Question

The Nut Shack sells hazelnuts for $6.80 per pound and peanuts nuts for $4.80 per pound. How much of each type should be used to make a 44 pound mixture that sells for $5.94 per pound?

Answer 1

18.92 pounds of peanut and 25.08 pounds of nut shack should be used to make the mixture

Explanation:

the cost per pound for the nut shack = $6.80

let the amount of pounds of nut shack used in the mixture = n

the cost per pound for the peanuts = $4.80

let the amount of pounds for the peanuts used in the mixture = p

We want to obtain 44 pounds of mixture which sells for $5.94 per pound

sum of pounds mixture = 44

amount of pounds of nut shack used in the mixture + amount of pounds for the peanuts used in the mixture = 44

$n+p=44\text{ }....\mleft(1\mright)$

cost per pound for the nut shack (amount used) + cost per pound for the peanuts (amount used) = cost per pound of the mixture (amount of mixture)

6.80(n) + 4.80(p) = 5.94(44)

$6.8n+4.8p=261.36\text{ }\ldots\mleft(2\mright)$

using substitution method:

from equation 1, we can make n the subject of formula

n = 44 - p

substitute for n in equation (2):

$\begin{gathered} 6.8(44\text{ - p) + 4.8p = 261.36} \\ 299.2\text{ - 6.8p + 4.8p = 261.3}6 \\ 299.2\text{ - 2p = 261.3}6 \end{gathered}$$\begin{gathered} collect\text{ like terms:} \\ 299.2\text{ - 261.36 - 2p = 0} \\ \text{add 2p to both sides:} \\ 37.84\text{ = 2p} \\ \text{divide both sides by 2:} \\ \frac{37.84}{2}\text{ = p} \\ p\text{ = 18.9}2 \end{gathered}$

substitute for p in equation 1:

$\begin{gathered} n\text{ + 18.92 = 44} \\ n\text{ = 44 - 18.9}2 \\ n\text{ = 25.0}8 \end{gathered}$

18.92 pounds of peanut and 25.08 pounds of nut shack should be used to make the mixture