Question

The Math Team sells crunchy and fruity trail mixes to raise money. The ratio of fruit to nuts in each mix and the price they charge for an 8-oz (or -cup) bag of each mix are given in the table below. [Hint: The ratio of fruit to nuts in the crunchy trail mix is 1:3. So, of each bag is fruit and of each bag is nuts.]Ratio of fruit to nuts Price of an 8-oz (or cup) bag

Crunchy trail mix 1:3 $2.50
Fruity trail mix 5:3 $3.50
They have 30 cups of fruit and 50 cups of nuts to use in their trail mixes. So, they need to know how many bags of each type of mix they should they make. They let c represent the number of bags of crunchy trail mix and f represent the number of bags of fruity trail mix. Which set of inequalities below represents the conditions?
A. c +f ≤ 80 and 3c + 5f ≤ 240
B. c + f ≥ 80 and 3c + 5f ≥ 240
C. c + f ≤80 and c + 5f ≤ 240
D. c + f ≥ 80 and c + 5f ≥ 240c

Answer 1

The correct set of inequalities representing the conditions for making crunchy and fruity trail mix bags using available fruit and nuts is Option A: c + f ≤ 80 and 3c + 5f ≤ 240. This is established by calculating the part of fruit and nuts each bag of trail mix contains and ensuring the total number of bags made does not exceed the available cups of fruit and nuts.

Step-by-step explanation:

The student is tasked with determining the correct set of inequalities to represent the conditions given for the Math Team's trail mix fundraiser. The team has 30 cups of fruit and 50 cups of nuts, and they need to decide how many 8-oz bags of crunchy and fruity trail mix they can make.

First, we identify the ratio of fruit to nuts for each type of trail mix: crunchy is 1:3 and fruity is 5:3. Next, we establish variables 'c' for the number of crunchy mix bags and 'f' for fruity mix bags. For crunchy mix, 1 out of 4 parts is fruit so each bag uses ¼ cup of fruit. For fruity mix, 5 out of 8 parts are fruit so each bag uses ⅘ cups of fruit.

Now let's express the constraints based on available fruit and nuts:
For fruit: ¼*c + ⅘*f ≤ 30 (Cups of fruit available)
Simplifying the expression, we get: c + 5f ≤ 120 (Multiplying both sides by 4 for easier calculation)
For nuts: ⅔*c + ⅘*f ≤ 50 (Cups of nuts available)
Simplifying this, we get: 3c + 3f ≤ 150 After simplifying the expressions, we see that the inequalities consist of two conditions about the resources (fruit and nuts) that must be satisfied. Thus, the correct answer is Option A: c + f ≤ 80 and 3c + 5f ≤ 240, because these inequalities correctly represent the conditions that must be met for using up all the fruit and nuts without exceeding the available quantities.