This question comprises three distinct parts requiring different approaches:
Keywords: Frequency distribution, mean, missing frequencies, work rate, ratio, mixture.
Required Approach: Part 1 is factual and analytical (requiring calculations). Part 2 is analytical (requiring ratio calculations). Part 3 is analytical (requiring equation solving).
Points to Remember:
- Understanding mean calculation in frequency distributions.
- Calculating individual work rates and total work.
- Applying ratio and proportion concepts for mixture problems.
Introduction:
This question tests the application of basic statistical concepts, work-rate calculations, and mixture problems. We will solve each part systematically, showing the steps involved and providing the final answers. The first part involves finding missing frequencies given the mean of a frequency distribution. The second involves calculating the share of earnings based on individual work rates. The third involves determining the ratio of two solutions to achieve a desired concentration.
Body:
Part 1: Finding Missing Frequencies
This part requires the complete frequency distributions to solve. Since the distributions are not provided, a general approach is outlined. Let’s assume the first distribution has frequencies f1, f2,… fx,… fn and the second has g1, g2,… y,… gm. The mean is calculated as the sum of (frequency * value) divided by the total frequency. To find x and y, we would need the complete data sets and use the formula for the mean to set up equations involving x and y, then solve these equations simultaneously. Without the full data, a numerical solution is impossible.
Part 2: Determining C’s Share of Earnings
- A’s work rate: 1/6 job per day
- B’s work rate: 1/8 job per day
- C’s work rate: 1/12 job per day
Let’s assume they work together for one day. Their combined work rate is:
1/6 + 1/8 + 1/12 = (4 + 3 + 2) / 24 = 9/24 = 3/8 job per day.
To complete the job, they would take 8/3 days. However, this is not needed to solve the problem directly. The ratio of their earnings is directly proportional to their individual work rates.
- A’s share: (1/6) / (3/8) * â¹23400 = (1/6) * (8/3) * â¹23400 = â¹10400
- B’s share: (1/8) / (3/8) * â¹23400 = (1/3) * â¹23400 = â¹7800
- C’s share: (1/12) / (3/8) * â¹23400 = (1/12) * (8/3) * â¹23400 = â¹5200
Therefore, C’s share is â¹5200.
Part 3: Mixing Solutions
Let x be the amount of 20% methyl alcohol solution and y be the amount of 50% solution. We want a 40% solution. The total amount of the mixture will be x + y.
The equation representing the amount of methyl alcohol is:
0.20x + 0.50y = 0.40(x + y)
Simplifying:
0.20x + 0.50y = 0.40x + 0.40y
0.10y = 0.20x
y = 2x
The ratio of the 20% solution to the 50% solution is x:y = x:2x = 1:2
Conclusion:
This question demonstrated the application of various mathematical concepts. Part 1 required knowledge of frequency distributions and mean calculation (which couldn’t be fully solved due to missing data). Part 2 successfully determined C’s share of earnings based on individual work rates, showing that earnings are directly proportional to individual contributions. Part 3 effectively used a simple equation to find the required ratio for mixing solutions to achieve the desired concentration. These examples highlight the importance of understanding fundamental mathematical principles in solving real-world problems. Further emphasis on clear problem-solving steps and the use of appropriate formulas is crucial for accurate results. A holistic approach to problem-solving, combining conceptual understanding with precise calculations, is key to success in such quantitative analyses.