8 men and 12 women can build a wall in 10 days. 6 men and 8 women can build the same wall in 14 days. How long will it take for 2 women and 1 man to build it together?
70 days
Work Rate Problem: Men and Women Building a Wall
This problem requires us to determine the time needed for a specific group (1 man and 2 women) to build a wall. We are given information about how long larger groups of men and women take. To solve this, we first need to understand the relative work efficiency of men and women.
Formulating Equations Based on Work Rates
Let '\(M\)' represent the amount of work one man completes in a single day, and let '\(W\)' represent the amount of work one woman completes in a single day. The total amount of work done is calculated by multiplying the combined rate of the workers by the number of days they work.
Scenario 1: 8 men and 12 women can build the wall in 10 days.
Total Work = \( (8M + 12W) \times 10 \) days
Scenario 2: 6 men and 8 women can build the same wall in 14 days.
Total Work = \( (6M + 8W) \times 14 \) days
Since the task (building the wall) is the same in both scenarios, the total work done must be equal. Therefore, we can set the two expressions equal to each other:
\( (8M + 12W) \times 10 = (6M + 8W) \times 14 \)
Deriving the Ratio of Work Efficiencies
To find the relationship between the work rate of a man and a woman, we simplify the equation:
\( 10 \times (8M + 12W) = 14 \times (6M + 8W) \)
Distribute the numbers:
\( 80M + 120W = 84M + 112W \)
Now, let's gather the terms involving '\(M\)' on one side and terms involving '\(W\)' on the other:
\( 120W - 112W = 84M - 80M \)
\( 8W = 4M \)
To find the ratio, we can divide both sides by 4:
\( 2W = M \)
This equation tells us that the work rate of one man (\(M\)) is equivalent to the work rate of two women (\(2W\)). In other words, a man is twice as efficient as a woman.
Calculating the Total Work Required
With the ratio known (\(M = 2W\)), we can now calculate the total amount of work required to build the wall. We can use either of the initial scenarios; let's use the first one:
Total Work = \( (8M + 12W) \times 10 \)
Substitute \( M = 2W \) into the equation:
Total Work = \( (8 \times (2W) + 12W) \times 10 \)
Total Work = \( (16W + 12W) \times 10 \)
Total Work = \( (28W) \times 10 \)
Total Work = \( 280W \)
This value, \( 280W \), represents the total work needed in terms of 'woman-days'.
Finding the Combined Rate of the Target Group
The question asks for the time it takes for a group consisting of 1 man and 2 women to build the wall. First, let's find their combined work rate:
Combined Rate = \( 1M + 2W \)
Using our relationship \( M = 2W \), we substitute:
Combined Rate = \( 1 \times (2W) + 2W \)
Combined Rate = \( 2W + 2W \)
Combined Rate = \( 4W \)
So, the combined rate of 1 man and 2 women is equivalent to the rate of 4 women working together.
Calculating the Time Taken by the Group
The time required for any group to complete a task is calculated by dividing the Total Work by their Combined Rate:
Time = \( \frac{\text{Total Work}}{\text{Combined Rate}} \)
Substitute the values we found:
Time = \( \frac{280W}{4W} \)
Time = \( 70 \)
The unit for time here is days.
| Step | Description | Mathematical Representation |
|---|---|---|
| 1 | Set up equations based on initial conditions | \( (8M + 12W) \times 10 = (6M + 8W) \times 14 \) |
| 2 | Calculate the ratio of work rates | \( 8W = 4M \implies M = 2W \) |
| 3 | Determine the total work required (using woman-days) | Total Work = \( (8(2W) + 12W) \times 10 = 280W \) |
| 4 | Calculate the combined work rate of 1 man and 2 women | Combined Rate = \( 1M + 2W = 1(2W) + 2W = 4W \) |
| 5 | Calculate the time needed by the group | Time = \( \frac{280W}{4W} = 70 \) days |
Therefore, it will take 70 days for 1 man and 2 women to build the wall together.
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