A candy shop owner is mixing up a batch of her candy medley. Overall the medley costs $2.95 per pound, and it is made of jelly beans costing $3.30 per pound and gumdrops costing $1.55 per pound. How many pounds of each should the owner use in a 30-pound batch of the medley?

Question

Write your answers as whole numbers or as decimals rounded to the nearest tenth.
pounds of jelly beans
pounds of gumdrops

Answer 1

Let \( x \) be the weight in pounds of jelly beans and \( y \) be the weight in pounds of gumdrops. We have the following system of equations based on the problem:

1. The total weight equation: \[ x + y = 30 \]

2. The total cost equation: \[ 3.30x + 1.55y = 2.95 \times 30 \]

First, calculate the total cost of the 30-pound batch of the medley: \[ 2.95 \times 30 = 88.5 \]

Now, substitute this into the cost equation: \[ 3.30x + 1.55y = 88.5 \]

Now we have the following system of equations:

1. \( x + y = 30 \) (Equation 1)
2. \( 3.30x + 1.55y = 88.5 \) (Equation 2)

From Equation 1, we can express \( y \) in terms of \( x \): \[ y = 30 - x \]

Substituting \( y \) in Equation 2 gives us: \[ 3.30x + 1.55(30 - x) = 88.5 \]

Expanding and simplifying the equation: \[ 3.30x + 46.5 - 1.55x = 88.5 \] \[ (3.30 - 1.55)x + 46.5 = 88.5 \] \[ 1.75x + 46.5 = 88.5 \]

Subtract \( 46.5 \) from both sides: \[ 1.75x = 42 \]

Now divide by \( 1.75 \): \[ x = \frac{42}{1.75} = 24 \]

Now, substitute \( x \) back into the equation \( y = 30 - x \): \[ y = 30 - 24 = 6 \]

So, the weight of jelly beans and gumdrops is:

• Pounds of jelly beans: \( x = 24 \)
• Pounds of gumdrops: \( y = 6 \)

Thus, the final answers are:

• 24 pounds of jelly beans
• 6 pounds of gumdrops