Problem 89 - Prealgebra

Question

Luke earns \(\$ 12\) per hour working as a math tutor. We can express the amount he earns each week for working \(x\) hours with the expression \(12 x .\) Indicate with a yes or no, which of the following could be one of Luke's paychecks. If you answer no, explain your answer. a. \(\$ 60\) for working five hours b. \(\$ 100\) for working nine hours c. \(\$ 80\) for working seven hours d. \(\$ 168\) for working 14 hours

Step-by-Step Solution

Answer

a. Yes; b. No; c. No; d. Yes.

Step 1: Verify Option A

Calculate the amount Luke earns for 5 hours: \( 12 \times 5 = 60 \). Since the calculated paycheck \( 60 \) matches the given paycheck \$60, the answer is yes for option a.

Step 2: Verify Option B

Calculate the amount Luke earns for 9 hours: \( 12 \times 9 = 108 \). Since the calculated paycheck \( 108 \) does not match the given paycheck \\(100, the answer is no for option b. This discrepancy is because 9 hours should result in more than \\)100.

Step 3: Verify Option C

Calculate the amount Luke earns for 7 hours: \( 12 \times 7 = 84 \). Since the calculated paycheck \( 84 \) does not match the given paycheck \\(80, the answer is no for option c. Luke would actually earn \\)84 if he worked 7 hours, not \$80.

Step 4: Verify Option D

Calculate the amount Luke earns for 14 hours: \( 12 \times 14 = 168 \). Since the calculated paycheck \( 168 \) matches the given paycheck \$168, the answer is yes for option d.

Key Concepts

Hourly Wage Calculation

Let's break down how you can determine an hourly wage payment and ensure it is correct. When someone is paid hourly, their total earnings for a period (like a week or a day) depend on how many hours they’ve worked.
The main formula involved is simple – multiply the hourly rate by the number of hours worked.
For example, if a math tutor, like Luke in our exercise, earns \(12 per hour, and he works for 5 hours, the calculation to find his total earnings would be:

• Hourly Rate: \)12
• Hours Worked: 5
• Total Earnings: \(12 \times 5 = 60\) dollars

By using this straightforward formula, you can figure out how much someone makes in a given time frame, based on their hourly wage. Calculating accurately is crucial not just for employees but also for employers managing payroll. It's a practical application of multiplication and basic arithmetic that directly applies to real-world scenarios.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations.
In our exercise, the expression \(12x\) represents the total earnings for Luke, where:

• 12 is the coefficient, representing the hourly rate, which is fixed.
• \(x\) is the variable, representing the number of hours worked, which can vary.

Expressions like this help us model real-world situations. They are powerful because they allow for adaptability – the expression \(12x\) can calculate the earnings for any number of hours, by substituting \(x\) with the actual hours worked.
Algebraic expressions are foundational in mathematics and help in encoding and solving practical problems, like determining total wages from hours worked and hourly pay, in a concise and flexible way.

Multiplication in Algebra

Multiplication is a fundamental operation in algebra that helps us deal with repeated addition quickly.
In algebra, multiplication is often used to scale up quantities, like calculating total earnings from hourly rates. This is evident in the exercise with the calculation of \(12 \times x\), where:

• 12 is the constant factor, representing Luke's hourly wage.
• \(x\) is the variable progressor, which can be substituted with exact numbers, like 5 hours or 9 hours.

Multiplying allows us to find the product, which, in this instance, gives us the total salary for Luke depending on how many hours he works. The application of multiplication in algebra helps to streamline and solve problems where additional factors change. It simplifies expressions and calculations, making it an indispensable tool in mathematics for both theoretical and practical applications. Understanding it helps in decoding real-life scenarios with efficiency and accuracy.