Question

A food store makes an $11-$pound mixture of peanuts, almonds, and raisins.

The cost of peanuts is $$1.50$ per pound, almonds cost $$3.00$ per pound, and raisins cost $$1.50$ per pound.

The mixture calls for twice as many peanuts as almonds.

The total cost of the mixture is $$21.00$.

How much of each ingredient did the store use?

Answer 1

Step-1: Make mathematical equations for the given condition:

Let $x$ be the amount of peanuts, $y$ be the amount of almonds, and $z$ be the amount of raisins,

The total amount of mixture is $11$ pounds. It can be expressed below :

$x+y+z=11...\left(1\right)$

Since the mixture calls for twice as many peanuts as almonds, it can be written as :

$x=2y...\left(2\right)$

The cost of peanuts is $$1.50$ per pound, almonds cost $$3.00$ per pound, raisins cost $$1.50$ per pound and the total cost of the mixture is $$21.00$, it can be written as below :

$1.5x+3y+1.5z=21...\left(3\right)$

Solve the above equations by using the substitution method.

Substitute $x=2y$ in equation $\left(1\right)$ and solve for $y$:

$\begin{array}{rcl}2y+y+z& =& 11\\ & \Rightarrow & 3y=11-z\\ \therefore y& =& \frac{11-z}{3}...\left(4\right)\end{array}$

Step-2: Solve for $z$:

Substitute $x=2y$ in equation (2) and solve for $y$:

$\begin{array}{rcl}1.5\left(2y\right)+3y+1.5z& =& 21\\ & \Rightarrow & 3y+3y=21-1.5z\\ & \Rightarrow & 6y=21-1.5z\\ \therefore y& =& \frac{21-1.5z}{6}...\left(5\right)\end{array}$

Equate the equation (4) and equation (5) and solve for $z$:

$\begin{array}{rcl}\frac{11-z}{3}& =& \frac{21-1.5z}{6}\\ & \Rightarrow & 66-6z=63-4.5z\left[\text{Cross multiply}\right]\\ & \Rightarrow & 6z-4.5z=66-63\\ & \Rightarrow & 1.5z=3\\ & \Rightarrow & z=\frac{3}{1.5}\left[\text{Divide by}1.5\right]\\ \therefore z& =& 2\end{array}$

Step-3 : Solve for $y$ and $x$:

Substitute $z=2$ in equation (4) and calculate the value of $y$ :

$\begin{array}{rcl}y& =& \frac{11-2}{3}\\ & \Rightarrow & y=\frac{9}{3}\\ \therefore y& =& 3\end{array}$

Now, substitute $y=3$ in equation (2) and calculate the value of $x$ :

$\begin{array}{rcl}x& =& 2\left(3\right)\\ \therefore x& =& 6\end{array}$

Therefore, the value of $x$ is $6$, $y$ is $3$ and $z$ is $2$.

Hence, the amount of peanuts is $\mathbf{6}$, the amount of almonds is $\mathbf{3}$, and the amount of raisins is $\mathbf{2}$.