Questions for: Sharing & Comparison
Three workers, P, Q, and R, collaborate on a project for which they jointly receive a total payment of $4200. Their working efficiencies are such that P's efficiency is three times Q's efficiency, and Q's efficiency is twice R's efficiency. P worked for 5 days, Q worked for 8 days, and R worked for 10 days. If the payment is distributed strictly proportional to the total work done by each person, what is the difference between Q's share and R's share?
✅ To determine the shares, we first establish the ratio of their efficiencies. Let R's efficiency be 'x'. Then Q's efficiency is '2x', and P's efficiency is '3 × 2x = 6x'. So, the efficiency ratio P:Q:R = 6:2:1.
Next, we calculate the total "work done" by each person, which is Efficiency × Days worked. P's work = 6x × 5 = 30x; Q's work = 2x × 8 = 16x; R's work = 1x × 10 = 10x. The ratio of work done P:Q:R simplifies to 30:16:10, or 15:8:5.
The total ratio parts for work done are 15 + 8 + 5 = 28. Using the sharing formula, Q's share = (8/28) × $4200 = (2/7) × $4200 = $1200. R's share = (5/28) × $4200 = $750.
The difference between Q's share and R's share is $1200 - $750 = $450.
❌ Option A ($350) is incorrect and would result from a significant miscalculation of individual shares or the total ratio. For example, if R's share was $850 and Q's $1200.
❌ Option B ($400) is incorrect. This might occur if there was an arithmetic error in the sharing calculation, or if the calculation was based on an incorrect total or individual work unit.
❌ Option D ($500) is incorrect. This value indicates a computational error in either calculating the individual shares or their difference. For instance, if Q's share was $1250 instead of $1200, given R's share of $750.
Three partners, A, B, and C, started a business venture. A invested $60,000 for the full 12 months. B joined later, investing a certain amount for 9 months, and C invested $75,000 for an unknown duration. At the end of the year, the total profit was $120,000. If A's share of the profit was $48,000 and C's share was $30,000, what is the simplified ratio of B's investment amount to C's investment period (in months)?
✅ The profit is proportional to the product of the investment amount and the investment period. First, calculate B's profit: $120,000 (Total) - $48,000 (A's share) - $30,000 (C's share) = $42,000.
The ratio of profits for A:B:C is $48,000 : $42,000 : $30,000, which simplifies to 8 : 7 : 5 by dividing by 6,000.
Let the product (Investment × Time) be 'P'. For A, P_A = $60,000 × 12 months = 720,000. Since A's profit ratio is 8, one unit of profit ratio corresponds to P_A / 8 = 720,000 / 8 = 90,000.
For C, P_C = C's investment × Y months = $75,000 × Y. C's profit ratio is 5, so P_C = 5 × 90,000 = 450,000. Thus, $75,000 × Y = 450,000, which means Y = 450,000 / 75,000 = 6 months.
For B, P_B = B's investment (X) × 9 months = 9X. B's profit ratio is 7, so P_B = 7 × 90,000 = 630,000. Thus, 9X = 630,000, which means X = 630,000 / 9 = $70,000.
The required ratio of B's investment (X) to C's investment period (Y) is $70,000 : 6, which simplifies to 35,000 : 3.
❌ Option B is the unsimplified ratio of X:Y before dividing both sides by 2.
❌ Option C uses B's profit ($42,000) as the first term and C's profit ratio (5) as the second term, which is an incorrect combination for the question asked.
❌ Option D uses A's investment ($60,000) as the first term and an arbitrary incorrect value (4) for the second term, failing to identify the correct values for X and Y.
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Three friends, Alex, Brian, and Cathy, shared a sum of money. Initially, the ratio of Alex's share to Brian's share was 3:4, and the ratio of Brian's share to Cathy's share was 2:3. Later, Brian gave $120 to Cathy. After this transfer, Alex's share was one-third of Cathy's share. What was the total sum of money shared by the three friends?
✅ To find the combined initial ratio Alex:Brian:Cathy, we align Brian's share from both ratios: Alex:Brian = 3:4 and Brian:Cathy = 2:3 (which becomes 4:6 when scaled by 2). This gives an initial ratio of Alex:Brian:Cathy = 3:4:6. Let the initial shares be 3x, 4x, and 6x respectively.
Brian gives $120 to Cathy, so their new shares are (4x - 120) and (6x + 120), while Alex's share remains 3x.
The problem states Alex's new share is one-third of Cathy's new share: 3x = (1/3)(6x + 120).
Solving the equation: 9x = 6x + 120 => 3x = 120 => x = 40.
The total initial sum of money was (3+4+6)x = 13x. Therefore, the total sum is 13 * $40 = $520.
❌ Option A ($480) would imply 13x = 480, resulting in x = 480/13, which is not an integer, indicating an incorrect calculation or assumption.
❌ Option C ($560) would imply 13x = 560, resulting in x = 560/13, which is not an integer, indicating an incorrect calculation or assumption.
❌ Option D ($600) would imply 13x = 600, resulting in x = 600/13, which is not an integer, indicating an incorrect calculation or assumption.
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Three siblings, P, Q, and R, initially share a sum of money such that P's share to Q's share is in the ratio 2:3, and Q's share to R's share is in the ratio 4:5. If Q gives $300 to R, the ratio of P's final share to R's final share becomes 4:9. What was the total amount of money initially shared among the siblings?
✅ To find the initial ratio P:Q:R, combine P:Q = 2:3 and Q:R = 4:5 by making Q's part common (LCM of 3 and 4 is 12); this yields P:Q:R = (2×4):(3×4) : (5×3) = 8:12:15.
Let the initial shares be 8x, 12x, and 15x, making the total initial money 35x.
After Q gives $300 to R, the shares become P = 8x, Q = 12x - 300, and R = 15x + 300.
The new ratio of P's final share to R's final share is 4:9, so we set up the proportion: (8x) / (15x + 300) = 4/9.
Cross-multiplying gives 9 * (8x) = 4 * (15x + 300), which simplifies to 72x = 60x + 1200, hence 12x = 1200, so x = 100.
Therefore, the total initial amount of money was 35x = 35 * $100 = $3500.
❌ Option A ($3000) would be incorrect, likely due to errors in combining ratios or arithmetic calculations.
❌ Option B ($3250) suggests a miscalculation in the combined total ratio or an error while solving for 'x'.
❌ Option D ($3750) could result from incorrect algebraic manipulation or summing the initial ratio parts.
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Three friends, Priya, Quentin, and Rishabh, started a business with a total initial investment of ₹120,000. The ratio of their individual investments was 3:4:5. After one year, Priya doubled her investment, Quentin increased his investment by 50%, and Rishabh reduced his investment by 20%. What is the new ratio of their investments?
✅ First, calculate the initial investments: The total ratio parts are 3 + 4 + 5 = 12. Priya's initial investment = (3/12) × ₹120,000 = ₹30,000; Quentin's = (4/12) × ₹120,000 = ₹40,000; Rishabh's = (5/12) × ₹120,000 = ₹50,000.
✅ Next, calculate the new investments: Priya doubled hers, so ₹30,000 × 2 = ₹60,000. Quentin increased by 50%, so ₹40,000 × 1.5 = ₹60,000. Rishabh reduced by 20%, so ₹50,000 × 0.8 = ₹40,000.
✅ The new investments are ₹60,000 : ₹60,000 : ₹40,000. To find the ratio, divide all by the greatest common divisor, which is ₹20,000. This simplifies to 3:3:2.
❌ Option B (6:8:10) is a scaled version of the initial ratio (3:4:5) and does not reflect the changes made to the investments.
❌ Option C (3:4:5) is the original ratio, completely ignoring all the specified changes in investments.
❌ Option D (9:10:8) would result from significant calculation errors in applying the percentage changes or forming the ratio.
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