Certainly! Let's go step-by-step to understand the relationship between the number of employees and the number of days it takes to complete a task.
1. Initial Information:
- It takes 4 employees 4 days to complete the task.
- It takes 2 employees 8 days to complete the same task.
2. Observing the Relationship:
- For 4 employees, the duration is 4 days.
- For 2 employees, the duration is 8 days.
3. Identifying the Pattern:
- When the number of employees is halved from 4 to 2, the number of days it takes doubles from 4 to 8.
- This suggests an inverse relationship because as the number of employees decreases, the number of days increases.
4. Inverse Variation Explanation:
- In an inverse variation, if one variable increases, the other decreases proportionally, and vice versa.
- The mathematical representation of an inverse variation is [tex]\( e \times d = k \)[/tex], where [tex]\( k \)[/tex] is a constant.
5. Calculating the Constant [tex]\( k \)[/tex]:
- For the given data:
- When [tex]\( e = 4 \)[/tex] and [tex]\( d = 4 \)[/tex]:
[tex]\( k = e \times d = 4 \times 4 = 16 \)[/tex]
- When [tex]\( e = 2 \)[/tex] and [tex]\( d = 8 \)[/tex]:
[tex]\( k = e \times d = 2 \times 8 = 16 \)[/tex]
- In both cases, the constant [tex]\( k \)[/tex] is 16.
6. Conclusion:
- The relationship is [tex]\( e \times d = 16 \)[/tex], indicating an inverse variation.
- Hence, the function that appropriately represents this relationship is:
[tex]\( \text{inverse variation; } de = k \)[/tex]
So, the correct function to represent [tex]\( e \)[/tex] (number of employees) and [tex]\( d \)[/tex] (number of days to complete the task) is:
inverse variation; [tex]\( de = k \)[/tex]
