The manager has one type of coffee that sells for $5 per pound and another type that sells for $11 per pound. The manager wishes to mix 90 pounds of the $11 coffee to get a mixture that will sell for $10 per pound. How many pounds of $5 coffee should be used?
To solve this problem, we can use the concept of weighted averages. Here's how we can approach it:
Let's assume that the manager needs to mix x pounds of the $5 coffee.
Step 1: Calculate the total cost of the mixture
The total cost of the mixture is the sum of the individual costs of the two types of coffee.
For the $11 coffee:
The manager wants to mix 90 pounds of the $11 coffee. Therefore, the cost of the $11 coffee is:
90 pounds * $11/pound = $990
For the $5 coffee:
The manager wants to mix x pounds of the $5 coffee. Therefore, the cost of the $5 coffee is:
x pounds * $5/pound = $5x
The total cost of the mixture is the sum of the cost of each type of coffee:
$990 + $5x
Step 2: Calculate the total weight of the mixture
The total weight of the mixture is the sum of the weights of the two types of coffee.
For the $11 coffee:
The manager wants to mix 90 pounds of the $11 coffee. Therefore, the weight of the $11 coffee is:
90 pounds
For the $5 coffee:
The manager wants to mix x pounds of the $5 coffee. Therefore, the weight of the $5 coffee is:
x pounds
The total weight of the mixture is the sum of the weight of each type of coffee:
90 pounds + x pounds
Step 3: Set up the equation
The average price per pound of the mixture is given as $10. Using the formula for average (sum of values divided by the total number of values), we can set up the equation:
Average price per pound = Total cost of mixture / Total weight of mixture
$10 = ($990 + $5x) / (90 pounds + x pounds)
Step 4: Solve the equation
To solve the equation, we can cross-multiply and simplify:
10(90 + x) = 990 + 5x
900 + 10x = 990 + 5x
10x - 5x = 990 - 900
5x = 90
x = 90 / 5
x = 18
Therefore, the manager should use 18 pounds of the $5 coffee to mix with the 90 pounds of the $11 coffee to get a mixture that will sell for $10 per pound.
Let's assume the amount of $5 coffee to be used is x pounds.
Since the manager wishes to mix 90 pounds of the $11 coffee to get a mixture that will sell for $10 per pound, the total weight of the mixture will be 90 pounds.
So, the weight of the $11 coffee used is 90 pounds.
Now, let's find the weight of the $5 coffee used.
The average price of the mixture will be the weighted average of the prices of the two types of coffee used.
10 = (11 * 90 + 5 * x) / (90 + x)
Multiplying both sides of the equation by (90 + x):
10(90 + x) = 11(90) + 5x
900 + 10x = 990 + 5x
Simplifying the equation:
10x - 5x = 990 - 900
5x = 90
Dividing both sides of the equation by 5:
x = 18
Therefore, the manager should use 18 pounds of $5 coffee.
let the number of pounds of the cheaper kind be x
solve for x
11(90) + 5x = 10(90+x)