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To solve this problem, let's analyze the information given:

- A garden hose fills a 2-gallon bucket in 5 seconds.
- The number of gallons \( g \) is proportional to the number of seconds \( t \).

From this, we can determine the rate at which the garden hose fills the bucket. Since it fills 2 gallons in 5 seconds, we calculate the rate:

\[
\text{rate} = \frac{2 \text{ gallons}}{5 \text{ seconds}} = 0.4 \text{ gallons per second}
\]

This rate means that the relationship between \( g \) and \( t \) can be written as:
\[
g = 0.4t
\]

Now, let's evaluate the answer choices to find which equations represent the relationship between \( g \) and \( t \):

1. **\( g = 0.4t \)** - This is correct, as it matches the proportional relationship we calculated.
2. **\( t = 0.4g \)** - This is incorrect. If we solved for \( t \) in terms of \( g \), we would get \( t = \frac{g}{0.4} = 2.5g \).
3. **\( g = 2.5t \)** - This is incorrect, as the constant of proportionality should be 0.4, not 2.5.
4. **\( t = 2.5g \)** - This is correct. Solving for \( t \) in terms of \( g \) gives \( t = \frac{g}{0.4} = 2.5g \).
5. **\( g = \frac{2}{5}t \)** - This is correct. The fraction \( \frac{2}{5} \) is equivalent to 0.4, so this equation also correctly represents the relationship between \( g \) and \( t \).

### Correct Answers:
- **\( g = 0.4t \)**
- **\( t = 2.5g \)**
- **\( g = \frac{2}{5}t \)**

Would you like more details, or do you have any questions?

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Here are 5 related questions for further practice:

1. If the rate were instead 3 gallons in 10 seconds, what would the equation relating \( g \) and \( t \) be?
2. How would the equation change if the hose filled 4 gallons in 8 seconds?
3. If the relationship were \( g = 0.6t \), how long would it take to fill a 3-gallon bucket?
4. How many gallons would fill in 7.5 seconds with a rate of \( g = 0.4t \)?
5. If the hose fills 5 gallons in 20 seconds, what is the rate of filling per second?

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**Tip:** When analyzing proportional relationships, identify the constant rate or unit rate first. This rate will be your proportionality constant in the equation.

A garden hose fills a 2-gallon bucket in 5 seconds. The number of gallons, g, is proportional to the number of seconds, t, that the water is running. Select all the equations that represent the relationship between g and t.

Understand the proportional relationship between the amount of water in gallons and the time in seconds for a garden hose filling a bucket. This problem covers concepts of proportionality and rates, suitable for Grade 7 students.

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