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:

The total work of building the house stays fixed, only the number of men changes, so you do not need to calculate total man days at all. Multiply the original number of days by the ratio of the old number of men to the new number of men, since more men always means fewer days for the same fixed job. Be careful not to flip the ratio the wrong way round, more men must always give a smaller number of days, never a larger one.

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.

Answer 2

Concept:

• The total amount of work needed to build the house never changes. Only the number of men working on it changes, so we can scale the number of days directly using a simple ratio, without writing out the man times days formula at all.

Step 1: Identify what stays fixed and what changes.
The work is fixed, one house needs to be built, and every man works at the same rate. Only the number of men changes, from 5 men to 9 men. More men working together means the same house gets finished in fewer days.

Step 2: Set up the ratio directly.
Since the number of men and the number of days move in opposite directions for the same fixed work, the new number of days equals the old number of days multiplied by the ratio of old men to new men.
$\text{New days} = \text{Old days} \times \dfrac{\text{Old men}}{\text{New men}}$

Step 3: Substitute the given numbers and simplify.
$\text{New days} = 27 \times \dfrac{5}{9}$
$\text{New days} = \dfrac{135}{9}$
$\text{New days} = 15$

Final Answer: 15 days