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Let’s go through each question on this page to find the correct answers.

### Question 1
*Which equation represents a direct proportion?*
- Direct proportion means the relationship between two variables where one is a constant multiple of the other. In other words, it has the form $ y = kx $, where $ k $ is a constant.
  
The correct answer is **D. $ y = 2x $**, since this equation represents a direct proportion with a constant of 2.

### Question 2
*Two trains, Train 1 and Train 2, are traveling at a constant rate of speed. The equation $ y = 122x $ shows the total distance $ y $ in miles for Train 1 over $ x $ hours. The graph shows the relationship between time and distance for Train 2.*

- From the graph, Train 2 reaches 480 miles in 4 hours. The speed of Train 2 is therefore:
  \[
  \frac{480 \text{ miles}}{4 \text{ hours}} = 120 \text{ miles per hour}
  \]
- Train 1’s speed is given by the equation $ y = 122x $, which indicates a speed of 122 miles per hour.

Since Train 1 has a higher speed than Train 2, the correct answer is **A. Train 1 is traveling at a faster speed than Train 2.**

### Question 3
*Sandra and Emil are house painters with different hourly rates. Sandra’s rate is given by $ y = 13x $ where $ y $ is the total charge in dollars for $ x $ hours. Emil’s rate is shown in the table.*

| $ x $ | 2  | 4  | 6  | 8  |
|---------|----|----|----|----|
| $ y $ | 26 | 52 | 78 | 104 |

- Emil’s rate can be found by dividing the total charge by the hours worked:
  \[
  \frac{26}{2} = 13, \quad \frac{52}{4} = 13, \quad \frac{78}{6} = 13, \quad \frac{104}{8} = 13
  \]
- Both Sandra and Emil charge $13 per hour.

The correct answer is **D. Sandra and Emil work for the same hourly rate.**

### Question 4
*Which table represents a direct proportion?*
- A direct proportion means that as $ x $ increases, $ y $ increases at a constant rate, represented as $ y = kx $.

Examining each table:
- **Table D** shows a constant rate (4, 8, 12, 16), which is a multiple of $ x $ by 4.

The correct answer is **D.**

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Would you like further details on any of these answers?

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### Related Questions
1. What is the difference between direct proportion and linear relationships?
2. How can you determine if two variables are in direct proportion from a table of values?
3. What are other real-life examples of direct proportionality?
4. How does constant speed relate to proportionality in real-world problems?
5. How can you derive an equation for direct proportion from a given table?

### Tip
To verify if a table represents a direct proportion, check if the ratio $ \frac{y}{x} $ is constant for each pair of values.

Which equation represents a direct proportion? Which statement is true about the speeds of two trains? What is true about Sandra and Emil's hourly painting rates? Which table represents a direct proportion?

Practice identifying direct proportion in equations, rates, and tables with these problems on constant speed and proportional relationships. Understand concepts of direct proportion and rate calculations, ideal for Grade 7 students.

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