Question: A coffee distributor needs to mix a(n) Guatemala Antigua coffee blend that normally sells for…

A coffee distributor needs to mix a(n) Guatemala Antigua coffee blend that normally sells for $10.70 per pound with a Arabian Mocha coffee blend that normally sells for $12.10 per pound to create 70 pounds of a coffee that can sell for $11.70 per pound. How many pounds of each kind of coffee should they mix?

A) Write an equation using the information as it is given above that can be solved to answer the question. Use \( x \) as your variable to represent the quantity of Guatemala Antigua coffee blend.

Equation: \[ 10.70x + 12.10(70 - x) = 11.70 \times 70 \]

B) Answer: They must mix \[ \text{pounds of the Guatemala Antigua Blend} \] \[ \text{pounds of the Arabian Mocha Blend.} \]

Round your answers to the nearest whole number of pounds.

Solution

A coffee distributor needs to mix two types of coffee: Guatemala Antigua at $10.70 per pound and Arabian Mocha at $12.10 per pound. They want to create a blend of 70 pounds that sells for $11.70 per pound. We need to find out how many pounds of each kind of coffee to mix. Let’s denote \( x \) as the pounds of Guatemala Antigua coffee blend. Then, the pounds of Arabian Mocha coffee blend will be \( 70 - x \). ### A) Write an equation The total cost of the mixed coffee should equal the cost of 70 pounds at $11.70 per pound: \[ 10.70x + 12.10(70 - x) = 11.70 \times 70 \] ### B) Solve the equation First, calculate the right side of the equation: \[ 11.70 \times 70 = 819 \] Now, substitute into the equation: \[ 10.70x + 12.10(70 - x) = 819 \] Distribute \(12.10\): \[ 10.70x + 847 - 12.10x = 819 \] Combine like terms: \[ -1.40x + 847 = 819 \] Subtract 847 from both sides: \[ -1.40x = -28 \] Divide by -1.40: \[ x = 20 \] So, 20 pounds of Guatemala Antigua coffee blend are needed. To find the pounds of Arabian Mocha: \[ 70 - x = 70 - 20 = 50 \] They must mix 20 pounds of the Guatemala Antigua Blend and 50 pounds of the Arabian Mocha Blend.