Respuesta :
The Ned's Nut shop sell two types of Nuts as follows:
[tex]\text{Peanuts and Cashews}[/tex]The selling price of each type of nut per pound are given to us as follows:
[tex]\begin{gathered} P(\text{Peanuts) = \$11} \\ P(Cashews\text{) = \$10} \end{gathered}[/tex]Ned tries to make a bag of mixture that consists of both types of nuts. The weights ( pounds ) of each type of Nut expected in the bag are as follows:
[tex]\begin{gathered} Weight\text{ of peanuts = x} \\ Weight\text{ of cashews = y} \end{gathered}[/tex]The total weight of a bag of mixture is given to be 20 pounds. We can go ahead and express this total weight in terms of induvidual weights of each type of nut as follows:
[tex]\begin{gathered} \text{Total weight of mix = Weight of peanuts + Weight of cashews} \\ 20\text{ = x + y }\ldots\text{ Eq1} \end{gathered}[/tex]Next Ned decides to sell this mixture of bag for the amount of $10.55 per pound. We can also express the selling price of the 20 pound bag of mixture in terms of weights and induvidual selling price of each type of nut as follows:
[tex]\begin{gathered} P(\text{Mixture) = P(Peanuts)}\cdot Weight\text{ of peanuts + P(Cashews)}\cdot Weight\text{ of Cashews} \\ \\ \frac{10.55}{pound}\cdot20\text{ pounds = }\frac{11}{pound}\cdot x\text{ + }\frac{10}{poound}\cdot y \\ \\ 211\text{ = 11x + 10y }\ldots\text{ Eq2} \end{gathered}[/tex]We have expressed two mathematical equations ( Eq1 and Eq2 ) relating the total weight and the selling price of the mixture with the induvidual weights of each type of nuts ( x and y ):
[tex]\begin{gathered} x\text{ + y = 20} \\ 11x\text{ + 10y = 211} \end{gathered}[/tex]We have two linear equations with two unknowns we can solve them simultaenously via Elimination method.
Multiply ( Eq1 ) with ( -10 ) and form ( Eq3 ) as follows:
[tex]-10x\text{ - 10y = -200 }\ldots\text{ Eq3}[/tex]Add ( Eq3 ) and ( Eq2 ):
[tex]\begin{gathered} 11x\text{ + 10y = 211} \\ -10x\text{ -10y = -200} \\ ============ \\ x\text{ = 11} \\ ============= \end{gathered}[/tex]Back substitute the value of ( x ) into ( Eq1 ) as follows:
[tex]\begin{gathered} x\text{ + y = 20} \\ 11\text{ + y = 20} \\ y\text{ = 9} \end{gathered}[/tex]Therefore the bag of mixtures will have the follwing quantity of each type of nuts by weights:
[tex]\begin{gathered} \text{Peanuts = 11 pounds} \\ \text{Cashews = 9 pounds} \end{gathered}[/tex]