Question

Juice Blends

The Juice Company offers three kinds of smoothies: Midnight Mango, Tropical Torrent, and Pineapple Power. Each smoothie contains the amounts of juices shown in the table.

On a particular day the Juice Company used 820 oz of mango juice, 690 oz of pineapple juice, and 450 oz of orange juice. How many smoothies of each kind were sold that day?

Short answer

Therefore, the number of Midnight Mango is 50, Tropical Torrent is 60andPineapple Power is 30 were sold on that particular day.

Step-by-step solution

Step 1. Given information

There are three kinds of smoothies- Midnight Mango, Tropical Torrent and Pineapple Power.

Step 2. Concept used

We will assume that x is the number of midnight smoothies sold, y is the number of tropical torrent smoothies sold andz is the number of pineapple powder smoothies sold.

To get the number of smoothies of each kind are sold on that day, convert all the facts into equation.

Step 3. Calculation

Equation for quantity of mango juice:-

$8x+6y+2z=820\left(1\right)$

Equation for quantity of pineapple juice:-

$3x+5y+8z=690\left(2\right)$

Equation for quantity of orange juice:-

$3x+3y+4z=450\left(3\right)$

Thus the system of equations is-

$\left\{\begin{array}{l}8x+6y+2z=820\\ 3x+5y+8z=690\\ 3x+3y+4z=450\end{array}\right.$

To get the complete solution of this system transform it into triangular system.

To get triangular system eliminate term $\text{x}$from equation $\left(2\right)$ and terms $\text{x}$and y from equation $\left(3\right)$.

Subtract equation $\left(3\right)$from equation $\left(2\right)$to eliminate terms $\text{x}$from equation $\left(2\right)$,

The equation $\left(4\right)$is new equation $\left(2\right)$for triangular system.

Multiply equation $\left(1\right)$by $-3$,

$\begin{array}{l}-3(8x+6y+2z=820)\\ -24x-18y-6z=-2460\left(5\right)\end{array}$

Multiply equation $\left(3\right)$by 8,

$\begin{array}{l}8(3x+3y+4z=450)\\ 24x+24y+32z=3600\left(6\right)\end{array}$

And add equation $\left(6\right)$and $\left(5\right)$to eliminate terms $\text{x}$from equation $\left(3\right)$as the coefficients of $\text{x}$in both equations are negative of each other,

$\begin{array}{l}24x+24y+32z=3600\\ \frac{-24x-18y-6z=-2460}{6y+26z=1140\left(7\right)}\end{array}$

Multiply equation $\left(4\right)$by $-3$,

$\begin{array}{l}-3(2y+4z=240)\\ -6y-12z=-720\left(8\right)\end{array}$

And add equation $\left(7\right)$and $\left(8\right)$to eliminate terms $y$from equation $\left(3\right)$as the coefficients of y in both equations are negative of each other,

$\begin{array}{l}6y+26z=1140\\ \frac{-6y-12z=-720}{14z=420}\\ z=\mathrm{30....}\left(9\right)\end{array}$

The equation $\left(9\right)$is new equation $\left(3\right)$for triangular system.

Thus, the triangular system is,

$\left\{\begin{array}{c}8x+6y+2z=820\\ 2y+4z=240\\ z=30\end{array}\right.$

into equation $\left(4\right)$to get the value of y,

$\begin{array}{l}2y+4z=240\\ 2y+4\left(30\right)=240\\ y=60\end{array}$

Further solving,

$\begin{array}{l}8x+6y+2z=820\\ 8x+6\left(60\right)+2\left(30\right)=820\\ 8x=400\\ x=50\end{array}$

We get final results as,

$\begin{array}{l}x=50\\ y=60\\ z=30\end{array}$