Question

The math team sells crunchy and fruity trail mixes to raise money. The ratio of fruit to nuts and the price for an 8-oz bag of each mix are given. They have 30 cups of fruit and 50 cups of nuts. 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 represents the conditions?

a) (1c + 3f ≤ 30), (5c + 3f ≤ 50), (c ≥ 0), (f ≥ 0)

b) (1c + 5f ≤ 60), (3c + 3f ≤ 100), (c ≥ 0), (f ≥ 0)

c) (0.25c + 0.625f ≤ 60), (0.75c + 0.375f ≤ 100), (c ≥ 0), (f ≥ 0)

d) (0.25c + 0.625f ≤ 30), (0.75c + 0.375f ≤ 50), (c ≥ 0), (f ≥ 0)

Answer 1

The correct set of inequalities representing the conditions for selling crunchy and fruity trail mixes, given the constraints of 30 cups of fruit and 50 cups of nuts, is option (d): (0.25c + 0.625f ≤ 30), (0.75c + 0.375f ≤ 50), (c ≥ 0), (f ≥ 0).

Step-by-step explanation:

To determine which set of inequalities represents the conditions for the math team selling crunchy and fruity trail mixes, we need to use the information that the math team has 30 cups of fruit and 50 cups of nuts. The variable (c) represents the number of bags of crunchy trail mix and (f) represents the number of bags of fruity trail mix. Each bag is an 8-oz (1 cup) bag. We are looking for inequalities that represent the limitations based on the fruit and nuts availability.

Options (a) to (d) suggest different ways of allocating fruits and nuts to the two types of trail mix. By examining the quantities given, the correct set must be able to ensure the ratios given in the problem are respected. Assuming each cup is used for one bag, the correct set of inequalities should directly reflect the available cups of fruits and nuts in relation to the bags of crunchy and fruity trail mixes.

Option (d) (0.25c + 0.625f ≤ 30), (0.75c + 0.375f ≤ 50), (c ≥ 0), (f ≥ 0) reflects the constraint that the team has 30 cups of fruit and 50 cups of nuts available. These inequalities ensure that the allocation of fruit and nuts to the bags of crunchy and fruity trail mix does not exceed the available supply, while also ensuring non-negative values for the number of each trail mix sold.