1. 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

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

2. 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

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

$$\eqalign{ & \left( {3 + 5 + 6} \right){\text{units}} = {\text{Rs}}{\text{. 5250}} \cr & {\text{14 units}} = {\text{Rs}}{\text{. 5250}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{. 375}} \cr & {\text{Share of A}} = 375 \times 3 \cr & = {\text{Rs}}{\text{. 1125}} \cr & {\text{Share of B}} = 375 \times 5 \cr & = {\text{Rs}}{\text{. 1875}} \cr & {\text{Share of C}} = 375 \times 6 \cr & = {\text{Rs}}{\text{. 2250}} \cr} $$

3. 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

Explanation: Let amount invested by A = Rs. x

A | : | B | |

Capital → | x | : | (x + 5000) |

$$\eqalign{ & {\text{Share of A in profit}} = \frac{{\left( {26000 - 6000} \right)}}{2} \cr & = {\text{Rs}}{\text{. 10000}} \cr & {\text{Share of B in profit}} = \left( {26000 - 10000} \right) \cr & = {\text{Rs}}{\text{. 16000}} \cr & {\text{By using formulas:}} \cr & {\frac{{{{\text{C}}_{\text{1}}} \times {{\text{T}}_1}}}{{{{\text{C}}_{\text{2}}} \times {{\text{T}}_{\text{2}}}}} = \frac{{{{\text{P}}_{\text{1}}}}}{{{{\text{P}}_2}}}} \cr & \frac{{x \times 5}}{{\left( {x + 5000} \right) \times 6}} = \frac{{10000}}{{16000}} \cr & 4x = 3x + 15000 \cr & x = {\text{Rs}}.15000 \cr & {\text{Required capital of A}} = {\text{Rs}}{\text{. 15000}} \cr & {\text{Required capital of B}} = \left( {15000 + 5000} \right) \cr & = {\text{Rs}}{\text{. 20000}} \cr} $$

4. 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

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

$$\eqalign{ & \frac{{3a}}{{11}} = 2460 \cr & 3a = 2460 \times 11 \cr & a = \frac{{2460 \times 11}}{3} \cr & a = {\text{Rs}}{\text{.}}\,9020 \cr} $$

5. 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

Explanation:

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

6. 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

Explanation: Let their initial investments be x, 3x and 5x respectively

$$ = {\text{A}}:{\text{B}}:{\text{C}}$$

$$ = \left( {x \times 4 + 2x \times 8} \right)$$ : $$\left( {3x \times 4 + \frac{{3x}}{2} \times 8} \right)$$ : $$\left( {5x \times 4 + \frac{{5x}}{2} \times 8} \right)$$

$$\eqalign{ & = 20x:24x:40x \cr & = 5:6:10 \cr} $$

7. 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

Explanation:

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

8. 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

Explanation:

A | B | C | |

Amounts invested | 14000 | ||

Time (in months) | 12 | 7 | 5 |

168000 |

$$\eqalign{ & {\text{Ratio of profits }}4:3:2 \cr & {\text{Let their profits }}4x:3x:2x \cr & 4x = 168000 \cr & x = 42000 \cr & \Rightarrow {\text{Profit share of C}} = 2x \cr & = 2 \times 42000 \cr & = {\text{Rs}}{\text{. 84000}} \cr & \Rightarrow {\text{Capital invested by C}} = \frac{{84000}}{5} \cr & = {\text{Rs. }}16800 \cr} $$

9. 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

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)

$$\eqalign{ & {\text{35 units}} = {\text{Rs}}{\text{. 1820}} \cr & {\text{1 unit}} = \frac{{1820}}{{35}} = {\text{Rs}}.52 \cr & {\text{Hence A's share in profit}} \cr & = 5{\text{ units}} \cr & = 52 \times 5 \cr & = {\text{Rs}}{\text{. 260}} \cr} $$

10. 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

Explanation: Let C's capital be Rs. x

A : B : C = 2560 : 2000 : x

$$\eqalign{ & {\text{A's share}} \cr & = {\text{Rs}}{\text{.}}\left( {1105 \times \frac{{2560}}{{4560 + x}}} \right) \cr & \therefore 1105 \times \frac{{2560}}{{4560 + x}} = 320 \cr & 320x + 1459200 = 2828800 \cr & 320x = 1369600 \cr & x = 4280 \cr} $$