Problem 14 - Elementary Algebra

Question

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question. Erica earned a total of \(\$ 50,450\) last year from her two jobs. The amount she earned from her job at the store was \(\$ 1,250\) more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

Step-by-Step Solution

Answer

Erica earned \( \$ 12,300 \) from her job at the college.

Step 1: Define the variables

Let the amount Erica earned from her job at the college be denoted as \( c \).

Step 2: Create an equation for the store earnings

The amount she earned from her job at the store was \( \$ 1,250 \) more than three times the amount she earned from her job at the college. Therefore, the earning from the store can be represented as \( 3c + 1,250 \).

Step 3: Write the total earnings equation

The total amount Erica earned from both jobs is \( \$ 50,450 \). Hence, the equation can be set up as follows: \[ c + (3c + 1,250) = 50,450 \]

Step 4: Simplify the equation

Combine like terms in the equation: \[ c + 3c + 1,250 = 50,450 \] This simplifies to: \[ 4c + 1,250 = 50,450 \]

Step 5: Isolate the variable

Subtract \( 1,250 \) from both sides of the equation: \[ 4c + 1,250 - 1,250 = 50,450 - 1,250 \] This reduces to: \[ 4c = 49,200 \]

Step 6: Solve for \( c \)

Divide both sides by 4 to isolate \( c \): \[ c = \frac{49,200}{4} \] \[ c = 12,300 \]

Step 7: Write the final answer

Erica earned \( \$ 12,300 \) from her job at the college.

Key Concepts

Defining Variables

When solving word problems, the first step is to define the variables. Variables represent unknown quantities in the problem. In Erica's earnings problem, we need a variable to represent the unknown amount she earned from her job at the college. We let this amount be denoted by the variable \( c \). Choosing clear and meaningful variables helps make the problem easier to understand and solve.

Creating Equations

Once the variables are defined, the next step is to create equations that express the relationships described in the problem. In this case, the problem states that the amount Erica earned from her job at the store is \( \$1,250 \) more than three times the amount she earned from her job at the college. We can represent this relationship with the equation: \( \text{store earnings} = 3c + 1,250 \). This way, we use the variable \( c \) to tie together the unknowns.

Simplifying Equations

After creating the equation, we need to simplify it to make solving easier. We already know that Erica’s total earnings from both jobs is \( \$50,450 \). Therefore, the total earnings equation is: \( c + (3c + 1,250) = 50,450 \). First, combine the like terms: \( c + 3c \), which results in: \( 4c \). This simplifies our equation to: \( 4c + 1,250 = 50,450 \). By simplifying the equation, we get closer to finding the value of the unknown variable.

Isolating Variables

Isolating the variable involves getting the unknown variable by itself on one side of the equation. In our simplified equation, \( 4c + 1,250 = 50,450 \), we isolate \( c \) by first subtracting \( 1,250 \) from both sides: \( 4c + 1,250 - 1,250 = 50,450 - 1,250 \). This operation simplifies to: \( 4c = 49,200 \). Isolating variables is a crucial step that prepares us to easily solve for the variable.

Solving for Variables

Finally, we solve for the variable. In our equation \( 4c = 49,200 \), we solve for \( c \) by dividing both sides of the equation by 4: \( c = \frac{49,200}{4} \). Performing the division, we get: \( c = 12,300 \). Therefore, Erica earned \( \$12,300 \) from her job at the college. Solving for the variable gives us the value needed to answer the question posed in the problem.