Sure, let's work through this problem step by step.
We have a situation where 5 people take 12 minutes to change all the light bulbs in a restaurant. We need to find out how long it would take for 3 people to do the same task.
1. Identify the relationship: This is a classic example of an inverse proportion. When more people are working together, they usually take less time to complete a task. Conversely, when fewer people are working, it takes more time.
2. Constant product method: In inverse proportion, the product of the number of people and the amount of time taken remains constant. Let's call this product the "constant."
- For 5 people taking 12 minutes, the constant would be:
[tex]\[
\text{constant} = \text{number of people} \times \text{time taken} = 5 \times 12
\][/tex]
So,
[tex]\[
\text{constant} = 60
\][/tex]
3. Set up the equation for the new situation: Now, we need to find the time it would take 3 people to complete the task. Let's call this time [tex]\( T \)[/tex].
- Using the constant from above, we set up the equation:
[tex]\[
\text{constant} = \text{number of people} \times \text{time taken} = 3 \times T
\][/tex]
- Plugging in the value of the constant:
[tex]\[
60 = 3 \times T
\][/tex]
4. Solve for [tex]\( T \)[/tex]:
[tex]\[
T = \frac{60}{3}
\][/tex]
[tex]\[
T = 20
\][/tex]
So, it would take 3 people 20 minutes to change all of the light bulbs in the restaurant.
