Question

Workmen John and Joe earn a total of \(\$ 24,60\) when John works 2 hours and Joe works 3 hours. If John works 3 hours and Joe works 2 hours, they get \(\$ 23.90 .\) Find their hourly rates.

Short answer

John's hourly rate is $4.50, and Joe's hourly rate is $5.20.

Step-by-step solution

Define Variables

Let \( J \) represent John's hourly rate in dollars, and \( O \) represent Joe's hourly rate in dollars. We need to determine the values of \( J \) and \( O \).

Write Equations from the Problem

The first scenario gives us: \( 2J + 3O = 24.60 \). The second scenario provides: \( 3J + 2O = 23.90 \). These are two equations we need to solve simultaneously.

Solve One of the Equations for a Single Variable

Let's solve the first equation \( 2J + 3O = 24.60 \) for \( J \):\[ J = \frac{24.60 - 3O}{2} \]

Substitute into the Second Equation

Substitute the expression for \( J \) from Step 3 into the second equation:\[ 3\left( \frac{24.60 - 3O}{2} \right) + 2O = 23.90 \]

Simplify and Solve for \( O \)

Multiply through by 2 to clear the fraction:\[ 3(24.60 - 3O) + 4O = 47.80 \]\[ 73.80 - 9O + 4O = 47.80 \]\[ 73.80 - 5O = 47.80 \]\[ 5O = 73.80 - 47.80 \]\[ 5O = 26.00 \]\[ O = 5.20 \]

Substitute \( O \) Back to Find \( J \)

Now that we know \( O = 5.20 \), substitute it back into the expression for \( J \):\[ J = \frac{24.60 - 3(5.20)}{2} \]\[ J = \frac{24.60 - 15.60}{2} \]\[ J = \frac{9.00}{2} \]\[ J = 4.50 \]

Verify the Solution

Substitute \( J = 4.50 \) and \( O = 5.20 \) back into both equations to verify.For \( 2J + 3O = 24.60 \):\[ 2(4.50) + 3(5.20) = 9.00 + 15.60 = 24.60 \] (True)For \( 3J + 2O = 23.90 \):\[ 3(4.50) + 2(5.20) = 13.50 + 10.40 = 23.90 \] (True)

Key concepts

Hourly Rate Calculation

Hourly rate calculation is an essential concept in this exercise. It involves determining the amount of money an individual earns per hour of work. To find the hourly rate, you usually need to know the total earnings and the number of hours worked.
In the case of John and Joe, we are comparing their total earnings when they work different hours. This exercise is an example of how hourly rate calculations are used in real-world scenarios.
Here's how it works:

• Identify the total income earned during the period.
• Divide the total income by the total number of hours worked by each individual.

Understanding hourly rate calculation helps break down earnings, allowing comparison and fair payment assessments based on work hours.

Algebraic Substitution

Algebraic substitution is a powerful technique used to simplify and solve equations. In this exercise, it helps solve for the unknown variable using another known equation. This is especially helpful when dealing with simultaneous equations.
To apply algebraic substitution, follow these steps:

• Solve one of the equations for one variable in terms of the other variables, if possible.
• Substitute this expression into another equation to find the value of one variable.
• This process helps in reducing complex equations into simpler ones.

You can then use the found value to determine other variables by substituting it back into the simplified expressions.

System of Linear Equations

A system of linear equations is a collection of two or more equations with the same variables. These systems are solved together because the equations share common values for the variables.
In the exercise with John and Joe, the system consists of:

• For John's 2 hours and Joe's 3 hours: \(2J + 3O = 24.60\)
• For John's 3 hours and Joe's 2 hours: \(3J + 2O = 23.90\)

The main objective is to find values for John and Joe's hourly rates where both conditions are true.
To solve the system:

• Express one variable in terms of another using one equation (usually the simpler one).
• Substitute the expression into the other equation and solve for one of the variables.
• Continue by substituting back to find the second variable.

This method ensures the solution fits both conditions of the original problem, making it a reliable problem-solving approach.