a new blend of coffee will cost $3.90 per pound by mixing 2 coffees that sell for $2.75 and $5 per pound. what amounts of each coffee should be blended to obtain the desired mixture?
(the total weight of the desired blend is 100 lbs)
How would I solve this?
The equation would 3.90 = 2.75x + 5x ?
3.90(100) = 2.75x + 5x
Let there be x lb of the expensive coffee and y lb of the inexpensive coffee.
x + y = 100
5x + 2.75 y = 3.90*100 = 390
Now solve for both x and y.
5x + 5y = 500
2.25 y = 110
y = 48.89
x = 51.11
It is almost a 50/50 blend
To solve this problem, you can set up a system of equations to represent the given information. Let's define two variables:
- Let x be the number of pounds of the $2.75 coffee.
- Let y be the number of pounds of the $5 coffee.
Since the total weight of the desired blend is 100 pounds, we know that the sum of x and y should equal 100:
x + y = 100 [Equation 1]
We also know that the cost per pound of the new blend is $3.90. To find the cost, we multiply the price of each coffee by the number of pounds used and sum them up:
2.75x + 5y = 3.90(100) [Equation 2]
Now we have a system of equations with two variables (x and y). We can solve this system using different methods, such as substitution or elimination. Let's use the substitution method here:
From Equation 1, we can express x in terms of y: x = 100 - y
Substitute this expression for x into Equation 2:
2.75(100 - y) + 5y = 3.90(100)
Now you can simplify the equation and solve for y:
275 - 2.75y + 5y = 390
Combine like terms:
2.25y = 115
Divide both sides by 2.25:
y = 115 / 2.25
y ≈ 51.11
Now, substitute the value of y back into Equation 1 to find x:
x + 51.11 = 100
x ≈ 100 - 51.11
x ≈ 48.89
Therefore, to obtain the desired mixture of 100 pounds of coffee, you should blend approximately 48.89 pounds of $2.75 coffee and 51.11 pounds of $5 coffee.