This is the detailed explanation for the Arithmetic Quiz 01. Each question’s solution is broken down step-by-step to help you understand the core concepts of profit-and-loss, partnership ratios, and proportional distribution.
1. In a business, A and C invested amounts in the ratio 2 : 1, whereas the ratio between amounts invested by A and B was 3 : 2. If Rs 157300 was their profit, how much amount did B receive?
Answer: Rs. 48400
- Step 1: Find a common ratio.
- We have A:C = 2:1 and A:B = 3:2. To combine these, we need to find a common value for A. The least common multiple of 2 and 3 is 6.
- Multiply the A:C ratio by 3 to get A:C = (2×3):(1×3) = 6:3.
- Multiply the A:B ratio by 2 to get A:B = (3×2):(2×2) = 6:4.
- Now, we have a combined ratio of A:B:C = 6:4:3.
- Step 2: Calculate B’s share of the profit.
- The total ratio is 6 + 4 + 3 = 13.
- B’s share is 4 out of 13 parts.
- B’s share = (4 / 13) * 157300 = Rs. 48400.
2. A, B, C subscribe Rs. 50,000 for a business. A subscribes Rs. 4000 more than B and B Rs. 5000 more than C. Out of a total profit of Rs. 35,000, A receives
Answer: Rs. 14,700
- Step 1: Determine the individual investments.
- Let C’s investment be x.
- B’s investment is C’s investment plus Rs 5000, so B = x + 5000.
- A’s investment is B’s investment plus Rs 4000, so A = (x + 5000) + 4000 = x + 9000.
- The total investment is A + B + C = 50,000.
- (x + 9000) + (x + 5000) + x = 50,000
- 3x + 14000 = 50,000
- 3x = 36000
- x = 12000.
- Therefore, C’s investment = Rs 12000, B’s investment = 12000 + 5000 = Rs 17000, and A’s investment = 12000 + 9000 = Rs 21000.
- Step 2: Find the ratio of investments.
- A:B:C = 21000:17000:12000 = 21:17:12.
- Step 3: Calculate A’s share of the profit.
- The sum of the ratios is 21 + 17 + 12 = 50.
- A’s share = (21 / 50) * 35000 = Rs. 14,700.
3. P, Q, R enter into a partnership. P initially invests 25 lakh & adds another 10 lakhs after one year. Q initially invests 35 lakh & withdrawal 10 lakh after 2 years and R invests Rs 30 Lakhs. In what ratio should the profit be divided at the end of 3 years?
Answer: 19:19:18
- Step 1: Calculate the total capital-months for each partner.
- A partner’s share in profit is proportional to their investment multiplied by the duration of that investment.
- P’s investment: (25 lakhs * 12 months) + (35 lakhs * 24 months) = 300 + 840 = 1140. (Wait, let’s re-read the question). “P initially invests 25 lakh & adds another 10 lakhs after one year.” Okay, so P’s investment is Rs 25 lakh for the first year and Rs 35 lakh for the next two years.
- P’s capital-months: (25 * 12) + (35 * 24) = 300 + 840 = 1140. (Mistake here: the question says “at the end of 3 years”, so the total duration is 36 months. P adds 10 lakhs after one year, so P’s capital is 25 lakhs for 12 months and (25+10) = 35 lakhs for the remaining 24 months. Correct.)
- Q’s investment: “Q initially invests 35 lakh & withdrawal 10 lakh after 2 years”. So Q’s capital is 35 lakhs for 24 months and (35-10) = 25 lakhs for the remaining 12 months.
- Q’s capital-months: (35 * 24) + (25 * 12) = 840 + 300 = 1140.
- R’s investment: “R invests Rs 30 Lakhs”. The period is not specified, so we assume R’s capital remains constant for the full 3 years (36 months).
- R’s capital-months: (30 * 36) = 1080.
- Step 2: Find the ratio of their capital-months.
- P:Q:R = 1140:1140:1080.
- Divide all numbers by 60 to simplify the ratio.
- 1140 / 60 = 19
- 1080 / 60 = 18
- The ratio is 19:19:18.
- Step 3: Match with the options.
- The ratio of profit sharing is 19:19:18.
4. A and B started a business in partnership investing Rs. 20,000 and Rs. 15,000 respectively. After six months, C joined them with Rs. 20,000. What will be B’s share in total profit of Rs. 25,000 earned at the end of 2 years from the starting of the business?
Answer: Rs. 7500
- Step 1: Calculate the total capital-months for each partner over 2 years (24 months).
- A invested for the full 24 months. A’s investment = 20000 * 24 = 480000.
- B invested for the full 24 months. B’s investment = 15000 * 24 = 360000.
- C joined after 6 months, so C invested for 24 – 6 = 18 months. C’s investment = 20000 * 18 = 360000.
- Step 2: Find the ratio of their investments.
- A:B:C = 480000:360000:360000.
- Divide all numbers by 10000 to simplify: 48:36:36.
- Divide by 12: 4:3:3.
- Step 3: Calculate B’s share of the profit.
- The sum of the ratios is 4 + 3 + 3 = 10.
- B’s share is 3 out of 10 parts.
- B’s share = (3 / 10) * 25000 = Rs. 7500.
5. A, B, C rent a pasture. A puts 10 oxen for 7 months, B puts 12 oxen for 5 months and C puts 15 oxen for 3 months for grazing. If the rent of the pasture is Rs. 175, how much must C pay as his share of rent?
Answer: Rs. 45
- Step 1: Calculate the effective use for each person.
- The rent should be divided based on the number of oxen and the duration they grazed.
- A’s share: 10 oxen * 7 months = 70.
- B’s share: 12 oxen * 5 months = 60.
- C’s share: 15 oxen * 3 months = 45.
- Step 2: Find the ratio of their shares.
- A:B:C = 70:60:45.
- Divide all by 5 to simplify: 14:12:9.
- Step 3: Calculate C’s share of the rent.
- The sum of the ratios is 14 + 12 + 9 = 35.
- C’s share = (9 / 35) * 175 = 9 * 5 = Rs. 45.
6. Three partners A, B, C start a business. B’s Capital is four times C’s capital and twice A’s capital is equal to thrice B’s capital. If the total profit is Rs 16500 at the end of a year, find out B’s share in it.
Answer: Rs. 6000
- Step 1: Express the capitals in terms of a single variable.
- B’s capital = 4 * C’s capital, so C = B/4.
- 2 * A’s capital = 3 * B’s capital, so A = (3/2) * B.
- Step 2: Find the ratio of their capitals.
- A:B:C = (3/2)B : B : (1/4)B.
- To get rid of the fractions, multiply everything by 4 (the least common multiple of the denominators).
- (3/2) * 4 : 1 * 4 : (1/4) * 4 = 6:4:1.
- Step 3: Calculate B’s share of the profit.
- The sum of the ratios is 6 + 4 + 1 = 11.
- B’s share is 4 out of 11 parts.
- B’s share = (4 / 11) * 16500 = 4 * 1500 = Rs. 6000.
7. Three partners shared the profit in a business in the ratio 5 : 7 : 8. They had partnered for 14 months, 8 months and 7 months respectively. What was the ratio of their investments?
Answer: 20 : 49 : 64
- Step 1: Understand the relationship between profit, investment, and time.
- Profit is proportional to (Investment * Time).
- Let the investments be I_A, I_B, and I_C.
- (I_A * 14) : (I_B * 8) : (I_C * 7) = 5:7:8.
- Step 2: Isolate the investment ratio.
- I_A = 5 / 14
- I_B = 7 / 8
- I_C = 8 / 7
- Step 3: Find a common denominator to express the ratio.
- The ratio of investments is (5/14) : (7/8) : (8/7).
- The least common multiple of 14, 8, and 7 is 56.
- Multiply each fraction by 56:
- (5/14) * 56 = 5 * 4 = 20
- (7/8) * 56 = 7 * 7 = 49
- (8/7) * 56 = 8 * 8 = 64
- The ratio of their investments is 20:49:64.
8. If 4 (P’s Capital) = 6 (Q’s Capital) = 10 (R’s Capital), then out of the total profit of Rs 4650, R will receive
Answer: Rs. 900
- Step 1: Find the ratio of the capitals.
- Let the equal value be k.
- 4P = k => P = k/4
- 6Q = k => Q = k/6
- 10R = k => R = k/10
- The ratio P:Q:R = k/4 : k/6 : k/10 = 1/4 : 1/6 : 1/10.
- To get whole numbers, multiply by the least common multiple of 4, 6, and 10, which is 60.
- (1/4) * 60 : (1/6) * 60 : (1/10) * 60 = 15:10:6.
- Step 2: Calculate R’s share of the profit.
- The sum of the ratios is 15 + 10 + 6 = 31.
- R’s share is 6 out of 31 parts.
- R’s share = (6 / 31) * 4650 = 6 * 150 = Rs. 900.
9. Kamal started a business investing Rs 9000. After five months, Sameer joined with a capital of Rs 8000. If at the end of the year, they earn a profit of Rs. 6970, then what will be the share of Sameer in the profit?
Answer: Rs 2380
- Step 1: Calculate the capital-months for each person.
- The total duration is one year (12 months).
- Kamal’s investment: 9000 for 12 months = 9000 * 12 = 108000.
- Sameer’s investment: Joined after 5 months, so invested for 12 – 5 = 7 months. Sameer’s investment = 8000 * 7 = 56000.
- Step 2: Find the ratio of their investments.
- Kamal:Sameer = 108000:56000.
- Divide by 1000: 108:56.
- Divide by 4: 27:14.
- Step 3: Calculate Sameer’s share of the profit.
- The sum of the ratios is 27 + 14 = 41.
- Sameer’s share is 14 out of 41 parts.
- Sameer’s share = (14 / 41) * 6970 = 14 * 170 = Rs 2380.
10. P and Q invested in a business. The profit earned was divided in the ratio 2 : 3. If P invested Rs 40000, the amount invested by Q is
Answer: Rs. 60000
- Step 1: Understand the relationship between investment and profit ratio.
- Profit is divided in the same ratio as the investment (assuming they invested for the same amount of time).
- P’s profit : Q’s profit = P’s investment : Q’s investment.
- 2 : 3 = 40000 : Q’s investment.
- Step 2: Set up a proportion and solve for Q’s investment.
- 2 / 3 = 40000 / x
- 2x = 3 * 40000
- 2x = 120000
- x = 120000 / 2 = Rs. 60000.
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