Answer:
Let's start by assigning variables to the number of quarters, dimes, and pennies, respectively, in the pile. We can represent these variables as q, d, and p. Then we can set up two equations based on the information given:
0.25q + 0.1d + 0.01p = 1.07 (total value of coins is $1.07)
q + d + p = n (total number of coins is some integer n)
We want to find the values of q, d, and p that satisfy these equations and also make it impossible to make change for a dollar (i.e., there is no combination of these coins that add up to exactly $1.00).
To maximize the number of coins, we want to minimize the value of each coin. Since we are dealing with quarters, dimes, and pennies, we can start by assuming that there are no nickels in the pile (since a nickel is worth more than two pennies). In other words, we want to express 1.07 as a sum of quarters, dimes, and pennies using the fewest possible coins.
One way to do this is to start with as many quarters as possible, then add dimes until we get as close to $1.07 as we can without going over, and then fill in the rest with pennies. If we start with 4 quarters (which is the most we can have without making change for a dollar), we have $1.07 - $1.00 = $0.07 left to make up with dimes and pennies. We can add one dime (since 2 dimes would put us over $0.07), which leaves us with $0.07 - $0.10 = $0.03 to make up with pennies. We can use three pennies to make up the remaining $0.03.
So the solution is q = 4, d = 1, and p = 3. We can check that this satisfies the equations:
0.25(4) + 0.1(1) + 0.01(3) = 1.07
4 + 1 + 3 = 8
Therefore, there are 4 quarters, 1 dime, and 3 pennies in the pile, and this is the largest number of coins that can be used to make a total value of $1.07 without being able to make change for a dollar.
As for why $1.19 is the greatest amount of money it is possible to have without being able to make change for a dollar, this is because any amount of money that is a multiple of $0.05 can be made up with nickels and pennies. For example, if we have $1.10, we can make change for a dollar with 2 dimes and 5 pennies, or with 1 nickel and 5 pennies. But if we have $1.15, we can make change for a dollar with 2 dimes and 5 pennies, or with 1 dime, 1 nickel, and 5 pennies. In other words, any amount of money between $1.05 and $1.20 can be made up with nickels, dimes, and pennies, so the largest amount of money we can have without being able to make change for a dollar is $1.19.