : 11 : 9
d) 10 : 11 : 19
Discussion
Explanation:
$$\eqalign{ & {\text{Let the total share}} = {\text{200 units}} \cr & {\text{Share of C}} = 200 \times \frac{1}{4} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 50{\text{ units}} \cr & {\text{Remaining share}} = \left( {200 - 50} \right) \cr & {\text{ = 150 units}} \cr & {\text{Share of A}} = \frac{{200}}{{3 + 2}} \times 3 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 120{\text{ units}} \cr & {\text{Share of B}} = \frac{{200}}{{3 + 2}} \times 2 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 80{\text{ units}} \cr} $$
C received equal amounts from A and B
$$\eqalign{ & {\text{A's remaining share}} = \left( {120 - 25} \right) \cr & = 95 \cr & {\text{B's remaining share}} = \left( {80 - 25} \right) \cr & = 55 \cr} $$
| A | : | B | : | C | |
| New Ratio → | 95 | : | 55 | : | 50 |
| 19 | : | 11 | : | 10 |
40. A, B and C share the profit in the ratio of 2 : 3 : 7. If the average gain is Rs. 8000, then B's share is ?
a) Rs. 2000
b) Rs. 1000
c) Rs. 1500
d) Rs. 6000
Discussion
Explanation:
| A | : | B | : | C | |
| Ratio of Profit → | 2 | : | 3 | : | 7 |
$$\eqalign{ & {\text{Average gain}} = \frac{{2 + 3 + 7}}{3} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4{\text{ units}} \cr & {\text{4 units}} = {\text{Rs}}{\text{. 8000}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{. 2000}} \cr & {\text{3 units}} = 3 \times 2000 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 6000}} \cr & {\text{Share of B}} = {\text{Rs}}{\text{. 6000}} \cr} $$
41. A, B and C are partners in a business partnership. A invested Rs. 4000 for whole year. B invested Rs. 6000 initially but increased this investment up to Rs. 8000 at the end of 4 months, while C invested Rs. 8000 initially, but withdraw Rs. 2000 at the end of 9 months. At the end of year total earned profit is Rs. 16950, find their share of profit ?
a) Rs. 3600, Rs. 6600, Rs. 6750
b) Rs. 2000, Rs. 3050, Rs. 5400
c) Rs. 2450, Rs. 2460, Rs. 1456
d) None of these
Discussion
Explanation: 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 |
$$\left( {24 + 44 + 45} \right){\text{units}}$$ = $${\text{Rs}}{\text{.}}\,{\text{16950}}$$
$$\eqalign{ & {\text{113 units}} = {\text{Rs}}{\text{. 16950}} \cr & 1{\text{ unit}} = \frac{{16950}}{{113}}{\text{ = Rs}}{\text{. 150}} \cr & {\text{Profit of A}} = 150 \times 24 = {\text{Rs}}{\text{.}}\,{\text{3600}} \cr & {\text{Profit of B}} = 150 \times 44 = {\text{Rs}}{\text{.}}\,{\text{6600}} \cr & {\text{Profit of C}} = 150 \times 45 = {\text{Rs}}{\text{.}}\,{\text{6750}} \cr} $$
42. Out of total capital required to start a business A invested 30%, B invested $$\frac{2}{5}$$ th and C invested the remaining capital. At the end of one year sum of Rs. 4000 is earned as a profit which is 20% of the capital given by B, then find how much C invested in the business ?
a) Rs. 25000
b) Rs. 10000
c) Rs. 15000
d) Rs. 12450
Discussion
Explanation: Total profit = Rs. 4000
$$\eqalign{ & {\text{20% of B's capital}} = {\text{Rs}}{\text{.}}\,{\text{4000}} \cr & {\text{1% of B's capital}} = {\text{Rs}}{\text{.}}\,\frac{{{\text{4000}}}}{{20}} \cr & {\text{B's total capital}} \cr & = {\text{Rs}}{\text{. }}\frac{{{\text{4000}}}}{{20}} \times 100 \cr & = {\text{Rs}}{\text{. }}20000 \cr} $$
Let total capital required for business = 100 units.
| A | : | B | : | C | |
| Capital | 30 | : | 40 | : | 30 |
| × 500 | : | × 500 | : | × 500 | |
| 15000 | : | 20000 | : | 15000 |
43. A and B started a business in partnership by investing in the ratio of 7 : 9. After 3 months A withdraw $$\frac{2}{3}$$ of its investment and after 4 months from the beginning B withdraw $$33\frac{1}{3}$$ % of its investment. If a total earned profit is Rs. 10201 at the end of 9 months, find the share of each in profit ?
a) Rs. 3535, Rs. 6666
b) Rs. 3055, Rs. 5555
c) Rs. 4503, Rs. 1345
d) Rs. 3545, Rs. 3333
Discussion
Explanation: Note :In such type of question we can assume ratio as per our need to avoid fraction
| Capital → | A 7 × 3 |
: | B 9 × 3 |
| New Ratio, → | A 21x |
: | B 27x |
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
$$\eqalign{ & = 27x \times 4 + 18x \times 5 \cr & = 198x \cr} $$
$$\eqalign{ & \left( {105x + 198x} \right) = {\text{Rs}}{\text{. 10201}} \cr & 303x = {\text{Rs}}{\text{. 10201}} \cr & x = \frac{{10201}}{{303}} \cr & {\text{Share of A}} = 105 \times \frac{{10201}}{{303}} = {\text{Rs}}{\text{.}}\,3535 \cr & {\text{Share of B}} = 198 \times \frac{{10201}}{{303}} = {\text{Rs}}{\text{.}}\,{\text{6666}} \cr} $$
44. A, B and C enter into a partnership, investing Rs. 6000. A invests Rs. 1000 and B and C in invests in the ratio of 2 : 3. Find the profit of C, when the annual profit is Rs. 2400 ?
a) Rs. 600
b) Rs. 1200
c) Rs. 1800
d) Rs. 1950
Discussion
Explanation:
$$\eqalign{ & {\text{investment of A}} = 1000 \cr & {\text{So, investment of B}} + {\text{C}} = {\text{6000}} - 1000 \cr & = 5000 \cr & {\text{B}}:{\text{C}} = 5000 \cr & 2:3 = 2000:3000 \cr} $$
| A | : | B | : | C |
| 1000 | : | 2000 | : | 3000 |
| 1 | : | 2 | : | 3 = 6 |
$$\eqalign{ & {\text{Profit of C}} = \frac{3}{6} \times 2400 = 1200 \cr} $$
45. 3 brothers divided 1620 among them in such a way that the share of second is equal to $$\frac{5}{{13}}$$ of share of other two, combined. What is the share of the second one ?
a) Rs. 1170
b) Rs. 450
c) Rs. 540
d) Rs. 500
Discussion
Explanation:
$$\eqalign{ & {\text{Given share of }}{{\text{2}}^{{\text{nd}}}} \cr & = \frac{5}{{13}}{\text{of}}\left( {{{\text{1}}^{{\text{st}}}} + {{\text{3}}^{{\text{rd}}}}} \right) \cr & {\text{or, }}\frac{{{{\text{2}}^{{\text{nd}}}}}}{{{{\text{1}}^{{\text{st}}}}{\text{ + }}{{\text{3}}^{{\text{rd}}}}}} = \frac{5}{{13}} \cr & \therefore {1^{{\text{st}}}} + {{\text{2}}^{{\text{nd}}}} + {{\text{3}}^{{\text{rd}}}} = 13 + 5 = 18 \cr & 18{\text{units}} = 1620 \cr & {\text{1 unit}} = \frac{{1620}}{{18}} \cr & 5{\text{ units}} = \frac{{1620}}{{18}} \times {\text{5}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4{\text{50}} \cr & {\text{Share of }}{{\text{2}}^{{\text{nd}}}} = {\text{Rs}}{\text{.}}\,{\text{450}} \cr} $$
46. Nonad, Vikas and and Manav enter into a partnership. Ninad invest some amount at the beginning. Vikas invest double the amount after 6 months and Manav invests thrice the amount invested by Ninad after 8 months. They earn a profit of Rs. 45000 at the end of the year. What is Manav's share in the profit ?
a) Rs. 9000
b) Rs. 12000
c) Rs. 15000
d) Rs. 25000
Discussion
Explanation:
$$\eqalign{ & {\text{Let Ninad's investment be Rs}}{\text{.}}x \cr & {\text{Ratio of capitals}} = \left( {x \times 12} \right):\left( {2x \times 6} \right):\left( {3x \times 4} \right) \cr & = 12x:12x:12x \cr & = 1:1:1 \cr & {\text{Manav's share}} = {\text{Rs}}{\text{.}}\left( {45000 \times \frac{1}{3}} \right) \cr & = {\text{Rs}}{\text{.15000}} \cr} $$
47. Four milkmen rented a pasture. A grazed 15 cows for 4 months, B grazed 12 cows for 2 months, C grazed 18 cows for 6 months and D grazed 16 cows for 5 months. If A's share of rent is Rs. 1020, what is C's share of rent ?
a) Rs. 916
b) Rs. 1360
c) Rs. 1836
d) Cannot be determined
Discussion
Explanation:
$$\eqalign{ & = {\text{A}}:{\text{B}}:{\text{C}}:{\text{D}} \cr & = 15 \times 4:12 \times 2:18 \times 6:16 \times 5 \cr & = 60:24:108:80 \cr & = 15:6:27:20 \cr & {\text{Let the total rent be Rs}}{\text{.}}x \cr & {\text{A's share}} = {\text{Rs}}{\text{.}}\left( {\frac{{15x}}{{68}}} \right) \cr & \therefore \,\frac{{15x}}{{68}} = 1020 \cr & x = \left( {\frac{{1020 \times 68}}{{15}}} \right) \cr & x = 4624 \cr & {\text{C's share}} = {\text{Rs}}{\text{.}}\left( {4624 \times \frac{{27}}{{68}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{1836}} \cr} $$
48. A, B and C are three partners in a business. The profit share of A is $$\frac{3}{{16}}$$ of the total profit and B's share is $$\frac{1}{{4}}$$ of the total profit. If C receives Rs. 243, then the amount received by B will be ?
a) Rs. 90
b) Rs. 96
c) Rs. 108
d) Rs. 120
Discussion
Explanation:
$$\eqalign{ & {\text{Let total profit}} = {\text{16 units }} \cr & {\text{Profit share of A}} = \frac{3}{{16}} \times 16{\text{ units}} \cr & = 3{\text{ units}} \cr & {\text{profit share of B }} {\text{ = }}\frac{1}{4} \times 16 = 4{\text{ units}} \cr & {\text{then profit share of C}} \cr & = \left[ {16 - \left( {4 + 3} \right)} \right] = 9{\text{ units}} \cr & {\text{But profit of C}} = {\text{Rs}}{\text{. 243 (given)}} \cr & {\text{9 units}} = {\text{Rs}}{\text{. 243}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{. 27}} \cr & {\text{Profit share of B}} \cr & = 4{\text{ units}} \cr & = 27 \times 4 \cr & = {\text{Rs}}{\text{. 108}} \cr} $$
49. A is a active partner and B is a inactive partner in business. A puts in Rs. 5000 and B puts in Rs. 6000. A received 15% of the profit for managing the business and the rest is divided in proportion to their capitals. The amounts received by A out of the profit of Rs. 880 in all is ?
a) Rs. 132
b) Rs. 340
c) Rs. 472
d) Rs. 492
Discussion
Explanation: Total profit = Rs. 880
Since A gets 15% of total profit for management
Remaining profit
$$\eqalign{ & = 880 - \frac{{880 \times 15}}{{100}} \cr & = {\text{Rs}}.\,748 \cr} $$
| A | B | ||
| Amount | 5000 | 6000 | |
| Ratio of Capital | 5 | : | 6 |
The remaining profit is being divided in the ratio of their capital
A's share of profit
$$\eqalign{ & = \frac{{748}}{{\left( {5 + 6} \right)}} \times 5 \cr & = {\text{Rs}}.\,340 \cr} $$
Total profit received by A = 340 + 132
= Rs. 472
50. A, B and C are partners in a business. Their shares are in the proportion of $$\frac{1}{3}:\frac{1}{4}:\frac{1}{5}$$ . A withdraws half of his capital after 15 months and after another 15 months, a profit of Rs. 4340 is divided. The share of C is ?
a) Rs. 1240
b) Rs. 1245
c) Rs. 1360
d) Rs. 1550
Discussion
Explanation: Ratio of initial investments
$$\eqalign{ & = \frac{1}{3}:\frac{1}{4}:\frac{1}{5} \cr & = 20:15:12 \cr} $$
Let their initial investments be 20x, 15x and 12x respectively
= A : B : C
= (20x × 15 + 10x × 15) : (15x × 30) : (12x × 30)
= 450x : 450x : 360x
= 5 : 5 : 4
$$\eqalign{ & {\text{C's share}} = {\text{Rs}}{\text{.}}\left( {4340 \times \frac{4}{{14}}} \right) \cr & = {\text{Rs}}{\text{.1240}} \cr} $$
51. 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
Discussion
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 |
$$ \left( {50x + 15x + 12x} \right)$$ = $${\text{Rs}}{\text{. 34650}}$$
$$\eqalign{ & 77x = {\text{Rs}}.{\text{ }}34650 \cr & x = {\text{Rs}}{\text{. }}\frac{{34650}}{{77}} \cr & x = {\text{Rs}}{\text{. }}450 \cr & {\text{Profit of A}} = {\text{Rs}}{\text{. }}450 \times 50 = {\text{Rs}}{\text{. 22500}} \cr & {\text{Profit of B}} = {\text{Rs}}{\text{. }}450 \times 15 = {\text{Rs}}{\text{. 6750}} \cr & {\text{Profit of C}} = {\text{Rs}}{\text{. }}450 \times 12 = {\text{Rs}}{\text{. 5400}} \cr} $$
52. 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
Discussion
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
53. 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
Discussion
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
$$\eqalign{ & \frac{{73x}}{{\left( {12 + 16 + 9} \right)}} \times 9 = {\text{Rs}}{\text{. }}65700 \cr & \frac{{657x}}{{37}} = {\text{Rs}}{\text{. }}65700 \cr & x = {\text{Rs}}{\text{. }}\frac{{65700 \times 37}}{{657}} \cr & x = {\text{Rs}}{\text{. 3}}700 \cr & {\text{Required profit}} = 100x \cr & = 100 \times 3700 \cr & = {\text{Rs}}{\text{. 3}}70000 \cr} $$
54. 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
Discussion
Explanation:
| A | : | B | : | C | |
| Capital | 500 | : | 400 | : | 800 |
| × | : | × | : | × | |
| Time | 12 | : | 10 | : | 6 |
| Profit | 6000 | : | 4000 | 4800 | |
| 15 | : | 10 | : | 12 |
$$\eqalign{ & \left( {15 + 10 + 12} \right){\text{units}} = {\text{Rs}}{\text{.}}\,{\text{444}} \cr & {\text{37 units}} = {\text{Rs}}{\text{. 444}} \cr & 1{\text{ unit}} = \frac{{444}}{{37}}{\text{ = Rs}}{\text{. 12}} \cr & {\text{Profit of A}} = 12 \times 15 = {\text{Rs}}{\text{.}}\,{\text{180}} \cr & {\text{Profit of B}} = 12 \times 10 = {\text{Rs}}{\text{.}}\,{\text{120}} \cr & {\text{Profit of C}} = 12 \times 12 = {\text{Rs}}{\text{.}}\,{\text{144}} \cr} $$
55. 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
Discussion
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
$$\eqalign{ & \frac{{{\text{X}} \times 22500}}{{3600 + 5100 + {\text{X}}}} = 8000 \cr & \frac{{{\text{X}} \times 22500}}{{8700 + {\text{X}}}} = 8000 \cr} $$
$$ 22500{\text{X}}$$ = $$69600000$$ + $$8000{\text{X}}$$
$$\eqalign{ & 14500{\text{X}} = 69600000 \cr & {\text{X}} = {\text{Rs}}.\,4800 \cr} $$
56. 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
Discussion
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
$$\eqalign{ & {\text{37 units}} = {\text{Rs}}{\text{. 7030}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{.}}\frac{{7030}}{{37}} \cr & {\text{Profit of Q}} = {\text{10 units}} \cr & = {\text{Rs}}.\frac{{7030}}{{37}} \times 10 \cr & = {\text{Rs}}.\,1900 \cr} $$
57. 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
Discussion
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
58. 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
Discussion
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
59. 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
Discussion
Explanation: Ratio of profit of Anil : Kamal : Vini
(8000 × 6) : (4000 × 8) : (8000 × 8)
= 48000 : 32000 : 64000
= 48 : 32 : 64
= 3 : 2 : 4
$$\eqalign{ & {\text{Kamal's share}} = {\text{Rs}}{\text{.}}\left( {4005 \times \frac{2}{9}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{890}} \cr} $$
60. 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
Discussion
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
61. Samaira, Mahira and Kiara rented a set of DVDs at a rent of Rs. 578. If they used it for 8 hours, 12 hours and 14 hours respectively, what is Kiara's share of rent to be paid ?
a) Rs. 192
b) Rs. 204
c) Rs. 215
d) Rs. 238
Discussion
Explanation:
$$\eqalign{ & {\text{Ratio of shares}} \cr & = 8:12:14 \cr & = 4:6:7 \cr & {\text{Kiara's share}} = {\text{Rs}}{\text{.}}\left( {578 \times \frac{7}{{17}}} \right) \cr & = {\text{Rs. }}{\text{238}} \cr} $$
62. P, Q and R invested Rs. 45000, Rs. 70000 and Rs. 90000 respectively to start a business. At the end of 2 years, they earned a profit of Rs.164000. What will be Q's share in the profit ?
a) Rs. 36000
b) Rs. 56000
c) Rs. 64000
d) Rs. 72000
Discussion
Explanation:
$$\eqalign{ & {\text{P}}:{\text{Q}}:{\text{R}} \cr & = 45000:70000:90000 \cr & = 9:14:18 \cr & {\text{Q's share}} = {\text{Rs}}{\text{.}}\left( {164000 \times \frac{{14}}{{41}}} \right) \cr & = {\text{Rs. }}{\text{56000}} \cr} $$
63. A working partner gets 20% as his commission of the profit after his commission is paid. If the working partner's commission is Rs. 8000, then what is the total profit in the business ?
a) Rs. 47000
b) Rs. 45000
c) Rs. 48000
d) None of these
Discussion
Explanation: Let the total profit = Rs. k
Remaining profit after paying 20% working
Partner's commission = (k - 8000)
$$\eqalign{ & \therefore \,\left( {k - 8000} \right) \times \frac{{20}}{{100}} = 8000 \cr & k = 48000 \cr & {\text{ Total profit}} = {\text{Rs}}{\text{. 48000}} \cr} $$
64. We have to divided a sum of Rs. 13950 among three persons A, B and C. B must get the double of A's share and C must get Rs. 50 less than the double of B's share. The share of A will be ?
a) Rs. 1950
b) Rs. 1981.25
c) Rs. 2000
d) Rs. 2007.75
Discussion
Explanation: Let the share of A = Rs. x
$$\eqalign{ & {\text{A}}:{\text{B}}:{\text{C}} \cr & {\text{Capital}} \to x:2x:\left( {4x - 50} \right) \cr & \left( {x + 2x + 4x - 50} \right) = 13950 \cr & 7x - 50 = 13950 \cr & 7x = 14000 \cr & x = 2000 \cr & {\text{Share of A}} = {\text{Rs}}{\text{. 2000}} \cr} $$
65. Prakash, Sachin and Anil started a business jointly investing Rs. 11 lakh, Rs. 16.5 lakh and Rs.8.25 lakh respectively. The profit earned by them in the business at the end of 3 years was Rs. 19.5 lakh. What will be 50% of Anil's share in the profit ?
a) Rs. 2.25 lakh
b) Rs. 2.50 lakh
c) Rs. 3.75 lakh
d) Rs. 4.50 lakh
Discussion
Explanation:
$$\eqalign{ & {\text{Ratio of shares}} \cr & = 11:16.5:8.25 \cr & = 4:6:3 \cr & {\text{Anil's share}} = {\text{Rs}}{\text{.}}\left( {19.5 \times \frac{3}{{13}}} \right) \cr & = {\text{Rs. }}{\text{4}}{\text{.5 lakh}} \cr & {\text{Required amount}} = 50\% {\text{ of 4}}{\text{.5 lakh}} \cr & = {\text{Rs. }}{\text{2}}{\text{.25 lakh}} \cr} $$
66. Arun started a business investing Rs. 38000. After 5 months Bakul joined him with a capital of Rs. 55000. At the end of the year the total profit was Rs. 22000. What is the approximate difference between the share of profits Arun and Bakul ?
a) Rs. 1007
b) Rs. 1192
c) Rs. 1568
d) Rs. 1857
Discussion
Explanation:
$$\eqalign{ & {\text{Arun}}:{\text{Bakul}} = \left( {38000 \times 12} \right):\left( {55000 \times 7} \right) \cr & = 456000:385000 \cr & = 456:385 \cr & {\text{Required difference}} = {\text{Rs}}{\text{.}}\,\left[ {22000 \times \left( {\frac{{456}}{{841}} - \frac{{385}}{{841}}} \right)} \right] \cr & = {\text{Rs}}{\text{.}}\left( {22000 \times \frac{{71}}{{841}}} \right) \cr & = {\text{Rs. }}{\text{1857}}{\text{.31}} \cr & = {\text{Rs. }}{\text{1857 }} \cr} $$
67. Goutam started a business with a sum of Rs. 60000. Jatin joined him 8 months later with a sum of Rs. 35000. At what respective ratio will the two share of profit after two years ?
a) 2 : 1
b) 3 : 1
c) 18 : 7
d) 37 : 14
Discussion
Explanation:
$$\eqalign{ & {\text{Gautam}}:{\text{Jatin}} = \left( {6000 \times 12} \right):\left( {35000 \times 8} \right) \cr & = 720000:280000 \cr & = 18:7 \cr} $$
68. Simran started a software business by investing Rs. 50000. After six months, Nanda joined her with a capital of Rs. 80000. After 3 years, they earned a profit of Rs. 24500. What was Simran's share in the profit ?
a) Rs. 9423
b) Rs. 10250
c) Rs. 12500
d) None of these
Discussion
Explanation:
$$\eqalign{ & {\text{Simran}}:{\text{Nanda}} = \left( {50000 \times 36} \right):\left( {80000 \times 30} \right) \cr & = 1800000:2400000 \cr & = 3:4 \cr & {\text{Simran's share}} = {\text{Rs}}{\text{.}}\left( {24500 \times \frac{3}{7}} \right) \cr & = {\text{Rs. }}{\text{10500 }} \cr} $$
69. Dilip, Ram and Avtar started a shop by investing Rs. 2700, Rs. 8100 and Rs. 7200 respectively. At the end of one year, the profit earned was distributed. If Ram's share was Rs. 3600, what was their total profit ?
a) Rs. 8000
b) Rs. 10800
c) Rs. 11600
d) Data inadequate
Discussion
Explanation:
$$\eqalign{ & {\text{Dilip}}:{\text{Ram}}:{\text{Avtar}} \cr & = 2700:8100:7200 \cr & = 3:9:8 \cr & {\text{Let the total profit be Rs}}.\,x \cr & {\text{Ram's share}} = {\text{Rs}}{\text{.}}\,\left( {\frac{9}{{20}}x} \right) \cr & \therefore \frac{9}{{20}}x = 3600 \cr & x = \left( {\frac{{3600 \times 20}}{9}} \right) \cr & x = 8000 \cr & {\text{Total profit}} = {\text{Rs}}{\text{. 8000}} \cr} $$
70. Aman started a business investing Rs. 70000. Rakhi joined him after six months with an amount of Rs. 105000 and Sagar joined them with Rs. 1.4 lakhs after another six months. The amount of profit earned should be distributed in what ratio among Aman, Rakhi and Sagar respectively, 3 years after Aman started the business ?
a) 7 : 6 : 10
b) 12 : 15 : 16
c) 42 : 45 : 56
d) Cannot be determined
Discussion
Explanation: Aman : Rakhi : Sagar
= (70000 × 36) : (105000 × 30) : (140000 × 24)
= 12 : 15 : 16
71. Yogesh started a business investing Rs. 45000. After 3 months, Pranab joined him with a capital of Rs. 60000. After another 6 months, Atul joined them with a capital of Rs. 90000. At the end of the year, they made a profit of Rs. 20000. What would be Atuls share in it?
a) Rs. 7000
b) Rs. 6000
c) Rs. 5000
d) Rs. 4000
Discussion
Explanation: Yogesh : Pranab : Atul
= 45000 × 12 : 60000 × 9 : 90000 × 3
= 2 : 2 : 1
Atul's share
$$\eqalign{ & = {\text{Rs}}{\text{. }}20000 \times \frac{1}{5} \cr & = {\text{Rs}}{\text{. }}4000 \cr} $$
72. Rahul and Bharti are partners in a business. Rahul contributes $$\frac{1}{4}$$th capital for 15 months and Bharti received $$\frac{2}{3}$$ of profit. For how long Bharti money was used
a) 8 months
b) 10 months
c) 11 months
d) 17 months
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{total}}\,{\text{profit}}\,{\text{be}}\,Rs.\,Z \cr & {\text{Bharti's}}\,{\text{share}} = {\frac{2}{3}} Z \cr & {\text{Rahul's}}\,{\text{share}} = Z - {\frac{2}{3}} Z \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{Z}{3} \cr & {\text{Rahul:Bharti}} \cr & = \frac{Z}{3}: {\frac{2}{3}} Z = 1:2 \cr & {\text{Now}}\,{\text{let}}\,{\text{the}}\,{\text{total}}\,{\text{capital}}\,{\text{be}}\,Rs\,X, {\text{and}}\,{\text{Bharti}}\,{\text{capital}}\,{\text{was}}\,{\text{used}}\,{\text{for}}\,Y\,{\text{months}}. \cr & {\text{then,}}\,{\text{Rahul}}\,{\text{capital}}\,{\text{will}}\,{\text{be}}\, {\frac{1}{4}} X \cr & {\text{and}}\,{\text{Bharti}}\,{\text{capital}}\,{\text{will}}\,{\text{be}}\,X - {\frac{1}{4}} X \cr & = \frac{{3X}}{4} \cr & \frac{{\frac{1}{4}X*15}}{{\frac{3}{4}X*Y}} = \frac{1}{2} \cr & Y = \frac{{15*2}}{3} = 10 \cr} $$
73. P and Q started a business investing Rs. 85000 and Rs. 15000 resp. In what ratio the profit earned after 2 years be divided between P and Q respectively.
a) 17 : 5
b) 17 : 3
c) 17 : 6
d) 17 : 7
Discussion
Explanation: P : Q = 85000 : 15000
= 85 : 15
= 17 : 3
74. A, B and C enter into a partnership investing Rs 35000, Rs 45000 and Rs 55000 resp. The respective share of A,B and C in an annual profit of Rs 40500 are.
a) Rs. 11500, Rs. 13500, Rs. 16500
b) Rs. 10500, Rs. 12500, Rs. 16500
c) Rs. 10500, Rs. 13500, Rs. 15500
d) Rs. 10500, Rs. 13500, Rs. 16500
Discussion
Explanation:
$$\eqalign{ & A:B:C \cr & = 35000:45000:55000 \cr & = 7:9:11 \cr & {\text{Now}}\,{\text{we}}\,{\text{are}}\,{\text{having}}\,{\text{the}}\,{\text{ratio}}\, {\text{to}}\,{\text{get}}\,{\text{the}}\,{\text{share,}}\,{\text{first}}\,{\text{make}}\,{\text{total}}\,{\text{of}}\,{\text{above}}\,{\text{ratio}}\, {\text{then}}\,{\text{get}}\,{\text{each}}\,{\text{share}}{\text{.}} \cr & {\text{A's}}\,{\text{Share}} = 40500*\frac{7}{{27}} = Rs.\,10500 \cr & {\text{B's}}\,{\text{Share}} = 40500*\frac{9}{{27}} = Rs.\,13500 \cr & {\text{C's}}\,{\text{Share}} = 40500*\frac{{11}}{{27}} = Rs.\,16500\, \cr} $$
75. Rahul, Arun and Sumit started a business. Rahul invested $$\frac{1}{2}$$ part, Arun $$\frac{1}{3}$$ part and rest of the capital was invested by Sumit. The ratio of their profits will be = ?
a) 2 : 3 : 1
b) 3 : 2 : 1
c) 2 : 3 : 6
d) 3 : 2 : 5
Discussion
Explanation: Let total capital be Rs. x
$$\eqalign{ & {\text{Rahul's share}} = {\text{Rs}}{\text{.}}\frac{x}{2} \cr & {\text{Arun's share}} = {\text{Rs}}{\text{.}}\frac{x}{3} \cr & {\text{Sumit's share}} \cr & = {\text{Rs}}{\text{.}}\left[ {x - \left( {\frac{x}{2} + \frac{x}{3}} \right)} \right] \cr & = {\text{Rs}}{\text{. }}\frac{x}{6} \cr & {\text{Required ratio}} \cr & = \frac{x}{2}:\frac{x}{3}:\frac{x}{6} \cr & = \frac{1}{2}:\frac{1}{3}:\frac{1}{6} \cr & = 3:2:1 \cr} $$
76. A, B and C subscribe Rs. 50000 for a business. A subscribes Rs. 4000 more than B and B Rs. 5000 more than C. Out of a total profit of Rs. 35000, A receives ?
a) Rs. 8400
b) Rs. 11900
c) Rs. 13600
d) Rs. 14700
Discussion
Explanation:
$$\eqalign{ & {\text{Let C}} = x \cr & {\text{B}} = x + 5000{\text{ and}} \cr & {\text{A}} = x + 5000 + 4000 = x + 9000 \cr & \Rightarrow x + x + 5000 + x + 9000 = 50000 \cr & 3x = 36000 \cr & x = 12000 \cr & {\text{A}}:{\text{B}}:{\text{C}} \cr & = 21000:17000:12000 \cr & = 21:17:12 \cr & {\text{A's share}} = {\text{Rs}}{\text{.}}\left( {35000 \times \frac{{21}}{{50}}} \right) \cr & = {\text{Rs.}}{\text{14700}} \cr} $$
77. A, B and C are three partners. They altogether invested Rs. 14000 in business. At the end of the year, A got Rs. 337.50, B Rs. 1125 and C Rs. 637.50 as profit. The difference between the investments of B and A was ?
a) Rs. 2200
b) Rs. 3200
c) Rs. 4200
d) Rs. 5250
Discussion
Explanation: Ratio of investments of A, B and C = Ratio of their profits
$$\eqalign{ & = 337.50:1125:637.50 \cr & = 9:30:17 \cr & {\text{A's investment}} = {\text{Rs}}{\text{.}}\left( {14000 \times \frac{9}{{56}}} \right) \cr & = {\text{Rs. }}{\text{2250}} \cr & {\text{B's investment}} = {\text{Rs}}{\text{.}}\left( {14000 \times \frac{{30}}{{56}}} \right) \cr & = {\text{Rs. }}{\text{7500}} \cr & {\text{Required difference}} = {\text{Rs}}{\text{.}}\left( {7500 - 2250} \right) \cr & = {\text{Rs. }}{\text{5250}} \cr} $$
78. A and B are two partners in a firm sharing the profit in the ratio 4 : 5. If the firm earns a profit of Rs. 14130, then profit to be received by B ?
a) Rs. 6280
b) Rs. 7850
c) Rs. 1570
d) Rs. 3140
Discussion
Explanation:
$$\eqalign{ & {\text{A}}:{\text{B}} \cr & 4:{\text{ }}5 \cr & \left( {4 + 5} \right){\text{units}} = {\text{Rs}}{\text{. 14130}} \cr & {\text{1 unit}} = \frac{{14130}}{9} = {\text{Rs}}{\text{. }}1570 \cr & 5{\text{ units}} = 5 \times 1570 = {\text{Rs}}{\text{. 7850}} \cr & {\text{Share of B}} = {\text{Rs}}{\text{. 7850}} \cr} $$
79. A and B started a business investing amounts in the ratio of 2 : 3. If A has invested an additional amount of Rs. 10000, their ratio of investment would have been 3 : 2. The amount invested by A was ?
a) Rs. 8000
b) Rs. 12000
c) Rs. 18000
d) Rs. 20000
Discussion
Explanation: Initial ratio of investments by A and B = 2 : 3
Let their respective investments are 2x and 3x
If A added Rs. 10000 to his investment
Then the new ratio = 3 : 2
$$\eqalign{ & \frac{{2x + 10000}}{{3x}} = \frac{3}{2} \cr & 4x + 20000 = 9x \cr & 5x = 20000 \cr & x = 4000 \cr} $$
Original investment by A = 2 × 4000 = Rs. 8000
80. A, B and C started a business investing amounts in the ratio of 5 : 6 : 8 respectively. After one year, C withdrew 50% of the amount and A invested an additional amount of 60% of the original amount invested by him. In what ratio, the profit earned at the end of 2 years should be distributed among A, B and C respectively ?
a) 2 : 3 : 3
b) 4 : 3 : 2
c) 13 : 12 : 12
d) Cannot be determined
Discussion
Explanation: Let the initial investments of A, B and C be Rs. 5x, Rs. 6x and Rs. 8x respectively
A : B : C
[5x × 12 + (160% of 5x) × 12] : (6x × 24) : (8x × 12 + 4x × 12)
= 156x : 144x : 144x
= 13 : 12 : 12
81. A, B and C enter into a partnership. A initially invests Rs. 25 lakhs and adds another Rs. 10 lakhs after one year. B initially invests Rs. 35 lakhs and withdraws Rs. 10 lakhs after 2 years and C invests Rs. 30 lakhs. In what ratio should the profits be divided at the end of 3 years ?
a) 10 : 10 : 9
b) 20 : 20 :19
c) 20 : 19 :18
d) None of these
Discussion
Explanation: A : B : C
= (25 lakhs × 1 + 35 lakhs × 2) : (35 lakhs × 2 + 25 lakhs × 1) : (30 lakhs × 3)
= 95 lakhs : 95 lakhs : 90 lakhs
= 19 : 19 : 18
82. Subhash starts a business by investing Rs. 25000. 6 months later Aditya joins him by investing Rs. 15000. After another 6 months Aditya invests an additional amount of Rs. 15000. At the end of 3 years they earn a profit of Rs. 247000. What is Aditya's share in the profit ?
a) Rs. 105000
b) Rs. 111500
c) Rs. 123000
d) Rs. 117000
Discussion
Explanation:
$$\eqalign{ & {\text{Subhas}}:{\text{Aditya}} = \left( {25000 \times 36} \right):\left( {15000 \times 6 + 30000 \times 24} \right) \cr & = 900000:810000 \cr & = 10:9 \cr & {\text{Aditya's share}} = {\text{Rs}}{\text{.}}\left( {247000 \times \frac{9}{{19}}} \right) \cr & = {\text{Rs}}{\text{.117000}} \cr} $$
83. A, B and C enter into a partnership with capitals in the ratio 5 : 6 : 8. At the end of the business term, they received the profit in the ratio 5 : 3 : 12. Find the ratio of time for which they contributed their capitals ?
a) 2 : 1 : 3
b) 1 : 2 : 3
c) 2 : 3 : 1
d) 3 : 2 : 1
Discussion
Explanation:
$$\eqalign{ & {\text{Here}}\,{{\text{p}}_1}:{{\text{p}}_2}:{{\text{p}}_3} = 5:3:12 \cr & {\text{and}}\,{{\text{x}}_1}:{{\text{x}}_2}:{{\text{x}}_3} = 5:6:8 \cr & {\text{Required}}\,{\text{ratio}} \cr & = \frac{{{{\text{p}}_1}}}{{{{\text{x}}_1}}}:\frac{{{{\text{p}}_2}}}{{{{\text{x}}_2}}}:\frac{{{{\text{p}}_3}}}{{{{\text{x}}_3}}} \cr & = \frac{5}{5}:\frac{3}{6}:\frac{{12}}{8} \cr & = 1:\frac{1}{2}:\frac{3}{2} \cr & = 2:1:3 \cr} $$
84. X and Z invest capital in the ratio of 2 : 1 while X and Y invest capital in the ratio of 3 : 2. If their annual profit is Rs. 157300 then what is Y share ?
a) Rs. 48400
b) Rs. 58809
c) Rs. 48810
d) Rs. 47782
Discussion
Explanation: X : Z = 2 : 1
X : Y = 3 : 2
X : Z = 2 : 1 (multiply with 3)
X : Y = 3 : 2 (multiply with 2)
i.e X : Z = 6 : 3
X : Y = 6 : 4
∴ X : Y : Z = 6 : 4 : 3
(6 + 4 + 3) units = Rs.157300
13 units = Rs.157300
1 unit = Rs.12100
4 units = Rs.12100 × 4 = Rs. 48400
Share of Y = Rs. 48400
85. A and B start a business with investments of Rs. 5000 and Rs. 4500 respectively. After 4 months, A takes out half of his capital. After two more months B takes out one-third of his capital while C joins them with a capital of Rs. 7000. At the end of a year, they earn a profit of Rs. 5080. Find the share of each member in the profit ?
a) A = Rs. 1400, B = Rs. 1900, C = Rs. 1780
b) A = Rs. 1600, B = Rs. 1800, C = Rs. 1680
c) A = Rs. 1800, B = Rs. 1500, C = Rs. 1780
d) A = Rs. 1680, B = Rs. 1600, C = Rs. 1800
Discussion
Explanation: A : B : C
(5000 × 4 + 2500 × 8) : (4500 × 6 + 3000 × 6) : (7000 × 6)
= 40000 : 45000 : 42000
= 40 : 45 : 42
$$\eqalign{ & {\text{A's share}} = {\text{Rs}}{\text{.}}\left( {5080 \times \frac{{40}}{{127}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{1600}} \cr & {\text{B's share}} = {\text{Rs}}{\text{.}}\left( {5080 \times \frac{{45}}{{127}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{1800}} \cr & {\text{C's share}} = {\text{Rs}}{\text{.}}\left( {5080 \times \frac{{42}}{{127}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{1680}} \cr} $$
86. A, B and C invested Rs. 47000 for a business. If A subscribes Rs. 7000 more than B and B Rs. 5000 more than C, then out of total profit of Rs. 4700, C receives ?
a) Rs. 1200
b) Rs. 4500
c) Rs. 1000
d) None of these
Discussion
Explanation: Let C subscribes the business = Rs. x
A : B : C
Capital → (x + 12000) : (x + 5000) : x
Note: Profit would be divide in the ratio of their capitals
$$\eqalign{ & \Rightarrow \left( {x + 12000} \right) + \left( {x + 5000} \right) + x = 47000 \cr & 3x + 17000 = 47000 \cr & 3x = 30000 \cr & x = 10000 \cr & {\text{ A }}:{\text{ B }}:{\text{ C}} \cr & {\text{Capital}} \to 22000:15000:10000 \cr & {\text{Profit}} \to {\text{ }}22{\text{ }}:{\text{ }}15{\text{ }}:{\text{ }}10 \cr & \left( {22 + 15 + 10} \right){\text{ units}} = {\text{4700}} \cr & {\text{1 unit}} = \frac{{4700}}{{47}} = 100 \cr & {\text{Share of C}} = 10{\text{ units}} = 10 \times 100 = {\text{Rs}}.1000 \cr & \cr} $$
87. A and B started a business investing amount of Rs. 185000 and Rs. 225000 respectively. If B's share in the profit earned by them is Rs. 9000 then what is the total profit earned by them together ?
a) Rs. 17400
b) Rs. 16400
c) Rs. 16800
d) Rs. 17800
Discussion
Explanation:
| A | B | ||
| Capital → | 1850000 | : | 225000 |
| Profit → | 37 | : | 45 |
| ↓×200 | : | ↓×200 | |
| 7400 | 9000 |
88. A and B stared a boutique investing amounts of Rs. 35000 and Rs. 56000 respectively. If A's share in the profit earned by them need is Rs. 45000, then what is the total profit earned ?
a) Rs. 81000
b) Rs. 127000
c) Rs. 72000
d) Rs. 117000
Discussion
Explanation:
| A | B | ||
| Capital → | 35000 | : | 56000 |
| Profit → | 5 | : | 8 |
| ↓×9000 | : | ↓×9000 | |
| 45000 | 72000 |
89. Sonia started a business investing Rs. 60000. After 6 months Vivek joined him with an amount of Rs. 140000. After 1 year Kirti also joined them with Rs. 120000. After 2 years the business yielded a total profit of Rs. 450000. What is the share of Vivek in the profit ?
a) Rs. 140000
b) Rs. 198500
c) Rs. 210000
d) Rs. 215000
Discussion
Explanation: Sonia : Vivek : Sagar
= (60000 × 24) : (140000 × 18) : (120000 × 12)
= 1440000 : 2520000 : 1440000
= 4 : 7 : 4
$$\eqalign{ & {\text{Vivek's share}} = {\text{Rs}}{\text{.}}\,\left( {450000 \times \frac{7}{{15}}} \right) \cr & = {\text{Rs}}{\text{. 210000}} \cr} $$
90. P and Q started a business in the ratio of 2 : 3. After 1 year P left the business but Q continues. After 2 years he had profit of Rs. 26000. What is the profit of Q?
a) Rs. 10400
b) Rs. 13000
c) Rs. 15600
d) None of these
Discussion
Explanation: Let the initial capital of P and Q be Rs. 2x and Rs. 3x respectively
Ratio of profits
$$\eqalign{ & = \left( {2x \times 12} \right):\left( {3x \times 24} \right) \cr & = 24x:72x \cr & = 1:3 \cr & {\text{Q's share}} = {\text{Rs}}{\text{.}}\left( {26000 \times \frac{3}{4}} \right) \cr & = {\text{Rs. }}{\text{19500}} \cr} $$
91. Two friends invested Rs.1500 and Rs. 2500 in a business. They earned a profit of Rs. 800. One-half of the profit was divided equally between them and the other half was divided in proportion to their capitals. How much did each of them receive ?
a) Rs. 350 and Rs. 450
b) Rs. 360 and Rs. 440
c) Rs. 370 and Rs. 430
d) Rs. 375 and Rs. 425
Discussion
Explanation:
$$\eqalign{ & {\text{Ratio of shares}} = 1500:2500 \cr & = 3:5 \cr & {\text{Share of first friend}} = {\text{Rs}}{\text{.}}\left[ {\frac{{400}}{2} + \left( {400 \times \frac{3}{8}} \right)} \right] \cr & = {\text{Rs}}.\left( {200 + 150} \right) \cr & = {\text{Rs}}{\text{. 350}} \cr & {\text{Share of second friend}} = {\text{Rs}}{\text{.}}\left[ {\frac{{400}}{2} + \left( {400 \times \frac{5}{8}} \right)} \right] \cr & = {\text{Rs}}.\left( {200 + 250} \right) \cr & = {\text{Rs}}{\text{. 450}} \cr} $$
92. Three persons stared a placement business with a capital of Rs. 3000. B invests Rs. 600 less than A and C invests Rs. 300 less than B. What is B's share in a profit of Rs. 886 ?
a) Rs. 443
b) Rs. 354.40
c) Rs. 265.80
d) Rs. 177.20
Discussion
Explanation: Let A's capital = Rs. x
B's capital = Rs. (x - 600)
C's capital = Rs. (x - 600) - 300 = Rs. (x - 900)
∴ x + (x - 600) + (x - 900) = 3000
3x = 4500
x = 1500
$$\eqalign{ & {\text{So, A}}:{\text{B}}:{\text{C}} = 1500:900:600 \cr & = 5:3:2 \cr & {\text{B's share}} = {\text{Rs}}{\text{.}}\left( {886 \times \frac{3}{{10}}} \right) \cr & = {\text{Rs}}{\text{. 265}}{\text{.80}} \cr} $$
93. X and Y are partners in a business. They invest in the ratio 5 : 6, at the end of 8 months X withdraws his capital. If they receive profits in the ratio of 5 : 9. Find how long Y's investment was used ?
a) 12 months
b) 10 months
c) 15 months
d) 14 months
Discussion
Explanation: Let Y's investment is used for T months → Now by using formula
$$\eqalign{ & \frac{{5 \times 8}}{{6 \times {{\text{T}}_2}}} = \frac{5}{9} \cr & {\text{T}} = 12{\text{ months}} \cr} $$
94. Four milkmen rented a pasture. M put to graze 16 cows for 3 months and N 20 cows for 4 months, O 18 cows for 6 months and P 42 cows for 2 months. If M's share of rent be Rs. 2400, the rent paid by O is ?
a) Rs. 3200
b) Rs. 4200
c) Rs. 4000
d) Rs. 5400
Discussion
Explanation: M graze 16 cows for 3 months
N graze 20 cows for 4 months
O graze 18 cows for 6 months
P graze 42 cows for 2 months
Ration of Rent = M : N : O : P
= (16 × 3) : (20 × 4) : (18 × 6) : (42 × 2)
= 48 : 80 : 108 : 84
= 12 : 20 : 27 : 21
$$\eqalign{ & {\text{12 units}} = {\text{Rs}}{\text{. 2400}} \cr & {\text{1 unit}} = \frac{{2400}}{{12}} \cr & 27{\text{ units}} = \frac{{2400 \times 27}}{{12}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}5400 \cr} $$
95. Shankar starts a business with an investment of Rs. 120000. After three months, Aniket joined him with an invesment of Rs. 190000. They earned a profit of Rs. 1750000 after one year. What is Aniket's share in the profit ?
a) Rs. 800000
b) Rs. 850000
c) Rs. 900000
d) Rs. 950000
Discussion
Explanation:
$$\eqalign{ & {\text{Shankar}}:{\text{Aniket}} = \left( {120000 \times 12} \right):\left( {190000 \times 9} \right) \cr & = 1440000:1710000 \cr & = 16:19 \cr & {\text{Aniket's share}} = {\text{Rs}}{\text{.}}\left( {1750000 \times \frac{{19}}{{35}}} \right) \cr & = {\text{Rs. }}{\text{950000 }} \cr} $$
96. Rs. 700 is divided among A, B, C so that A receives half as much as B and B half as much as C. Then C's share is
a) Rs. 200
b) Rs. 300
c) Rs. 400
d) Rs. 500
Discussion
Explanation: Let C = x
B = $$\frac{{\text{x}}}{2}$$
A = $$\frac{{\text{x}}}{4}$$
A : B : C = 1 : 2 : 4
C's share = $${\text{Rs}}{\text{.}}\left( {\frac{4}{7} \times 700} \right) = 400$$
97. Anand and Deepak started a business investing Rs.22,500 and Rs.35,000 respectively. Out of a total profit of Rs. 13,800. Deepak's share is
a) Rs. 9600
b) Rs. 8500
c) Rs. 8450
d) Rs. 8400
Discussion
Explanation: Ratio of their shares = 22500 : 35000
= 9 : 14
Deepak's share
$$\eqalign{ & = {\text{Rs}}{\text{.}}\left( {13800 \times \frac{{14}}{{23}}} \right) \cr & = {\text{Rs}}{\text{. 8}}400 \cr} $$
98. A started a business with Rs. 21,000 and is joined afterwards by B with Rs. 36,000. After how many months did B join if the profits at the end of the year are divided equally?
a) 4
b) 5
c) 6
d) 7
Discussion
Explanation: Suppose B joined after x months
21000 × 12 = 36000 × (12 - x)
36x = 180
x = 5
99. Nirmal and Kapil started a business investing Rs. 9000 and Rs. 12000 respectively. After 6 months, Kapil withdrew half of his investment. If after a year, the total profit was Rs. 4600, what was Kapil’s share initially ?
a) Rs. 2300
b) Rs. 2400
c) Rs. 2500
d) None of above
Discussion
Explanation: Nirmal : Kapil = 9000 × 12 : (12000 × 6 + 6000 × 6)
= 1 : 1
Kapils share
$$\eqalign{ & = {\text{Rs}}{\text{.}}\left( {4600 \times \frac{1}{2}} \right) \cr & = {\text{Rs}}{\text{. 2300}} \cr} $$
100. Manoj received Rs. 6000 as his share out of the total profit of Rs. 9000 which he and Ramesh earned at the end of one year. If Manoj invested Rs.120000 for 6 months, whereas Ramesh invested his amount for the whole year, what was the amount invested by Ramesh
a) Rs. 2000
b) Rs. 30000
c) Rs. 4000
d) Rs. 5000
Discussion
Explanation: Suppose Ramesh invested Rs. x.
Manoj : Ramesh = 120000 × 6 : x × 12.
$$\eqalign{ & \frac{{720000}}{{12{\text{x}}}}:\frac{{6000}}{{3000}} \cr & {\text{x}} = \frac{{720000}}{{24}} \cr & {\text{x}} = 30000 \cr} $$