If x pounds of cashews valued at $6.00 per pound are to be mixed with y pounds of peanuts valued at $4.00 per pound to obtain a 40 pound cashew-peanut mixture.
Find x and y so that when the mix is sold for $4.75 per pound, the same amount of revenue
is collected from the sale of the mix as would have been collected from the sale of each type of nut individually. Set up an appropriate system of linear equations and solve it to determine x and y.
obvious first equation:
x + y = 40
2nd equation deals with value of the nuts
6x + 4y = 4.75(40)
6x + 4y = 190
3x + 2y = 95 **
double the first one: 2x + 2y = 80 ***
subtract *** from **
x = 15
in your head looking at x+y = 40
y = 25
check:
6(15) + 4(25) = 190
4.75(40) = 190
To solve this problem, let's first set up the equations based on the given information.
Let's assume that the selling price of the cashew-peanut mixture is $4.75 per pound, and the amount of cashews in the mixture is x pounds while the amount of peanuts is y pounds. The selling price of cashews is $6.00 per pound, and the selling price of peanuts is $4.00 per pound.
To find the revenue from selling the mixture and each nut individually, we need to consider the pounds sold and the selling price.
Revenue from selling the cashew-peanut mixture:
Revenue = Selling price × Pounds sold
Revenue = $4.75 × 40
Revenue from selling cashews individually:
Revenue = Selling price × Pounds sold
Revenue = $6.00 × x
Revenue from selling peanuts individually:
Revenue = Selling price × Pounds sold
Revenue = $4.00 × y
Since we want the same amount of revenue from both cases, we can set up the equation:
$4.75 × 40 = $6.00 × x + $4.00 × y
Now, we have the equation:
190 = 6x + 4y
However, we also have another constraint which is that the total weight of the mixture is 40 pounds:
x + y = 40
We now have a system of linear equations:
6x + 4y = 190
x + y = 40
To solve this system, we can use either substitution or elimination method.
Using the elimination method, we can multiply the second equation by -6 to cancel out the x term:
-6(x + y) = -6(40)
-6x - 6y = -240
Adding this equation to the first equation, we get:
(6x + 4y) + (-6x - 6y) = 190 + (-240)
-2y = -50
y = 25
Substituting the value of y back into the second equation, we can solve for x:
x + 25 = 40
x = 15
Therefore, the solution to the system of equations is x = 15 and y = 25. This means that 15 pounds of cashews and 25 pounds of peanuts should be used to obtain the desired mixture.