### 2.3 The following table shows the number of men working on a job and the time in hours it takes to complete the job.
| Number of men | 1 | 2 | 3 | 4 | 6 |
|---------------|-------|-------|-------|-------|-------|
| Time in hours | 48 | 24 | 16 | 12 | [tex]$a$[/tex] |
#### 2.3.1 Is this an example of a direct or indirect proportion? Motivate your answer.
(2)
This is an example of an indirect (inverse) proportion. In an indirect proportion, when one variable increases, the other variable decreases such that their product is constant. In this case, as the number of men increases, the time taken to complete the job decreases. For instance:
- When 1 man works, the job takes 48 hours.
- When 2 men work, the job takes 24 hours.
- When 3 men work, the job takes 16 hours.
- When 4 men work, the job takes 12 hours.
Since the product of the number of men and the time taken (men [tex]\([n]\)[/tex] \* time [tex]\([t]\)[/tex]) remains constant, it demonstrates an indirect proportionality.
#### 2.3.2 Calculate how long it will take 6 men to complete the job.
(2)
To calculate how long it will take 6 men to complete the job, we first need to determine the constant product of the number of men and the time taken for any given pair in the table. We can use the information for 1 man and 48 hours to find this constant.
[tex]\[
\text{Constant} = \text{Number of men} \times \text{Time in hours} = 1 \times 48 = 48
\][/tex]
Given that the product is constant, we can use this value to find the time taken when 6 men are working. Let [tex]\(a\)[/tex] represent the time taken for 6 men:
[tex]\[
\text{Constant} = \text{Number of men} \times \text{Time in hours}
\][/tex]
[tex]\[
48 = 6 \times a
\][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{48}{6} = 8 \text{ hours}
\][/tex]
Therefore, it will take 6 men 8 hours to complete the job.
