Joy is a tech geek and loves her new smartphone, but she is frustrated by how slowly the battery charges. After coming home from school, Joy notices that the battery level is low, so she plugs in her phone to charge the battery.

This table shows the relationship between the total time (in hours) that Joy charges her phone, 

*x*,

 and the battery level (as a percent of its full capacity), 

*y*.

*x* (hours)

*y* (percent)

1

15

2

25

3

35

4

45

According to the values in the table, do *x* and *y* have a proportional relationship?

yes

no





To determine whether \( x \) (time in hours) and \( y \) (battery level in percent) have a proportional relationship, we need to check if the ratio between \( y \) and \( x \) remains constant.

The formula for a proportional relationship is:
\[
\frac{y}{x} = \text{constant}
\]
Let's check if the ratios for the given pairs of \( (x, y) \) are the same:

1. For \( x = 1 \) and \( y = 15 \):
\[
\frac{y}{x} = \frac{15}{1} = 15
\]

2. For \( x = 2 \) and \( y = 25 \):
\[
\frac{y}{x} = \frac{25}{2} = 12.5
\]

3. For \( x = 3 \) and \( y = 35 \):
\[
\frac{y}{x} = \frac{35}{3} \approx 11.67
\]

4. For \( x = 4 \) and \( y = 45 \):
\[
\frac{y}{x} = \frac{45}{4} = 11.25
\]

Since the ratios are not constant, \( x \) and \( y \) do **not** have a proportional relationship.

---

Would you like more details or have any questions?

Here are 5 related questions:
1. What is a proportional relationship in mathematics?
2. How can you recognize proportional relationships in tables or graphs?
3. Can you describe a situation where two variables are proportional?
4. How would the battery charge look if \( x \) and \( y \) had a proportional relationship?
5. How can we model non-proportional relationships using functions?

**Tip:** In a proportional relationship, the graph is always a straight line passing through the origin.

Joy is a tech geek and loves her new smartphone, but she is frustrated by how slowly the battery charges. After coming home from school, Joy notices that the battery level is low, so she plugs in her phone to charge the battery. This table shows the relationship between the total time (in hours) that Joy charges her phone, x, and the battery level (as a percent of its full capacity), y. x (hours) y (percent) 1 15 2 25 3 35 4 45 According to the values in the table, do x and y have a proportional relationship?

Explore the relationship between charging time and battery level with Joy's smartphone example. Determine if the time (in hours) and battery level (in percent) form a proportional relationship by calculating the ratio of y/x. This problem is designed for students in Grades 6-8, covering key concepts in ratios and proportional relationships.

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