Sarah blends coffee for Tasti-Delight. She needs to prepare 170 lbs of blended coffee beans selling for $3.59 per pound. She plans to do this by blending together a high-quality bean costing $5.00 per pound and a cheaper bean at $2.00 per pound. To the nearest pound, find how much high-quality coffee bean and how much cheaper coffee bean she should blend.
Step 1: Let's assume that Sarah needs to blend x pounds of high-quality coffee beans and y pounds of cheaper coffee beans.
Step 2: The total weight of the blended coffee beans is 170 pounds, so we can write the equation: x + y = 170.
Step 3: The price per pound of the high-quality coffee beans is $5.00, and the price per pound of the cheaper coffee beans is $2.00. The average price per pound of the blended coffee beans is $3.59. This gives us the equation: (5x + 2y) / 170 = 3.59.
Step 4: We can solve the system of equations by substitution. Rearrange the first equation to solve for x: x = 170 - y.
Step 5: Substitute this expression for x in the second equation: (5(170 - y) + 2y) / 170 = 3.59.
Step 6: Simplify the equation: (850 - 5y + 2y) / 170 = 3.59.
Step 7: Combine like terms: (850 - 3y) / 170 = 3.59.
Step 8: Multiply both sides of the equation by 170 to eliminate the fraction: 850 - 3y = 3.59 * 170.
Step 9: Simplify the equation: 850 - 3y = 611.3.
Step 10: Subtract 850 from both sides of the equation: -3y = 611.3 - 850.
Step 11: Simplify the equation: -3y = -238.7.
Step 12: Divide both sides of the equation by -3 to solve for y: y = -238.7 / -3.
Step 13: Calculate y: y = 79.57.
Step 14: Since we need to find the nearest whole number, let's round y to the nearest pound: y ≈ 80 pounds.
Step 15: Substitute the value of y back into the first equation to find x: x + 80 = 170.
Step 16: Solve for x: x = 170 - 80.
Step 17: Calculate x: x = 90 pounds.
Therefore, Sarah needs to blend approximately 90 pounds of high-quality coffee beans and 80 pounds of cheaper coffee beans.
To determine how much high-quality coffee bean and cheaper coffee bean Sarah should blend, we can set up an equation based on the given information.
Let's assume Sarah needs to blend x pounds of high-quality coffee beans and y pounds of cheaper coffee beans.
The total weight of the blended coffee beans is 170 lbs, so we have the equation:
x + y = 170 (Equation 1)
We also know that the cost per pound for the blended coffee beans is $3.59. To calculate the cost per pound, we can set up another equation based on the individual costs per pound and the quantity of each type of coffee bean used:
5.00x + 2.00y = 3.59 * 170 (Equation 2)
Now we have a system of two equations:
x + y = 170 (Equation 1)
5.00x + 2.00y = 3.59 * 170 (Equation 2)
We can solve this system of equations to find the values of x and y.
One way to solve this system is by substitution. We can solve Equation 1 for x and substitute it into Equation 2.
From Equation 1, we have:
x = 170 - y
Substituting this into Equation 2, we get:
5.00(170 - y) + 2.00y = 3.59 * 170
Now we can simplify and solve for y:
850 - 5y + 2y = 610.3
Combine like terms:
-3y = -239.7
Divide both sides by -3:
y = 79.9
Since we want the answer rounded to the nearest pound, y ≈ 80.
Now we can substitute the value of y back into Equation 1 to find x:
x + 80 = 170
x = 170 - 80
x = 90
Therefore, Sarah should blend approximately 90 pounds of high-quality coffee beans and 80 pounds of cheaper coffee beans to prepare 170 pounds of blended coffee beans.
Blend:
X lbs of hi-quality beans.
Y lbs of cheaper beans.
Eq1: X + Y = 170 lbs.
Eq2: 5x + 2y = 3.59*170 = $610.30.
Multiply Eq1 by -2 and add the Eqs:
-2x - 2y = -340
+5x + 2y = 610.3
Sum: 3x = 270.3,
X = 90 lbs.
90 + y = 170,
Y = 170 - 90 = 80 lbs.