Question

Althea uses this rule to figure out how much to charge for baby-sitting: Charge \(\$ 5\) per hour for one child, plus \(\$ 2\) per hour for each additional child. a. Last Saturday she watched the Newsome twins for 3 hours. How much money did she earn? Explain how you found your answer. b. Mr. Foster hires Althea to watch his three children for 2 hours. How much will she charge? c. Does Althea earn more for watching two children for 3 hours or three children for 2 hours? d. Althea hopes to earn \(\$ 25\) next weekend to buy her sister a birthday present. Describe two ways she could earn at least \(\$ 25\) baby-sitting.

Short answer

a. \( \$ 21 \) b. \( \$ 18 \) c. Two children for 3 hours d. Two options: 2 children for 4 hours or 3 children for 3 hours.

Step-by-step solution

Understanding the Rule

Althea charges \(5\) per hour for one child and \(2\) per hour for each additional child. This means for two children, the charge per hour is \(5 + 2 = 7\), and for three children, the charge per hour is \(5 + 2 + 2 = 9\).

Part (a) - Calculate Earnings for Two Children in 3 Hours

For two children, the hourly charge is \(5 + 2 = 7\). So, for 3 hours, she earned \(3 \times 7 = 21\). Therefore, she earned \( \$ 21 \) watching the Newsome twins for 3 hours.

Part (b) - Calculate Earnings for Three Children in 2 Hours

For three children, the hourly charge is \(5 + 2 + 2 = 9\). So, for 2 hours, she will charge \(2 \times 9 = 18\). Therefore, she will charge Mr. Foster \( \$ 18 \).

Part (c) - Compare Earnings

From part (a), we know she earns \( \$ 21\) for two children for 3 hours. From part (b), she earns \( \$ 18 \) for three children for 2 hours. Thus, she earns more by watching two children for 3 hours.

Part (d) - Describe Two Ways to Earn At Least \( \$ 25 \)

One way she could earn at least \( \$ 25 \) is by watching two children for 4 hours: \(4 \times 7 = 28\)\. Another way is by watching three children for 3 hours: \(3 \times 9 = 27\)\.

Key concepts

Hourly Rate Calculation

Understanding how to calculate hourly rates is essential in various jobs. For Althea, her rates are structured based on the number of children she babysits. She charges \(\text{\$5}\) per hour for one child. For each additional child, she charges \(\text{\$2}\) extra per hour.

Let's break it down with examples:

• For two children: \(5 + 2 = 7\) per hour.
  
• For three children: \(5 + 2 + 2 = 9\) per hour.

This structure helps her to easily calculate earnings no matter the number of children she’s babysitting.

Comparing Earnings

Comparing earnings involves looking at different scenarios to see which one results in more money. In Althea's case, let's compare the earnings for different situations:

• Watching two children for 3 hours: The hourly rate is \(7\). Multiply this by 3 hours: \(3 \times 7 = 21\). She earns \(\text{\$21}\).
  
• Watching three children for 2 hours: The hourly rate is \(9\). Multiply this by 2 hours: \(2 \times 9 = 18\). She earns \(\text{\$18}\).

Hence, she earns more with the first scenario of watching two children for 3 hours (\(\text{\$21}\)) compared to the second scenario of three children for 2 hours (\(\text{\$18}\)). This comparison helps in making better choices about how to use her time most effectively.

Linear Equations

Linear equations can be a powerful tool to simplify and solve problems involving hourly rates. Using Althea’s rates, we can form an equation to represent her earnings:

\( \text{Earnings} = 5x + 2y\) \
Where \( x \) is the number of hours for one child, and \( y \) is the number of additional children.

For example:

• Watching 2 children for 3 hours: The equation is \( \text{Earnings} = 5 \times 3 + 2 \times 3 = 21\).
  
• Watching 3 children for 2 hours: The equation is \( \text{Earnings} = 5 \times 2 + 2 \times 2 \times 2 = 18\).

Linear equations help visualize and calculate various earning scenarios easily, making complex calculations straightforward. This aids in better financial planning and decision-making.