Section 1.8, #10

A pound of dried pineapple bits sells for \$5.89, a pound of dried banana chips sells for \$5.09, and a pound of raisins sells for \$2.79. Two pounds of raisins are to be mixed with equal amounts of pineapple and banana to create a trail mix that will sell for \$3.59 a pound. How many pounds of pineapple and banana chips should be used? (Round your answer to the nearest hundredth of a pound.)

Let $p$ be the number of pounds of pineapple bits required. Let $b$ be the number of pounds of banana chips required. We are told that we will be using 2 pounds of raisins at \$2.79/pound.

We are also told that equal amounts (numbers of pounds) of pineapple and banana are going to be used, so that means $p = b$. The total value of the mixture is to be \$3.59 per pound.

The value of the pineapple bits is: (# pounds of bits)(price per pound of bits) = $p (5.89) = 5.89 p$

The value of the banana chips is: (# pounds of chips)(price per pound of chips) = $b (5.09) = 5.09 b = 5.09 p$, since $b = p$. Note that we are trying to get the equation in terms of a single variable rather than two different variables.

The value of the raising is: (# pounds of raisins)(price per pound of raisins) = $2 (2.79) = 5.58$

The total value of the mixture is: (# pounds of mixture)(price per pound of mixture) = $(p + b + 2) (3.59) = (p + p + 2) (3.59) = (2p + 2) (3.59)$. Note again that the first equality follows, since $b = p$.

Adding up the total values of all the ingredients gives the total value of the mixture: $5.89 p + 5.09 p + 5.58 = (2p + 2)(3.59)$.

Solve this last equation for $p$ and you will know how many pounds of pineapple chips must be used. You will also know how many pounds of banana chips must be used since $b = p$.

\begin{equation} x^2 + \frac{2x - 1}{x^2 + 1} \end{equation}