Let's solve this problem step by step using the concept of "man-days" (the total amount of work that can be done by one man in one day).

Step 1: Calculate total work required to build the house.

The total amount of work can be represented as the product of the number of men, the number of days, and the rate at which they work. Since the house was completed in 48 days, the total amount of work $W$ is the same regardless of how many workers were employed or added later.

Thus, the total amount of work is:

$W = \text{Total Men-Days} = 120 \times 32 + (120 + 90) \times 16$

Breaking it down:

• For the first 32 days, 120 men worked. So the work done in the first 32 days is: $120 \times 32 = 3840 \text{ man-days}.$

• For the remaining 16 days, 120 original men plus 90 additional men (total 210 men) worked. The work done in the last 16 days is: $210 \times 16 = 3360 \text{ man-days}.$

Now, the total amount of work required to build the house is: $W = 3840 + 3360 = 7200 \text{ man-days}.$

Step 2: Calculate the number of days if no additional men were employed.

If no additional men were employed, the builder would have continued with only the initial 120 men. To find how many days $D$ it would take to finish the same total work (7200 man-days) with only 120 men, we use the formula:

$D = \frac{W}{\text{Men}} = \frac{7200}{120} = 60 \text{ days}.$

Step 3: Calculate the delay.

If the builder had not employed the additional 90 men, the house would have taken 60 days instead of 48 days. The delay is:

$\text{Delay} = 60 - 48 = 12 \text{ days}.$

Conclusion:

If the builder had not employed the additional 90 men, the construction would have been finished 12 days behind schedule.

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Would you like a more detailed explanation, or do you have any questions?

Here are five related questions for further practice:

1. What would have happened if only 60 men were hired from the start?
2. How would the schedule be affected if the additional 90 men were hired after 40 days instead of 32?
3. What is the formula for calculating the total work done in terms of man-days?
4. How many men would be required to finish the project in 40 days instead of 48?
5. How does adding more workers affect the total project time?

Tip: Always check how the distribution of work over time changes when additional resources are employed.