Question

Suppose you want to make a certain kind of tropical punch, using bananas, oranges, and papayas. You don't know how many of each to put in the punch, but you know that there are seven pieces of fruit in the mix, and there are twice as many oranges as bananas. You also know that the seven pieces of fruit cost $5.25, where bananas cost $.50 each, oranges cost $.75 each, and papayas cost $1.25 each.

Answer 1

Suppose we need to put $x$ pieces of bananas, $y$ pieces of oranges and $z$ pieces of papaya. We know that mix contains of 7 pieces of fruits, so $x+y+z=7$. There are twice as many oranges as bananas, so $y=2x$. Bananas cost in punch is $ $.5x$, orranges -- $ $.75y$, papapyas -- $$1.25z$. The total cost of mix is $$5.25$ that leads to equation $.5x+ .75y+1.25z=5.25$. Let's solve the system.

$\begin{cases} x+y+z=7\\ y=2x \\ \ .5x+ \ .75y+1.25z=5.25\\ \end{cases}$

$\begin{cases} z=7-x-y\\ y=2x \\ \ .5x+ \ .75y+1.25z=5.25\\ \end{cases}$

$\begin{cases} z=7-x-2x\\ y=2x \\ \ .5x+ \ .75y+1.25z=5.25\\ \end{cases}$

$\begin{cases} z=7-3x\\ y=2x \\ \ .5x+ \ .75y+1.25z=5.25 \ | \ :\ .25\\ \end{cases}$

$\begin{cases} z=7-3x\\ y=2x \\ 2x+ 3y+ 5z=21\\ \end{cases}$

$\begin{cases} z=7-3x\\ y=2x \\ 2x+ 3*2x+ 5*(7-3x)=21 \end{cases}$

$\begin{cases} z=7-3x\\ y=2x \\ 2x+6x+35-15x=21 \end{cases}$

$\begin{cases} z=7-3x\\ y=2x \\ -7x=21-35 \end{cases}$

$\begin{cases} z=7-3x\\ y=2x \\ -7x=-14 \end{cases}$

$\begin{cases} z=7-3x\\ y=2x \\ x=-14:(-7) \end{cases}$ $\begin{cases} x=2 \\ z=7-3 *2\\ y=2*2 \\ \end{cases}$ $\begin{cases} x=2\\ y=4\\ z=1 \end{cases}$

So, there are 2 pieces of bananas, 4 pieces of orranges and 1 piece of papaya in tropical punch.

Answer: 2 bananas, 4 orranges, 1 papaya