1. 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

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

$$ = \left( {50000 \times 12} \right)$$ : $$\left[ {60000 \times \left( {12 - x} \right)} \right]$$ : $$\left[ {70000 \times \left( {12 - x} \right)} \right]$$

$$\eqalign{ & = 60:6\left( {12 - x} \right):7\left( {12 - x} \right) \cr & {\text{But ratio of profits}} \cr & = 20:18:21 \cr & = 60:54:63 \cr} $$

$$ 60:\left( {72 - 6x} \right):\left( {84 - 7x} \right)$$ = $$60$$ : $$54$$ : $$63$$

$$\eqalign{ & {\text{So, }}72 - 6x = 54 \cr & 6x = 18 \cr & x = 3 \cr} $$

2. 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

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

$$ \left( {5x \times 12} \right)$$ + $$\left( {6x \times 12} \right)$$ + $$\left( {6x \times 6} \right)$$ = $$490000 \times 12$$

$$\eqalign{ & 168x = 490000 \times 12 \cr & x = \left( {\frac{{490000 \times 12}}{{168}}} \right) \cr & x = 35000 \cr & {\text{R's investment}} = {\text{Rs}}{\text{. }}6x \cr & = {\text{Rs}}{\text{.}}\left( {6 \times 35000} \right) \cr & = {\text{Rs}}{\text{. 210000}} \cr} $$

3. 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

Explanation: Let their investments be Rs. 3x for p months

Rs. 4x for q months and Rs. 6x for r months respectively

$$\eqalign{ & 3xp:4xq:6xr = 1:1:1 \cr & 3p:4q:6r = 1:1:1 \cr & {\text{So, }}3p = 4q \cr & \Leftrightarrow q = \frac{{3p}}{4} \cr & 4q = 6r \cr & \Leftrightarrow r = \frac{{2q}}{3} = \left( {\frac{2}{3} \times \frac{3}{4}p} \right) = \frac{p}{2} \cr & \therefore p:q:r \cr & = p:\frac{{3p}}{4}:\frac{p}{2} \cr & = 4:3:2 \cr} $$

4. 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

Explanation: Suppose, Swati invested Rs. 5x for 7 months

Rajni invested Rs. 6x for y months

$$\eqalign{ & \frac{{5x \times 7}}{{6x \times y}} = \frac{5}{9} \cr & 30y = 315 \cr & y = 10\frac{1}{2} \cr} $$

5. 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

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

$$\eqalign{ & \frac{{\frac{1}{3}x \times 9}}{{\frac{2}{3}x \times y}} = \frac{3}{2} \cr & 18x = 6xy \cr & y = 3 \cr} $$

6. 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

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

$$\eqalign{ & {\text{B's capital}} = {\text{Rs}}{\text{.}}\frac{x}{2} \cr & {\text{C's capital}} = {\text{Rs}}{\text{.}}\left[ {x - \left( {\frac{x}{3} + \frac{x}{2}} \right)} \right] \cr & = {\text{Rs}}{\text{.}}\,\frac{x}{6} \cr & \therefore {\text{A}}:{\text{B}}:{\text{C}} = \frac{x}{3}:\frac{x}{2}:\frac{x}{6} \cr & = 2:3:1 \cr & {\text{C's share}} = {\text{Rs}}{\text{.}}\left( {900 \times \frac{1}{6}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{150}} \cr} $$

7. 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

Explanation: Suppose B invested Rs. x for y months

A invested Rs. 3x for 2y months

$$\eqalign{ & {\text{A}}:{\text{B}} = \left( {3x \times 2y} \right):\left( {x \times y} \right) \cr & = 6xy:xy \cr & = 6:1 \cr & {\text{B's profit}}:{\text{Total profit}} = 1:7 \cr & {\text{Let the total profit is Rs}}{\text{. }}x \cr & \frac{1}{7} = \frac{{4000}}{x} \cr & x = 28000 \cr} $$

8. 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

Explanation:

$$\eqalign{ & {\text{Suppose B joined for }}x{\text{ months}} \cr & \frac{{85000 \times 12}}{{42500 \times x}} = \frac{3}{1} \cr & x = \frac{{85000 \times 12}}{{42500 \times 3}} \cr & x = 8 \cr} $$

9. 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

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 |

10. 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

Explanation:

A | : | B | : | C | |

Ratio of Profit → | 2 | : | 3 | : | 7 |

$$\eqalign{ & {\text{Average gain}} = \frac{{2 + 3 + 7}}{3} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4{\text{ units}} \cr & {\text{4 units}} = {\text{Rs}}{\text{. 8000}} \cr & {\text{1 unit}} = {\text{Rs}}{\text{. 2000}} \cr & {\text{3 units}} = 3 \times 2000 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 6000}} \cr & {\text{Share of B}} = {\text{Rs}}{\text{. 6000}} \cr} $$