I want to order a pizza with 3/4 slices of the slices having bacon topping, 5/8 with mushrooom and 4/16 with pepperoni. How many slices can have all three toppings, given that every slice must have at least one topping?

Question

I said no slice could have all three. Couldn't show my work though. My reasoning was that if one slice had all three, there would be at least one slice without any toppings at all, and this would not be allowed.

I have a feeling that was not the right answer!

Now I am thinking the answer is 3 slices could have all 3, as that would leave 9/16 of the slices with bacon, 7/16 with mushrooms and 1/16 with pepperoni.

Aargh! How do you answer a question like this?

Answer 1

First we have to decide how many slices we cut the pizza!
It could be 8, but inspiring from the mushroom of 4/16 slices, I assume it is a large pizza cut into 16 slices.
Then we add up the fractions to get toppings to cover
3/4+5/8+4/16=26/16 of the pizza.

Evidently, if we allow 5/16 of the slices get all three topings, and the rest get just one, we have
3(5/16)+11/16=26/16.

So mathematically, we can have 5 slices out of 16 to have all three toppings.

Let's see if this can be done.
We have sufficient pepperoni to cover 4 of the 16 slices, so we can never have 5 slices (out of 16) to have 3 toppings.
So we end up with
4 slices with all three
2 slices with two
10 slices with one.
4*3+2*2+10*1=26 portions of toppings over 4+2+10=16 slices.
That's the best we can do.

Answer 2

Thank you, Mathmate! I don't know where my brain was at.