Partnership Questions and Answers

Answer: C

They received ₹ 1,100 and ₹ 1,700.

Half the profit, ₹ 1,400, is shared equally: ₹ 700 each. The other ₹ 1,400 is shared in the ratio of capitals, \(3000 : 7500 = 2 : 5\):

\[ \begin{aligned} \text{first} &= 1400 \times \tfrac{2}{7} = 400 \\ \text{second} &= 1400 \times \tfrac{5}{7} = 1000 \end{aligned} \]

Totals: \(700 + 400 = 1100\) and \(700 + 1000 = 1700\), which add to the profit of ₹ 2,800. Every option adds to 2,800, so the sum alone does not decide it; only this split gives the right shares.

Answer: B

The ratio of the times is 15 : 12 : 10. Profit is proportional to the capital multiplied by the time, so the time is proportional to profit divided by capital.

\[ t_X : t_Y : t_Z = \frac{6}{8} : \frac{3}{5} : \frac{2}{4} \]

\[ = 0.75 : 0.6 : 0.5 = 15 : 12 : 10 \]

The options 13 : 8 : 5, 18 : 5 : 9 and 20 : 15 : 8 do not match these three values, so the second option is the answer. As a check, capitals of 8, 5 and 4 units for times of 15, 12 and 10 give products of 120, 60 and 40, which are in the ratio 6 : 3 : 2 of the profits.

Answer: D

A gets Rs. 15,400.

The ratio 7/3 : 9/4 : 12/5 becomes 140 : 135 : 144 over 60. After 6 months C raises his share by 75%, so C's investment is 144 for 6 months and 252 for the next 6 months. The shares are in proportion to investment times months: A 140 × 12 = 1680, B 135 × 12 = 1620, C 144 × 6 + 252 × 6 = 2376, a total of 5676.

\[ 52030\times\frac{1680}{5676}=15400 \]

The other amounts do not match this proportion.

Using the ratio of investment times time is the standard method for partnerships. The profit is divided in the ratio 1680 : 1620 : 2376, which simplifies to 140 : 135 : 198 and totals 473 parts. Each part is 52030 / 473 = 110, so A gets 140 × 110 = 15400.

Answer: D

The total profit is ₹ 38,000. Profit is shared in the ratio of capital times period. P's capital is 1 part for 3 units of time and Q's is 2 parts for 1 unit of time. \[ P : Q = 1\times 3 : 2\times 1 = 3 : 2 \] P's share is \(\frac{3}{5}\) of the profit, so \[ \text{Total} = 22800\times\frac{5}{3} = 38000 \] Reversing the periods would give a ratio of 1 : 6 and a very different total, and ₹ 36,000, ₹ 36,900 and ₹ 42,000 do not match P's share of 3 out of 5 parts.

Answer: B

The value of \(x\) is ₹ 3000.

P's total investment is \(5000 \times 6 = 30\,000\) and Q's is \(8x\). Q's share of the profit is its share of the total investment:

\[ \begin{aligned} \frac{8x}{30\,000 + 8x} &= \frac{4}{9} \\ 72x &= 120\,000 + 32x \\ x &= 3000 \end{aligned} \]

Check: Q invests 24,000, P 30,000, and \(24\,000/54\,000 = 4/9\). 2500 gives Q only 20,000 (40%); 4687 and 8437 give Q more than 4/9.

Answer: D

The ratio is 63 : 86 : 110.

Take the monthly investments as 3, 4 and 5 units. For the second six months they rise by 10%, 15% and 20% of those amounts, to 3.3, 4.6 and 6 units. Since both periods last six months, the yearly totals are proportional to the sums:

\[ \begin{aligned} &3 + 3.3 : 4 + 4.6 : 5 + 6 \\ &= 6.3 : 8.6 : 11 = 63 : 86 : 110 \end{aligned} \]

33 : 46 : 60 is the second half-year alone, and 22 : 23 : 24 and 22 : 33 : 50 do not follow from either period.

Answer: D

Manju joined after 6 months.

Profits are shared in the ratio of capital multiplied by time. Equal shares mean equal products. Bharati invested for 12 months; let Manju invest for \(m\) months:

\[ \begin{aligned} 18000 \times 12 &= 36000 \times m \\ m &= 6 \end{aligned} \]

Manju's money was at work for 6 months, so she joined 12 - 6 = 6 months after the start. Joining after 10, 5 or 8 months would leave her capital working for 2, 7 or 4 months, giving products of 72,000, 252,000 or 144,000 against Bharati's 216,000.

Answer: A

B and C receive ₹ 405 and ₹ 486.

Let the total profit be \(P\). A gets \(P/10\) as manager, and the remaining \(0.9P\) is shared 1500 : 2500 : 3000 = 3 : 5 : 6:

\[ \begin{aligned} 0.1P + 0.9P \times \tfrac{3}{14} &= 369 \\ P\left(\tfrac{14 + 27}{140}\right) &= 369 \Rightarrow P = 1260 \end{aligned} \]

The shared part is 1,134. B gets \(1134 \times 5/14 = 405\) and C gets \(1134 \times 6/14 = 486\). Option C lists the same amounts the other way round, giving C the smaller share though C invested more.

Answer: A

Kalu invested Rs 42000.

Profit is shared in proportion to capital multiplied by time. Raju's Rs 36000 was in for 12 months; Kalu joined after 4 months, so his capital \(x\) was in for 8 months.

\[ \begin{aligned} \frac{36000 \times 12}{8x} &= \frac{9}{7} \\ 72x &= 36000 \times 12 \times 7 \\ x &= 42000 \end{aligned} \]

Check: \(36000 \times 12 = 432000\) and \(42000 \times 8 = 336000\), in the ratio \(432 : 336 = 9 : 7\).

Rs 36000 is what you get by ignoring the time and taking equal capitals, which would give a 3 : 2 split. Rs 56000 would give \(432000 : 448000\), a larger share for Kalu, the reverse of 9 : 7.

Answer: C

The investments are in the ratio 8 : 9 : 12.

Profit is proportional to investment times duration, so investment is proportional to profit divided by duration:

\[ \begin{aligned} \frac{4}{3} : \frac{3}{2} : \frac{14}{7} &= \frac{4}{3} : \frac{3}{2} : 2 \\ &= 8 : 9 : 12 \quad (\times 6) \end{aligned} \]

Check: \(8 \times 3 : 9 \times 2 : 12 \times 7 = 24 : 18 : 84 = 4 : 3 : 14\). Multiplying profit by duration instead of dividing gives a different, wrong ratio.