What is the inverse proportion of this question?

If we want to find the ratio of the production rates \(a : b : c\) when they are **inversely proportional** to the original production times, the process is as follows:

### Original Production Times:
- Machine A takes 3 hours.
- Machine B takes 2 hours.
- Machine C takes 6 hours.

### Inverse Proportionality:
If the adjusted production rates \(a\), \(b\), and \(c\) are **inversely proportional** to the original times, we have:
\[
a \propto \frac{1}{3}, \quad b \propto \frac{1}{2}, \quad c \propto \frac{1}{6}
\]
Therefore, we can write:
\[
a = \frac{k}{3}, \quad b = \frac{k}{2}, \quad c = \frac{k}{6}
\]
where \(k\) is a constant of proportionality.

### Find the Ratio \(a : b : c\):
1. To express \(a : b : c\) in a simpler form, multiply each expression by the least common multiple (LCM) of the denominators (3, 2, and 6). The LCM of 3, 2, and 6 is 6.

2. Multiply each term by 6:
   \[
   6a = 6 \times \frac{k}{3} = 2k, \quad 6b = 6 \times \frac{k}{2} = 3k, \quad 6c = 6 \times \frac{k}{6} = k
   \]

3. Therefore, the ratio is:
   \[
   6a : 6b : 6c = 2k : 3k : k
   \]

4. Simplify by dividing each term by \(k\):
   \[
   a : b : c = 2 : 3 : 1
   \]

### Final Answer:
The ratio of the production rates \(a : b : c\) when they are **inversely proportional** to the original production times is **2 : 3 : 1**.

Would you like any more details or have further questions?

### Relative Questions:
1. How do you interpret the concept of inverse proportionality in real-world scenarios?
2. What are the applications of inverse proportionality in production planning?
3. Can you explain why the ratio changes when switching from direct to inverse proportionality?
4. How does inverse proportionality affect the efficiency of the machines?
5. What if the inverse proportion also involved additional factors like machine efficiency or maintenance?
6. How would the results change if the production times of the machines were different?
7. What is the relationship between direct and inverse proportionality?
8. Can you provide an example of inverse proportionality in a different context, such as finance or physics?

### Tip:
In problems involving rates and times, understanding when to apply direct versus inverse proportionality is crucial, as it directly impacts the solution and its interpretation.

Explore how inverse proportionality affects production rates using real-world examples. Learn the process of determining production ratios when machines operate inversely proportional to their production times. Suitable for students in Grades 7-9.

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