Question

Consider a picnic. if you want to buy enough hot dogs and buns without having any leftovers, you need to balance the number of packages of buns (which usually contain 8 buns) with the number of packages of hot dogs (which usually contain 10 dogs). you would get 5 and 4 packages. Now you need to feed at least 500 people. If you want to feed everyone, but still have equal numbers of buns and hot dogs, what is the minimum number of packages of buns and hot dogs you need, respectively?Hint: you determined that you can make 40 servings from five packages of buns and four packages of hot dogs without anything left over. This means that you can also make 80, 120, 160, and so forth, servings without anything left over. These numbers are called multiples of 40.

Answer 1

We are given information about how many buns and dogs are contained in each package and are asked which is the minimum number of packages needed.

By using the hint given to us, we know that we need 5 packages of buns and 4 packages of hot dogs to feed 40 people, therefore, let's use multiples of 40 to get closer to the magic number of 500, as follows:

`\begin{gathered} 40\cdot12=480,\text{ which is the closest we can get to 500 using multiples of 40} \\ Therefore,\text{ }in\text{ }order\text{ }to\text{ }feed\text{ }480\text{ }people\text{ }we\text{ }need: \\ 5\cdot12=60\text{ }packages\text{ }of\text{ }buns \\ 4\cdot12=48\text{ }packages\text{ }of\text{ }hot\text{ }dogs \\ \end{gathered}`

Now, we have already fed 480 people, we need at least 20 more, but remember that we don't want to have any leftovers and we want the same number of buns and dogs, therefore, we can't feed exactly 500, we have to feed 520, which means we have to go over another multiple of 40, by buying 5 more packages of buns and 4 more packages of hot dogs.

Therefore, the correct answer is 65 packages of buns and 52 packages of hotdogs:

`\begin{gathered} 60+5=65\text{ }packages\text{ }of\text{ }buns \\ 48+4=52\text{ }packages\text{ }of\text{ }hotdogs \end{gathered}`