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If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7 men and 10 women take to do it?
Let's think about this in everyday terms first. We have two different work scenarios with different combinations of men and women, and we know how long each group takes to complete the same job. From this information, we need to figure out how productive each man and each woman is individually, then use that to predict how long a third group will take.
Think of it like this: if you know that 2 bakers and 3 decorators can finish 100 cakes in 5 hours, and 3 bakers and 1 decorator can do the same 100 cakes in 4 hours, you could figure out how fast each baker works versus each decorator, right?
In mathematical terms, we're looking for:
- The work rate of each man per day (let's call this \(\mathrm{m}\))
- The work rate of each woman per day (let's call this \(\mathrm{w}\))
- The total work = 1 complete job
Process Skill: TRANSLATE - Converting the real-world work scenario into mathematical relationships
Now let's translate our two scenarios into mathematical equations. Remember, the fundamental principle is: Rate × Time = Work Done
Scenario 1: 12 men and 16 women complete the job in 5 days
- Combined daily rate = \((12\mathrm{m} + 16\mathrm{w})\)
- Time = 5 days
- Work completed = 1 job
- So: \((12\mathrm{m} + 16\mathrm{w}) \times 5 = 1\)
- This gives us: \(60\mathrm{m} + 80\mathrm{w} = 1\) ... (Equation 1)
Scenario 2: 13 men and 24 women complete the job in 4 days
- Combined daily rate = \((13\mathrm{m} + 24\mathrm{w})\)
- Time = 4 days
- Work completed = 1 job
- So: \((13\mathrm{m} + 24\mathrm{w}) \times 4 = 1\)
- This gives us: \(52\mathrm{m} + 96\mathrm{w} = 1\) ... (Equation 2)
Now we have two equations with two unknowns. Let's solve this systematically:
From our equations:
\(60\mathrm{m} + 80\mathrm{w} = 1\) ... (1)
\(52\mathrm{m} + 96\mathrm{w} = 1\) ... (2)
To eliminate one variable, let's multiply equation (1) by 6 and equation (2) by 5 to get the same coefficient for w:
\(360\mathrm{m} + 480\mathrm{w} = 6\) ... (1×6)
\(260\mathrm{m} + 480\mathrm{w} = 5\) ... (2×5)
Subtracting the second from the first:
\((360\mathrm{m} - 260\mathrm{m}) + (480\mathrm{w} - 480\mathrm{w}) = 6 - 5\)
\(100\mathrm{m} = 1\)
\(\mathrm{m} = \frac{1}{100} = 0.01\)
Now substituting \(\mathrm{m} = 0.01\) back into equation (1):
\(60(0.01) + 80\mathrm{w} = 1\)
\(0.6 + 80\mathrm{w} = 1\)
\(80\mathrm{w} = 0.4\)
\(\mathrm{w} = \frac{0.4}{80} = 0.005\)
So each man completes 0.01 of the job per day, and each woman completes 0.005 of the job per day.
Process Skill: MANIPULATE - Using algebraic techniques to solve the system of equations efficiently
Now we can answer the original question: How long will 7 men and 10 women take?
Daily work rate for 7 men and 10 women:
\(= 7\mathrm{m} + 10\mathrm{w}\)
\(= 7(0.01) + 10(0.005)\)
\(= 0.07 + 0.05\)
\(= 0.12\) of the job per day
If they complete 0.12 of the job each day, the time needed is:
Time = Total work ÷ Daily rate
\(\text{Time} = 1 \div 0.12\)
Time = 8.333... days
Time ≈ 8.3 days
The answer is (C) 8.3 days.
To verify: Our solution shows that 7 men and 10 women working together have a combined daily rate of 0.12 jobs per day, which means they need approximately 8.33 days to complete one full job. This matches answer choice (C) 8.3 days.
Students often confuse whether \(\mathrm{m}\) and \(\mathrm{w}\) represent the total work done by all men/women or the work rate per individual man/woman. The solution correctly defines m as the work rate of each individual man per day, but students might think m represents the combined work rate of all men in the group.
2. Setting up incorrect rate equationsStudents may struggle with the fundamental work formula (Rate × Time = Work) and incorrectly set up equations like \("12\mathrm{m} + 16\mathrm{w} = 5"\) instead of \("(12\mathrm{m} + 16\mathrm{w}) \times 5 = 1"\). They might forget that the combined daily rate needs to be multiplied by the number of days to equal the total work completed.
3. Confusion about the work unitStudents often get confused about what constitutes "1 unit of work." They might try to assign arbitrary values to the total work instead of understanding that since it's the same piece of work in both scenarios, we can standardize it as 1 complete job.
When eliminating variables, students frequently make errors in choosing appropriate multipliers for the equations. In this problem, they might incorrectly multiply the equations or make sign errors when subtracting one equation from another, leading to wrong values for m and w.
2. Arithmetic calculation mistakesStudents often make computational errors when calculating fractions like \(\mathrm{w} = \frac{0.4}{80} = 0.005\), especially when dealing with decimals. They might also incorrectly substitute values back into the original equations during verification.
3. Incorrect back-substitutionAfter finding one variable (like \(\mathrm{m} = 0.01\)), students may substitute it into the wrong equation or make errors in the substitution process when solving for the second variable, leading to an incorrect value for w.
Students might calculate the final answer as 8.333... days but incorrectly round it to 8.33 or 8.4 days instead of recognizing that 8.3 is the closest answer choice. They may also not understand which level of precision is expected in the final answer.
2. Reciprocal confusionWhen calculating the final time, students might confuse whether to divide the work by the rate \((1 \div 0.12)\) or multiply them together, especially since they've been working with rate × time = work throughout the problem. This could lead them to select an answer that's off by a factor related to this confusion.