Q. 142 - Intermediate 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 twenty- 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

City Roast Columbian coffee= 12 pounds        French Roast Columbian coffee= 8 pounds 

Step 1. Given Information

City Roast Columbian coffee costs  $7.80 per pound.

French Roast Columbian coffee cost  $8.10 per pound.

In total, we have to make 20 pound blend.

Blend costs   $7.92 per pound.

Step 2. Formation of two linear Equations

Let number of pounds of City Roast Columbian coffee  = $x$.

Let number of pounds of French Roast Columbian coffee   = $y$.

As joseph has to make 20 pounds blend in total so,

$x+y=20$

And if we will balance the total amount of money then,

$$\begin{gathered} 7.80x+8.10y=20\times 7.92 \\ 7.80x+8.10y=158.4 \end{gathered}$$

Now we have 2 equations as 

$$\begin{gathered} x+y=20 \\ 7.80x+8.10y=158.4 \end{gathered}$$

Step 3. Solving the equations

From the first equation,

$x=20-y$

Now we will substitute this value in the second equation.

$$\begin{gathered} 7.80x+8.10y=158.4 \\ 7.80\left(20-y\right)+8.10y=158.4 \\ 156-7.80y+8.10y=158.4 \\ 0.3y=158.4-156 \\ 0.3y=2.4 \\ 3y=24 \\ y=8 \end{gathered}$$

Therefore the pounds of French Roast Columbian coffee = 8 pounds.

Now put this value in the equation 

$$\begin{gathered} x=20-y \\ x=20-8 \\ x=12 \end{gathered}$$

Therefore the pounds of French Roast Columbian coffee = 12 pounds.

Step 4. Check the solution

Substitute $12$ for $x$ and $8$ for $y$ in the first equation formed.

$\begin{array}{rcl}x+y& =& 20\\ 12+8& =& 20\\ 20& =& 20\end{array}$

It is a true statement.

Again, substitute the values in the second equation formed.

$\begin{array}{rcl}7.80x+8.10y& =& 158.4\\ 7.80·12+8.10·8& =& 158.4\\ 93.6+64.8& =& 158.4\\ 158.4& =& 158.4\end{array}$

This is also a true statement.

So the point satisfies both the equations.