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81. 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

Discuss

Solution:

Let their initial investments be x, 3x and 5x respectively
Then,

= A : B : C

= (x \times 4 + 2 x \times 8)

(3 x \times 4 + \frac{3 x}{2} \times 8)

(5 x \times 4 + \frac{5 x}{2} \times 8)

\begin{aligned} = 20 x : 24 x : 40 x \\ = 5 : 6 : 10 \end{aligned}

82. 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

Discuss

Solution:

\begin{aligned} Suppose, \\ A invests Rs . \frac{x}{6} for \frac{y}{6} months \\ Then, \\ B invests Rs . \frac{x}{3} for \frac{y}{3} months \\ C invests [x - (\frac{x}{6} + \frac{x}{3})] i . e ., Rs . \frac{x}{2} for y months \\ ∴ A : B : C \\ = (\frac{x}{6} \times \frac{y}{6}) : (\frac{x}{3} \times \frac{y}{3}) : (\frac{x}{2} \times y) \\ = \frac{1}{36} : \frac{1}{9} : \frac{1}{2} \\ = 1 : 4 : 18 \\ Hence, B's share \\ = Rs . (4600 \times \frac{4}{23}) \\ = Rs . 800 \end{aligned}

83. 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

Discuss

Solution:

	A	B	C
Amounts invested	14000
Time (in months)	12	7	5
	168000

\begin{aligned} Ratio of profits 4 : 3 : 2 \\ Let their profits 4 x : 3 x : 2 x \\ \Leftrightarrow 4 x = 168000 \\ \Leftrightarrow x = 42000 \\ \Rightarrow Profit share of C \\ = 2 x \\ = 2 \times 42000 \\ = Rs . 84000 \\ \Rightarrow Capital invested by C \\ = \frac{84000}{5} \\ = Rs. 16800 \end{aligned}

84. 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

Discuss

Solution:

Let the total capital of A, B and C = 15 units
Let total time for investment = 12 units
Now, According to the question,

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)

\begin{aligned} 35 units = Rs . 1820 \\ 1 unit = \frac{1820}{35} = Rs .52 \\ Hence A's share in profit \\ = 5 units \\ = 52 \times 5 \\ = Rs . 260 \end{aligned}

85. 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

Discuss

Solution:

Let C's capital be Rs. x
Then,
A : B : C = 2560 : 2000 : x

\begin{aligned} A's share \\ = Rs . (1105 \times \frac{2560}{4560 + x}) \\ ∴ 1105 \times \frac{2560}{4560 + x} = 320 \\ \Rightarrow 320 x + 1459200 = 2828800 \\ \Rightarrow 320 x = 1369600 \\ \Rightarrow x = 4280 \end{aligned}

86. 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

Discuss

Solution:

\begin{aligned} Let total capital = Rs . x \\ Then, A's capital = Rs . \frac{x}{3} \end{aligned}

B's capital = (A + C)'s capital ⇒ 2(B's capital)
B's capital = (A + B + C)'s capital = Rs. x

\begin{aligned} \Rightarrow B's capital = Rs . \frac{x}{2} \\ \Rightarrow C's capital \\ = Rs . [x - (\frac{x}{3} + \frac{x}{2})] \\ = Rs . \frac{x}{6} \\ ∴ A : B : C \\ = \frac{x}{3} : \frac{x}{2} : \frac{x}{6} \\ = 2 : 3 : 1 \\ So C's share \\ = Rs . (900 \times \frac{1}{6}) \\ = Rs . 150 \end{aligned}

87. 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

Discuss

Solution:

Suppose B invested Rs. x for y months
Then, A invested Rs. 3x for 2y months

\begin{aligned} So, A : B \\ = (3 x \times 2 y) : (x \times y) \\ = 6 x y : x y \\ = 6 : 1 \\ ∴ B's profit : Total profit = 1 : 7 \\ Let the total profit is Rs . x \\ Then, \frac{1}{7} = \frac{4000}{x} \\ \Leftrightarrow x = 28000 \end{aligned}

88. 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

Discuss

Solution:

\begin{aligned} Suppose B joined for x months \\ Then, \\ \Rightarrow \frac{85000 \times 12}{42500 \times x} = \frac{3}{1} \\ \Rightarrow x = \frac{85000 \times 12}{42500 \times 3} \\ \Rightarrow x = 8 \\ So, B joined for 8 months \end{aligned}

89. 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

Discuss

Solution:

\begin{aligned} Let the total share = 200 units \\ ∴ Share of C = 200 \times \frac{1}{4} \\ = 50 units \\ Remaining share \\ = (200 - 50) \\ = 150 units \\ ∴ Share of A = \frac{200}{3 + 2} \times 3 \\ = 120 units \\ Share of B = \frac{200}{3 + 2} \times 2 \\ = 80 units \end{aligned}

According to the question,
C received equal amounts from A and B

\begin{aligned} ∴ A's remaining share \\ = (120 - 25) \\ = 95 \\ B's remaining share \\ = (80 - 25) \\ = 55 \end{aligned}

	A	:	B	:	C
New Ratio →	95	:	55	:	50
	19	:	11	:	10

90. 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

Discuss

Solution:

	A	:	B	:	C
Ratio of Profit →	2	:	3	:	7

\begin{aligned} Average gain = \frac{2 + 3 + 7}{3} \\ = 4 units \\ According to the question, \\ 4 units = Rs . 8000 \\ 1 unit = Rs . 2000 \\ 3 units = 3 \times 2000 \\ = Rs . 6000 \\ ∴ Share of B = Rs . 6000 \end{aligned}

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