Problem 255 - Elementary Algebra

Question

Julia and her husband own a coffee shop. They experimented with mixing a City Roast Columbian coffee that cost \(\$ 7.80\) per pound with French Roast Columbian coffee that cost \(\$ 8.10\) per pound to make a 20 pound blend. Their blend should cost them \(\$ 7.92\) per pound. How much of each type of coffee should they buy?

Step-by-Step Solution

Answer

12 pounds of City Roast, 8 pounds of French Roast.

Step 1 - Define variables

Let \( x \) be the number of pounds of City Roast coffee, and \( y \) be the number of pounds of French Roast coffee.

Step 2 - Set up the equation for the total weight

Julia and her husband want a total of 20 pounds of coffee. Therefore, the equation for the total weight is: \[ x + y = 20 \]

Step 3 - Set up the cost equation

The total cost of the coffee blend should be equal to 20 pounds multiplied by the cost per pound \(\$ 7.92\): \[ 7.80x + 8.10y = 7.92 \times 20 \]

Step 4 - Simplify the cost equation

Calculate \( 7.92 \times 20 \): \[ 7.80x + 8.10y = 158.4 \]

Step 5 - Solve for one variable using the weight equation

From the equation \( x + y = 20 \), solve for \( y \): \[ y = 20 - x \]

Step 6 - Substitute \( y \) in the cost equation

Replace \( y \) in the cost equation with \( 20 - x \): \[ 7.80x + 8.10(20 - x) = 158.4 \]

Step 7 - Simplify and solve for \( x \)

Distribute and combine like terms: \[ 7.80x + 162 - 8.10x = 158.4 \] Combine \( x \) terms: \[ -0.30x + 162 = 158.4 \] Solve for \( x \): \[ -0.30x = 158.4 - 162 \] \[ -0.30x = -3.6 \] \[ x = 12 \]

Step 8 - Solve for \( y \)

Substitute \( x \) back into the equation \( y = 20 - x \): \[ y = 20 - 12 \] \[ y = 8 \]

Step 9 - Verify the solution

Check the total weight and cost: \[ x + y = 12 + 8 = 20 \] and \[ 7.80(12) + 8.10(8) = 93.6 + 64.8 = 158.4 \]

Key Concepts

System of Equations

Understanding a system of equations is crucial for solving mixture problems in algebra. A system of equations consists of two or more equations with the same set of variables. In this exercise, we have two variables. Letting \( x \) and \( y \) represent the pounds of City Roast and French Roast, respectively, we set up two key equations:

1. The total weight equation: \( x + y = 20 \)
2. The cost equation: \( 7.80x + 8.10y = 158.4 \)

The goal is to find the specific values of \( x \) and \( y \) that satisfy both equations simultaneously. We do this by using substitution or elimination methods. These methods allow us to manipulate the equations to isolate one variable, which can then be solved. Once one variable is known, it can be substituted back into one of the original equations to find the other variable.

Linear Equations

Linear equations are the foundation of the system of equations used in this problem. A linear equation is an equation of the first degree, meaning it has no variables raised to a power higher than one. In this exercise, the equations are:

1. Weight equation: \( x + y = 20 \)
2. Cost equation: \( 7.80x + 8.10y = 158.4 \)

These equations are linear because both their variables, \( x \) and \( y \), are to the first power. Linear equations can be graphed as straight lines, and the solution to a system of linear equations is the point where these lines intersect.

To solve the system, we first manipulate one equation to express one variable in terms of the other. From the weight equation, we find:
\( y = 20 - x \)
We then substitute this expression into the cost equation to find the value of \( x \).

Once the value of \( x \) is determined, we use it to find the value of \( y \), completing the solution.

Cost Calculations

Cost calculations are an essential part of mixture problems like the one Julia and her husband face. In our problem, the cost per pound of mixing City Roast and French Roast must average out to a specific price per pound that matches the target blend. Here's how we set this up:

First, identify the costs:

• City Roast costs \( \$ 7.80 \) per pound.
• French Roast costs \( \$ 8.10 \) per pound.

Our total blend must weigh 20 pounds and cost \( \$ 7.92 \) per pound. Therefore, the total cost of the blend is:

\[ 7.92 \times 20 = 158.4 \]

We use this information to formulate our cost equation:

\( 7.80x + 8.10y = 158.4 \)
This equation states that the combined cost of \( x \) pounds of City Roast and \( y \) pounds of French Roast must sum up to \( \$ 158.40 \).

By solving this system of equations, we ensure that the total weight is 20 pounds and the combined cost matches the specified target. This helps Julia and her husband determine exactly how much of each type of coffee to mix to achieve the desired blend at the right price.