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Part 1

Type​ A, type​ B, and type C lightbulbs are lasting longer today than ever before.  On​ average, the number of bulb hours for a type C bulb is 

1717 

times the number of bulb hours for a type B bulb.  The number of bulb hours for a type A bulb is 

25002500 

less than the type B bulb.  If the total number of bulb hours for the three types of lightbulbs is 

54 comma 50054,500​, 

find the number of bulb hours for each type.

Let's define the variables and set up equations based on the given problem:

### Step 1: Define the variables
- Let \( x \) represent the number of bulb hours for the type B bulb.
- The number of bulb hours for the type C bulb is \( 1717x \) (since it's 1717 times the number of bulb hours for type B).
- The number of bulb hours for the type A bulb is \( x - 2500 \) (since it's 2500 less than the number of bulb hours for type B).

### Step 2: Set up the equation
The total number of bulb hours for the three types of lightbulbs is given as 54,500. Therefore, we can write the equation:
\[
x + (x - 2500) + 1717x = 54,500
\]

### Step 3: Simplify the equation
Combine like terms:
\[
x + x - 2500 + 1717x = 54,500
\]
\[
1719x - 2500 = 54,500
\]

### Step 4: Solve for \( x \)
Add 2500 to both sides of the equation:
\[
1719x = 57,000
\]

Now, divide by 1719 to solve for \( x \):
\[
x = \frac{57,000}{1719} \approx 33.15
\]

### Step 5: Calculate the bulb hours for each type
- **Type B bulb hours**: \( x = 33.15 \) (approximately)
- **Type C bulb hours**: \( 1717x = 1717 \times 33.15 \approx 56925.55 \)
- **Type A bulb hours**: \( x - 2500 = 33.15 - 2500 \approx -2466.85 \)

### Final Result
The calculations indicate that something is wrong since we are getting negative hours for the Type A bulb. This suggests that the values or relationships provided might have an error. Specifically, the large multiplier for type C makes the sum difficult to reconcile with the total of 54,500. 

Could you confirm the figures, or shall we revisit the setup?

Here are five related questions:
1. How would the equation change if the relationship between the bulb types was different?
2. What could cause a scenario where hours are negative in a problem like this?
3. How can you verify if your results make logical sense in a real-world context?
4. Could there be a different approach to solve this problem more efficiently?
5. What is the importance of units in solving word problems like this?

**Tip:** Always check the realism of the numbers obtained in your solution—if they seem unreasonable (like negative bulb hours), recheck the problem's conditions and the calculations.

Explore how to solve equations involving different types of lightbulbs with varying hours. Understand the relationships between bulb types and calculate the hours for each type based on given conditions. Suitable for students in Grades 7-9.

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