Final answer:
The collection contains 25 quarters, and the number of nickels, represented as y, must satisfy the inequalities y ≥ 17 and y ≤ 35, resulting in 18 possible amounts of nickels.
Step-by-step explanation:
The question involves finding the possible number of nickels in a collection of coins consisting of quarters and nickels. Given that the collection has at least 42 coins and the total value does not exceed $8.00, we can set up inequalities to solve for the maximum number of nickels.
Let x be the number of quarters and y be the number of nickels. We are given that there are 25 quarters in the collection, so x = 25. Each quarter is worth 25 cents and each nickel is worth 5 cents. The total number of coins is at least 42, and the total value is at most $8.00, which translates to 800 cents.
Therefore, the inequalities we can set up are:
x + y ≥ 42
25x + 5y ≤ 800
Substituting x = 25 into the inequalities:
25 + y ≥ 42 (y ≥ 17)
25(25) + 5y ≤ 800 (625 + 5y ≤ 800, 5y ≤ 175, y ≤ 35)
So the number of nickels, y, can range from 17 to 35. This means there can be 18 different amounts of nickels in the collection.