Question

Caleb made the table below to show the number of hours he works and the amounts he earns over a four-week period.


Caleb’s Earnings
Hours
12
15
18
21
Earnings
(in dollars)
140
170
200
230

Which describes whether the relationship between the earnings and the number of hours represents a direct variation?
He earns 20 more than 10 times the number of hours he works each week, so the earnings vary directly with the hours.
The earnings increase by 30 for every increase of 3 hours, so the earnings vary directly with the hours.
The ratios of earnings to hours are not the same each week, so the earnings do not vary directly with the hours.
The earnings increase at a steady rate, so the earnings do not vary directly with the hours.

Answer 1

To determine if Caleb's earnings vary directly with the number of hours he works, we need to analyze the relationship given in the table. Direct variation implies that there is a constant ratio between the two quantities.

1. Let's examine the ratios of earnings to hours:
- For 12 hours: \( \frac{140}{12} \approx 11.67 \)
- For 15 hours: \( \frac{170}{15} \approx 11.33 \)
- For 18 hours: \( \frac{200}{18} \approx 11.11 \)
- For 21 hours: \( \frac{230}{21} \approx 10.95 \)

2. The ratios are not the same for each week. As shown above, the ratios of earnings to hours are decreasing. Therefore, the relationship between earnings and hours does not have a constant ratio.

Now let's evaluate the statements:

- **He earns 20 more than 10 times the number of hours he works each week, so the earnings vary directly with the hours.** (This is inaccurate since direct variation does not have an additive constant like 20.)

- **The earnings increase by 30 for every increase of 3 hours, so the earnings vary directly with the hours.** (This statement is misleading. Direct variation requires a constant ratio, regardless of how much money is earned or hours worked. Adjusting hours with fixed dollar increases does not denote direct variation.)

- **The ratios of earnings to hours are not the same each week, so the earnings do not vary directly with the hours.** (This is true. Since the ratios are different, Caleb's earnings do not vary directly with hours.)

- **The earnings increase at a steady rate, so the earnings do not vary directly with the hours.** (This is incorrect; while there is a consistent increase, the key point for direct variation is the constant ratio, not just the increase.)

Given this analysis, the correct conclusion is:

**The ratios of earnings to hours are not the same each week, so the earnings do not vary directly with the hours.**