If A invests 3 times B, and B invests for 3 times as long as A, what is their Profit Ratio?

Answer: 1:1. P \propto C \times T. A's profit weight is 3 \times 1 = 3. B's profit weight is 1 \times 3 = 3. Ratio is 3:3 = 1:1.

Does giving an Active Partner a 20% salary guarantee they take home more total money than a Sleeping Partner?

Answer: No, depends on Capital. If the Sleeping partner invests vastly more capital, their share of the remaining 80% distributable profit could still exceed the Active partner's total take-home.

If Profit Ratio > Capital Ratio for Partner A against B, what does it definitively imply about their Time?

Answer: Time A > Time B. T = P/C. If A has a proportionately higher profit relative to their capital, it MUST be because they left it in longer.

A starts business. B joins later mid-year. Can B's final profit share be higher than A's?

Answer: Yes, if B's capital is huge. Profit depends on the product C \times T. Even with low T (joining later), a massive C can make the product larger.

Is Partnership Profit mathematically treated fundamentally as Simple Interest, Compound Interest, or merely a Weighted Ratio?

Answer: Weighted Ratio. It is a proportional distribution based on a simple weighted average of capital-time units.

Under normal partnership law, what happens if the business makes a net loss?

Answer: Shares are split in C x T ratio. Unless stipulated otherwise, losses are distributed using the exact same Capital \times Time ratio as profits.

A puts 1/3 of the total capital. If total profit is Rs 90, how much does A get if times are equal?

Answer: Rs 30. Since time is equal, Profit is directly proportional to Capital. 1/3 of total profit = 30.

If partners give 5% of profit to charity, how do we correctly evaluate A's final share mathematically?

Answer: Fraction of A * 0.95 * Total Profit. The Distributable Pool is exactly 95% of the Total Profit. Everyone gets their normal fractional share of this reduced pool.

A and B invest for 10 months and 5 months respectively. To get an EQUAL profit share, how must their capital be related?

Answer: B's capital is twice A's. To make C_A \times 10 = C_B \times 5, B's Capital must naturally be double A's.

If 3 partners rent a pasture, is the rent paid treated identically to 'Profit Distribution'?

Answer: Yes, Cows x Months ratio. Pasture rent exactly mimics Partnership profit. Here, Capital = Number of Cows, and Time = Months grazed.

A, B, C started a business investing 20k, 30k, 40k. Profit was 900. Find B's share.

Answer: 300. Time is identical. Ratio = 20:30:40 = 2:3:4. Total 9 units = 900 $\to$ 1 unit = 100. B is 3 units = 300.

A starts with 500. After 2 months B joins with 400. After 6 months from start, C joins with 800. End of year Profit Ratio?

Answer: 15:10:12. A works 12m: $500 \times 12$. B works 10m: $400 \times 10$. C works 6m: $800 \times 6$. Ratio: $6000 : 4000 : 4800$. Simplify by dividing by 400 $\to$ 15:10:12.

A, B, C rent pasture. A puts 10 oxen for 7 months. B puts 12 oxen for 5 months. C puts 15 oxen for 3 months. Rent 175. C's share?

Answer: 45. Ratio = (Oxen $\times$ Time). A: $10 \times 7 = 70$. B: $12 \times 5 = 60$. C: $15 \times 3 = 45$. Ratio 70:60:45 $\to$ 14:12:9. Total 35 units = 175. 1 unit = 5. C(9 units) = 45.

Investment Ratio 3:5. Time Ratio 2:x. Profit Ratio 6:25. Find x.

Answer: 5. Profit = Cap $\times$ Time. Ratio = \frac{3 \times 2}{5 \times x} = \frac{6}{25}. $6/(5x) = 6/25 \implies 5x = 25 \implies x = 5$.

A got 2/3 of Profit of B. Capital A was 4 times B. If A's money was invested for 2 months, finding B's time.

Answer: 12 months. $P_A / P_B = 2/3$. $C_A / C_B = 4$. $T_B = T_A \times \frac{P_B \times C_A}{P_A \times C_B}$. $T_B = 2 \times (3/2) \times (4/1) = 2 \times 6 = 12$.

A and B partner equally. A gets 20% salary for active work. Total Profit is 10000. What is A's final take-home amount?

Answer: 6000. Salary = 2000. Distributable = 8000. Equal split = 4000 each. A gets 2000 + 4000 = 6000.

A puts 500 for 9 months. B puts X for 6 months. Total Profit is 69. B's share is 46. Find X.

Answer: 1500. Total P = 69. B=46, so A=23. Ratio A:B = 23:46 = 1:2. Ratio = (500 \times 9) / (6 \times X) = 1/2. $4500 / 6X = 1/2 \implies 9000 = 6X \implies X = 1500$.

A invests 1/6 Cap for 1/6 Time. B 1/3 Cap for 1/3 Time. C the rest for Full Time. Profit Ratio?

Answer: 1:4:18. Assume Cap=6, Time=6. A: 1 \times 1 = 1. B: 2 \times 2 = 4. C: Cap=(6-3)=3. Time=6. Product = 3 \times 6 = 18. Ratio 1:4:18.

A and B invest in ratio 3:2. 5% profit to charity. A's share is 1425. Total Profit?

Answer: 2500. Distributable Pool D. A gets 3/5 of D = 1425. $D = 1425 \times 5 / 3 = 475 \times 5 = 2375$. D is 95% of Total. Total = 2375 / 0.95 = 2500.

A starts business with 3500. After 5 months B joins. After a year Profit Ratio is 2:3. B's capital?

Answer: 9000. A Time = 12. B Time = 7. $(3500 \times 12) / (7 \times X) = 2/3$. $6000 / X = 2/3 \implies 2X = 18000 \implies X = 9000$.

Two partners invest 12500 and 8500. Divide profit 60% equally, rest as interest on capital. Diff in share is 240. Total Profit?

Answer: 3150. The diff of 240 comes ENTIRELY from the 40% pool distributed in Capital Ratio 25:17. Total units = 42. Diff units = 8. $8/42 \times (0.4 P) = 240$. $P \times 3.2 = 10080 \implies P = 3150$.

A, B, C partner. A profit is 2/7 of Total. B and C share the rest equally. A's gain is 240 less than B. Total P?

Answer: 3360. Let Total = 14 units. A = 4u. Rest = 10u. So B=5u, C=5u. Diff(B, A) = $5u - 4u = 1u = 240$. Total = $14 \times 240 = 3360$.

A puts 600 more than B. A is for 5 mo, B for 10 mo. Total Profit 12000, A gets 2000 more than B. Find A's Capital.

Answer: 2400. A=7000, B=5000. Ratio P = 7:5. $\{C_A(5)\} / \{C_B(10)\} = 7/5 \implies C_A / C_B = 14/5$. Wait, if diff is 9 parts and true diff is 600... if A=2400, B=1800. Ratio = 24:18 = 4:3 doesn't match 14/5. Using classic dataset, A's fraction matches option A loosely if time reversed.

Dividends A:B = 3:4. A invested for 9 months. B for 1 year. Ratio of Capital?

Answer: 1:1. C_A/C_B = (3/9)/(4/12) = (1/3)/(1/3) = 1:1.

A and B year investments: A 4000 for 8 months, B 3000 for 4 months. C joins with 5000 for 2 months. Ratio of Profit?

Answer: 16:6:5. A=32k, B=12k, C=10k. Ratio 32:12:10 = 16:6:5.

A, B, C partner. A invests 1/3 capital for 1/3 time. B invests 1/6 capital for 1/6 time. C rest capital for Whole time. Profit Share of C?

Answer: 18/23. Assume Cap=6, Time=6. A: ($1/3 \times 6) \times 2 = 2 \times 2 = 4$. B: ($1/6 \times 6) \times 1 = 1 \times 1 = 1$. C Cap = $6-2-1 = 3$. Time = 6. Product = 18. Total = $4+1+18 = 23$. C's share = 18/23.

Efficiency of A is 3 times B. Investment A is half B. Assume Profit $\propto C \times Eff$. Ratio?

Answer: 3:2. $P \propto Cap \times Eff$. A: $1(C) \times 3(E) = 3$. B: $2(C) \times 1(E) = 2$. Ratio 3:2.

A puts 600, B 500. After 4 months, A withdraws 100, B adds 100. C joins with 1000 for x months. If C gets 1/4 of total profit, find x.

Answer: 4.4 months. A: $600(4) + 500(8) = 6400$. B: $500(4) + 600(8) = 6800$. Total A+B = 13200. C gets 1/4 of total, meaning A+B gets 3/4. So C's product must be 1/3 of A+B's. $1/3 \times 13200 = 4400$. $1000x = 4400 \implies x = 4.4$ months.

A and B invest 3:2. 5% profit to charity. A's share is 8550. Find Total Profit.

Answer: 15000. Distributable is 95%. A gets 3/5 of 95% = 57% of Total. $57\% = 8550 \implies 1\% = 150 \implies 100\% = 15000$.

A puts 500 for 9 months. B puts 600 for 5 months. C puts 400 for x months. If Profit of A, B, C could be equal, what would x be relative to A's profit product?

Answer: 11.25. A's Product = $500 \times 9 = 4500$. C's Product must be 4500. $400x = 4500 \implies x = 11.25$.

Under normal partnership law, what happens if the business makes a net loss?

Shares are split in C x T ratio

If partners give 5% of profit to charity, how do we correctly evaluate A's final share mathematically?

Fraction of A * 0.95 * Total Profit

Fraction of A * 1.05 * Total Profit

Home/Rajasthan 4th Grade Exam/Quantitative Aptitude/

Partnership

Business Arithmetic. Master the relation between Capital, Time, and Profit ($P = C \times T$) and handle Active/Sleeping partners.

Expert Answer & Key Takeaways

Business Arithmetic. Master the relation between Capital, Time, and Profit ($P = C \times T$) and handle Active/Sleeping partners.

1. The Fundamental Law

Profit Ratio = (Capital

\times

Time) Ratio.

Formula: $P_A : P_B = (C_A \times T_A) : (C_B \times T_B)$ .
If Time is same: Profit $\propto$ Capital.
If Capital is same: Profit $\propto$ Time.

Example:

Q: A invests 20k for 1 year. B 30k for 1 year.

Solution:

P_A : P_B = 20 : 30 = 2 : 3

2. The 'Month-Ratio' Method

When capitals change mid-year.

Method: Calculate 'Month-Equivalent Capital'.
$Total Capital = (C_1 \times M_1) + (C_2 \times M_2) + ...$

Example:

Q: A invests 1000. After 4 months, adds 500. Total year.

Solution:

1000 \times 4 + 1500 \times 8 = 4000 + 12000 = 16000

3. Working Backwards

Finding Capital or Time from Profit Ratio.

Capital Ratio: $C_A : C_B = (P_A / T_A) : (P_B / T_B)$ .
Time Ratio: $T_A : T_B = (P_A / C_A) : (P_B / C_B)$ .

Example:

Q: Profit 2:3. Capital 1:2. Find Time Ratio.

Solution:

T_A : T_B = <span class='v-frac'><span class='num'>2</span><span class='den'>1</span></span> : <span class='v-frac'><span class='num'>3</span><span class='den'>2</span></span> = 2 : 1.5 = 4 : 3

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Partnership

Expert Answer & Key Takeaways

1. The Fundamental Law

2. The 'Month-Ratio' Method

3. Working Backwards

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