Question

How many pounds of each should I buy if I have $\$15$ and wish to buy 5 pounds of mixed nuts for a party where peanuts cost $\$2.2$ per pound and cashew cost $\$4.7$ per pound?

Answer 1

Hint: We assume the weight distribution for the peanuts and cashew. We find individual prices for peanuts and cashews when the cost is $\$2.2$ per pound for peanuts and $\$4.7$ per pound for cashews. We equate the weights and the cost and find two equations. We solve them to find the solution.

Complete step by step solution:
Let the weights for peanuts and cashew for the party be $x$ and $y$ respectively.
The total amount in money form available is $\$15$. Total mixed weight for peanuts and cashew has to be 5 pounds.
This gives our first condition for the variables $x$ and $y$ where $x+y=5....(i)$.
The individual prices for peanuts and cashews are $\$2.2$ per pound and $\$4.7$ per pound.
For $x$ pound peanuts and $y$ pound cashews the cost will be separately
$2.2\times x=2.2x$ for peanuts and $4.7\times y=4.7y$ for cashews.
Total cost will be $2.2x+4.7y$ which is equal to $\$15$.
So, $2.2x+4.7y=15....(ii)$.
We have two equations and two unknowns
We multiply $2.2$ to the both sides of the first equation and get
$\begin{align}
  & 2.2\times \left( x+y \right)=2.2\times 5 \\
 & \Rightarrow 2.2x+2.2y=11 \\
\end{align}$
We take the equation as $2.2x+2.2y=11.....(iii)$.
Now we subtract the equation (iii) from equation (ii) and get
$\left( 2.2x+4.7y \right)-\left( 2.2x+2.2y \right)=15-11$.
We take the variables together and the constants on the other side.
Simplifying the equation, we get
$\begin{align}
  & \left( 2.2x+4.7y \right)-\left( 2.2x+2.2y \right)=15-11 \\
 & \Rightarrow 2.5y=4 \\
 & \Rightarrow y=\dfrac{4}{2.5}=1.6 \\
\end{align}$
The value of $y$ is $1.6$. Now putting the value in the equation $x+y=5....(i)$, we get
$\begin{align}
  & x+y=5 \\
 & \Rightarrow x=5-1.6=3.4 \\
\end{align}$.
Therefore, the buy was $3.4$ pound peanuts and $1.6$ pound cashew.

Note: We can also find the value of one variable $y$ with respect to $x$ based on the equation
$x+y=5$ where $y=5-x$. We replace the value of $y$ in the second equation of
$2.2x+4.7y=15$ and get
\[\begin{align}
  & 2.2x+4.7y=15 \\
 & \Rightarrow 2.2x+4.7\left( 5-x \right)=15 \\
 & \Rightarrow 2.2x+23.5-4.7x=15 \\
\end{align}\]
We get the equation of $x$ and solve
$\begin{align}
  & 2.2x+23.5-4.7x=15 \\
 & \Rightarrow -2.5x=15-23.5=-8.5 \\
 & \Rightarrow x=\dfrac{-8.5}{-2.5}=3.4 \\
\end{align}$
Putting the value of $x$ we get $x+y=5\Rightarrow y=5-3.4=1.6$.
Therefore, the values are $x=3.4,y=1.6$.