14 men can complete a work in 15 days. If 21 men are employed, then in how many days will they complete the same work?
10
Solving Work and Time Problems: Men and Days
This question involves a classic work and time problem where the amount of work is constant, and the number of workers and the time taken to complete the work are related.
When the number of workers increases, the time taken to complete the same work decreases, assuming all workers work at the same rate. This is an example of inverse variation.
Understanding Inverse Variation in Work Problems
In problems involving men and days to complete a fixed amount of work, the relationship is typically inverse variation. This means that if the number of men increases, the number of days required to complete the work decreases proportionally, and vice versa. The total amount of work done remains constant.
The fundamental principle can be expressed with the formula:
\( \text{Number of Men} \times \text{Number of Days} = \text{Total Work} \)
If we have two scenarios (Scenario 1 and Scenario 2) for completing the same work, we can write:
\( M_1 \times D_1 = M_2 \times D_2 \)
Where:
- \( M_1 \) = Number of men in Scenario 1
- \( D_1 \) = Number of days in Scenario 1
- \( M_2 \) = Number of men in Scenario 2
- \( D_2 \) = Number of days in Scenario 2
Applying the Formula to the Given Problem
Let's identify the known values from the problem statement:
- Scenario 1: 14 men complete a work in 15 days.
- So, \( M_1 = 14 \) and \( D_1 = 15 \).
We are asked to find the number of days ( \( D_2 \) ) required if 21 men are employed to complete the same work.
- Scenario 2: 21 men are employed.
- So, \( M_2 = 21 \) and we need to find \( D_2 \).
Using the inverse variation formula \( M_1 \times D_1 = M_2 \times D_2 \):
\( 14 \times 15 = 21 \times D_2 \)
Calculating the Number of Days
Now, we need to solve the equation for \( D_2 \):
\( 14 \times 15 = 21 \times D_2 \)
First, calculate the total work (which is \( 14 \times 15 \)):
\( 14 \times 15 = 210 \)
So, the total work is 210 "man-days".
Now, substitute this back into the equation:
\( 210 = 21 \times D_2 \)
To find \( D_2 \), divide the total work by the number of men in Scenario 2:
\( D_2 = \frac{210}{21} \)
\( D_2 = 10 \)
So, 21 men will complete the same work in 10 days.
Summary of Calculation Steps
- Identify the given quantities: \( M_1 = 14 \), \( D_1 = 15 \), \( M_2 = 21 \).
- Recognize that this is an inverse variation problem for fixed work.
- Use the formula: \( M_1 \times D_1 = M_2 \times D_2 \).
- Substitute the values: \( 14 \times 15 = 21 \times D_2 \).
- Calculate the product \( 14 \times 15 \): \( 210 \).
- Set up the equation: \( 210 = 21 \times D_2 \).
- Solve for \( D_2 \): \( D_2 = \frac{210}{21} = 10 \).
Comparing Scenarios
| Scenario | Number of Men (M) | Number of Days (D) | Total Work (M × D) |
|---|---|---|---|
| Scenario 1 | 14 | 15 | \( 14 \times 15 = 210 \) |
| Scenario 2 | 21 | \( D_2 \) | \( 21 \times D_2 = 210 \) |
As shown in the table, the total work remains constant at 210 man-days. With more men (21 compared to 14), fewer days (10 compared to 15) are required to complete the same amount of work.
Revision Table: Work and Time Concepts
| Concept | Description | Formula Example |
|---|---|---|
| Inverse Variation (Men & Days) | If work is constant, Men & Days are inversely proportional. More men, less days. | \( M_1 D_1 = M_2 D_2 \) (for same work) |
| Direct Variation (Work & Days) | If men are constant, Work & Days are directly proportional. More work, more days. | \( \frac{W_1}{D_1} = \frac{W_2}{D_2} \) (for same number of men) |
| Combined Variation (Men, Days, Work) | Relates Men, Days, and Work done. | \( \frac{M_1 D_1}{W_1} = \frac{M_2 D_2}{W_2} \) |
Additional Information: Work Rate
Another way to think about these problems is in terms of work rate. If 14 men complete a work in 15 days, the total work is proportional to \( 14 \times 15 \) units. We can consider 1 "man-day" as one unit of work.
- Total work = 14 men × 15 days = 210 man-days.
Now, if 21 men are employed to do these 210 man-days of work, the number of days required is:
- Number of Days = Total Work / Number of Men
- Number of Days = 210 man-days / 21 men = 10 days.
This confirms the result obtained using the inverse variation formula. The work rate of one man is constant throughout the problem.
Important Questions from Work Efficiency
A and B together can complete a certain work in 20 days whereas B and C together can complete it in 24 days. If A is twice as good a workman as C, then in what time will B alone do 40% of the same work?
A can do a certain work in 15 days, while B can do the same work in 21 days. If they work together, then in how many days will the same work be completed?
To do a certain work, A and B work on alternate days with B beginning the work on the first day. A alone can complete the same work in 24 days. If the work gets completed in \(11 \frac{1}{3}\) days, then B alone can complete \(\rm \frac{7}{9}^{th}\) part of the original work in:
Two men and 7 women can complete a work in 28 days whereas 6 men and 16 women can do the same work in 11 days. In how many days can 7 men complete the same work?
Five men and 2 boys can do in 30 days as much work as 7 men and 10 boys can do in 15 days. How many boys should join 40 men to do the same work in 4 days?
