Problem 90 - Prealgebra

Question

Kelly earns \(\$ 15\) per hour working as a graphic designer. We can express the amount she earns each week for working \(x\) hours with the expression \(15 x .\) Indicate with a yes or no which of the following could be one of Kelly's paychecks. If you answer no, explain your answer. a. \(\$ 75\) for working five hours b. \(\$ 125\) for working nine hours c. \(\$ 90\) for working six hours d. \(\$ 500\) for working 35 hours

Step-by-Step Solution

Answer

a. yes, b. no, c. yes, d. no.

Step 1: Calculate Paycheck for Option A

To determine Kelly's paycheck for working five hours, calculate the expression \(15 \times 5\). This results in \(75\). Since the paycheck is \(\$75\), which matches the calculation, the answer for option A is **yes**.

Step 2: Calculate Paycheck for Option B

To determine Kelly's paycheck for working nine hours, calculate the expression \(15 \times 9\). This results in \(135\). Since the paycheck is \(\$125\), which does not match the calculation, the answer for option B is **no**.

Step 3: Calculate Paycheck for Option C

To determine Kelly's paycheck for working six hours, calculate the expression \(15 \times 6\). This results in \(90\). Since the paycheck is \(\$90\), which matches the calculation, the answer for option C is **yes**.

Step 4: Calculate Paycheck for Option D

To determine Kelly's paycheck for working 35 hours, calculate the expression \(15 \times 35\). This results in \(525\). Since the paycheck is \(\$500\), which does not match the calculation, the answer for option D is **no**.

Key Concepts

Hourly Wage Calculation

When estimating earnings for work, understanding hourly wage calculations is vital. In the exercise provided, Kelly earns a fixed rate of $15 per hour working as a graphic designer. To determine her weekly earnings, you simply multiply the number of hours worked by the hourly wage. This calculation helps in setting expectations for earnings based on the number of hours planned to work.

For example, if Kelly works 5 hours in a week, she earns \(15 \times 5 = 75\) dollars. This principle of hourly wage calculation is that the earnings directly correlate with the time spent working, assuming the same hourly rate.

Hourly wage calculations are fundamental when budgeting and managing personal or household finances. This method of calculation applies to many employment scenarios where the hours worked may vary weekly.

Multiplication of Integers

Multiplication of integers is a crucial mathematical operation. It's the foundation for many real-life applications, including wage calculations. In this context, multiplying Kelly's hourly rate by the hours worked means multiplying integers: her hourly wage (an integer) and her hours (another integer).

For instance, multiplying her hourly wage of 15 dollars by 9 hours involves calculating \(15 \times 9 = 135\) dollars. This operation shows how each hour contributes equally to the total pay. Multiplying integers involves adding the integer to itself repeatedly, for example, 15 added 9 times.

When managing large numbers, multiplying integers ensures accuracy and efficiency. This method is also essential in handling larger datasets or when scaling calculations to bigger numbers, such as yearly or monthly salaries.

Problem-Solving Steps

Problem-solving involves breaking down a complex task into manageable steps. This approach is evident in the provided wage calculation exercise. Each option (A, B, C, D) presents a different set of hours worked and paycheck received, requiring verification through specific calculation steps.

To solve the problem:

• Identify the hourly wage and hours worked.
• Calculate the total expected earnings using multiplication.
• Compare the calculated earnings with the given paycheck.

Each step is precise and methodical, designed to reveal discrepancies, such as incorrect paychecks or work hour calculations.

Honing problem-solving skills not only helps in academic scenarios but also develops critical thinking for everyday situations, such as financial planning or troubleshooting issues at work.