A company mixes peanuts, cashews, and almonds to obtain 22 oz package worth 6.91 dollars. If peanuts, cashews, and almonds cost 18, 45, and 50 cents per oz respectively, and the amount of cashews is equal to the amount of almonds, how much of each is in the package?
p+c+a = 22
a=c, so
p+2a=22
p = 22-2a
18p+45c+50a = 691
18(22-2a)+45a+50a = 691
a = 5, so
5 oz almonds
5 oz cashews
12 oz peanuts
To solve this problem, we need to set up a system of equations based on the given information.
Let's represent the amount of peanuts as P, the amount of cashews as C, and the amount of almonds as A. We know that the total weight of the package is 22 oz, so we can write the equation:
P + C + A = 22 (equation 1)
We also know that the cost of peanuts is 18 cents per oz, cashews cost 45 cents per oz, and almonds cost 50 cents per oz. So, we can write the equation for the total cost:
18P + 45C + 50A = 691 (equation 2)
Given that the amount of cashews is equal to the amount of almonds, we can write the equation:
C = A (equation 3)
Now we have a system of three equations (equations 1, 2, and 3). We can solve this system to find the values of P, C, and A.
First, let's substitute equation 3 into equations 1 and 2:
P + (C) + (C) = 22
18P + 45(C) + 50(C) = 691
Simplifying these equations, we get:
P + 2C = 22 (equation 4)
18P + 95C = 691 (equation 5)
Now we can solve this system of equations (4 and 5) using various methods such as substitution or elimination, but for the simplicity of explanation, we'll use the substitution method.
Solve equation 4 for P:
P = 22 - 2C
Substitute this value into equation 5:
18(22 - 2C) + 95C = 691
Simplifying the equation:
396 - 36C + 95C = 691
Combine like terms:
59C = 295
Divide both sides by 59:
C = 5
Now substitute the value of C back into equation 4 to find P:
P + 2(5) = 22
P + 10 = 22
P = 12
So, the amount of peanuts (P) is 12 oz, the amount of cashews (C) is 5 oz, and the amount of almonds (A) is also 5 oz.