Question

If five men can build a house in 27 days, how long will it take nine men to build the same house, assuming they all work at the same rate?

Hint:

In work-related problems, always determine if the relationship is one of direct or inverse proportion. More workers for the same job means less time (inverse), while more work for the same number of workers means more time (direct).

Answer 1

Step 1: Understanding the Question:
This is a classic work and time problem. The amount of work (building one house) is constant. The number of men is inversely proportional to the number of days required to complete the work. This means that if you increase the number of men, the time taken will decrease.
Step 2: Key Formula or Approach:
The formula for inverse proportion in work problems is:
\[ M_1 \times D_1 = M_2 \times D_2 \] Where:
- \(M_1\) is the initial number of men.
- \(D_1\) is the initial number of days.
- \(M_2\) is the new number of men.
- \(D_2\) is the new number of days we need to find.
Alternatively, we can calculate the total "man-days" required for the job.
Step 3: Detailed Explanation:
Method 1: Using the formula
Given values:
\(M_1 = 5\) men
\(D_1 = 27\) days
\(M_2 = 9\) men
We need to find \(D_2\).
Substituting the values into the formula:
\[ 5 \times 27 = 9 \times D_2 \] \[ 135 = 9 \times D_2 \] \[ D_2 = \frac{135}{9} \] \[ D_2 = 15 \] Method 2: Using the Man-Days concept
First, calculate the total work required in "man-days":
Total Work = (Number of men) \(\times\) (Number of days)
Total Work = \(5 \times 27 = 135\) man-days.
This means the job requires 135 man-days of effort to complete.
Now, calculate how many days it will take for 9 men to complete this amount of work:
Days required = \(\frac{\text{Total Work}}{\text{New Number of men}}\)
Days required = \(\frac{135}{9} = 15\) days.
Step 4: Final Answer:
It will take nine men 15 days to build the same house.