Final answer:
Mr. Hamm should buy at least 23 packages of hot dogs and 5 packages of buns to avoid leftovers and meet the minimum requirement of 180 hot dogs, as this ensures he will have the same number of hot dogs and buns without leftovers.
Step-by-step explanation:
The objective here is for Mr. Hamm to purchase hot dogs and buns in such amounts that he has no leftovers and both the hot dogs and the buns run out at the same time. Since hot dogs come in packages of 8 and buns come in packages of 10, we are looking for the least common multiple (LCM) of these two numbers to ensure that there will be no leftovers.
To start, we know that Mr. Hamm needs to prepare at least 180 hot dogs daily. Therefore, the number of packages of hot dogs needed is at least 180 divided by 8, which is 22.5. Since we can't have half a package, we need to round up to the next whole number, which gives us 23 packages of hot dogs (or 184 hot dogs).
Now, to find the LCM of 8 and 10, we list multiples of each until we find a common one: The multiples of 8 are 8, 16, 24, 32, 40, ..., and of 10 are 10, 20, 30, 40,... The least common multiple is 40. This means Mr. Hamm should plan in batches of 40. With 40 being a multiple of both 8 and 10, he has to purchase 5 packages of buns (since 40 divided by 10 is 4).
Finally, to meet the minimum requirement of 180 hot dogs, we need to find out how many complete sets of 40 we can get from 184 hot dogs. This is 184 divided by 40, which gives us 4.6. Rounding up means that Mr. Hamm should prepare for 5 sets to meet his minimum without leftovers. Thus, Mr. Hamm needs to purchase a minimum of 23 packages of hot dogs and 5 packages of buns to ensure that there are no leftovers, while also satisfying the minimum requirement of preparing at least 180 hot dogs.