Question

In the following exercises, translate to a system of equations and solve.

Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost \(4.89 per bag with peanut butter pieces that cost \)3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use?

Short answer

The number of choclate pieces is 10 bags and number of Peanut butter pieces is 15 bags.

Step-by-step solution

Step 1. Given Information   

The given data is that Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost $3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make.

Step 2. Explanation   

Let number of units of Choclate pieces be x and number of units of Peanut butter pieces be y.

The mixed candy cost is $$4.23$and total number of units of candy is 25 bags.

Thus, total value of candy is $25\times \$4.23=\$105.75$

Total value of mixture can be expressed as $4.89x+3.79y=105.75--\left(1\right)$

Total units of mixed candy can be expressed as$x+y=25--\left(2\right)$

Step 3. Calculation   

Multiply equation (2) with number $4.89$and write the revised equation.

$$\begin{gathered} 4.89(x+y)=4.89\left(25\right) \\ 4.89x+4.89y=122.25--\left(3\right) \end{gathered}$$

Solve the equation (3) and (1) by subtracting equation (1) from equation (3).

$$\begin{gathered} 4.89x+4.89y-4.89x-3.79y=122.25-105.75 \\ 1.1y=16.5 \\ y=15 \end{gathered}$$

Substitute the value of y in equation (2) to find the value of x.

$$\begin{gathered} x+y=25 \\ x+15=25 \\ x=10 \end{gathered}$$