- Analyze the definition of proportionality, which requires a constant ratio \(\frac{y}{x} = k\).
- Calculate the ratio of Pay (\(y\)) to Hours Worked (\(x\)) for all given data points.
- Observe that the ratio \(\frac{y}{x} = 8\) remains consistent for every entry.
- Conclude that the relationship is proportional with a constant of proportionality of \(8\), expressed as \(\boxed{\text{Yes}}\).
Explanation
- Analysis of Proportionality
The problem asks to determine if the pay received for babysitting is proportional to the number of hours worked, based on a provided schedule of rates. Two variables, \(x\) (Hours Worked) and \(y\) (Pay), are considered proportional if there exists a non-zero constant \(k\), known as the constant of proportionality, such that \(y = kx\), or equivalently, \(\frac{y}{x} = k\) for all pairs of \((x, y)\). The data provided is: \((1, 8), (2, 16), (3, 24), (4, 32), (4.5, 36), (5, 40), (6, 48), (6.5, 52)\).
- Calculating the Proportionality Constant
To verify if the relationship is proportional, we calculate the ratio \(\frac{y}{x}\) for each pair provided in the table:
- For 1 hour: \(\frac{8}{1} = 8\)
- For 2 hours: \(\frac{16}{2} = 8\)
- For 3 hours: \(\frac{24}{3} = 8\)
- For 4 hours: \(\frac{32}{4} = 8\)
- For 4.5 hours: \(\frac{36}{4.5} = 8\)
- For 5 hours: \(\frac{40}{5} = 8\)
- For 6 hours: \(\frac{48}{6} = 8\)
- For 6.5 hours: \(\frac{52}{6.5} = 8\)
As shown, for every data point, the ratio of pay to hours worked is consistently \(8\).
- Conclusion
Since the ratio \(\frac{y}{x}\) is constant for all given pairs, we can conclude that the pay is indeed directly proportional to the number of hours worked. The constant of proportionality is \(k = 8\), which represents the hourly rate of \(\$8\) per hour. The relationship can be expressed by the linear equation \(y = 8x\), where \(y\) is the total pay and \(x\) is the number of hours worked. Therefore, the answer to whether the pay is proportional to hours worked is yes, as evidenced by the constant ratio. \(\boxed{\text{Yes}}\)
Examples
Proportional relationships are common in daily life, such as when you buy groceries by weight. If you are buying apples that cost \(\$2\) per pound, the total cost is proportional to the weight in pounds (\(C = 2 \times w\)). If you buy 1 pound, you pay \(\$2\); if you buy 3 pounds, you pay \(\$6\). Recognizing these patterns helps you budget accurately and understand unit rates, which are fundamental in shopping, construction, and managing finances.