Question

A grocer wishes to blend two different coffee brands in order to make a blend of $400$ pounds to sell at $\$3.00$ a pound.

If he uses a brand of coffee worth $\$2.70$ a pound with another brand worth $\$3.20$ a pound, how many pounds of each does he use?

Answer 1

Find how many pounds of each coffee brand use.

Let $x$ is the amount of $\$2.70/\mathrm{lb}$ coffee and $y$ is the amount of $\$3.20/\mathrm{lb}$ coffee.

The total amount is computed by the following equation as follows:

$x+y=400$

The total price is computed by the following equation as follows:

$$\begin{gathered} 2.7x+3.2y=3.00\times 400 \\ 2.7x+3.2y=1200 \end{gathered}$$

So the system of equation is :

$$\begin{gathered} x+y=400 \\ 2.7x+3.2y=1200 \end{gathered}$$

Solve the system of equation by substitution:

$$\begin{gathered} x+y=400 \\ \Rightarrow x=400-y \end{gathered}$$

Substitute $x=400-y$ into second equation.

$$\begin{gathered} 2.7x+3.2y=1200 \\ \Rightarrow 2.7(400-y)+3.2y=1200 \\ \Rightarrow 1080-2.7y+3.2y=1200 \\ \Rightarrow 0.5y=120 \\ \Rightarrow y=240 \end{gathered}$$

So,further simplify as shown:

$$\begin{gathered} x=400-y \\ \Rightarrow x=400-240 \\ \Rightarrow x=160 \end{gathered}$$

Therefore, the grocer uses $160$ pounds of the $\$2.70/\mathrm{lb}$ coffee and $240$ pounds of the $\$3.20/\mathrm{lb}$ coffee.