Cut each pizza into thirds, so that would make 15 slices (5×3 = 15).

Then the fifth pizza, you cut each third into six so that pie has six slices.

So each person gets two of the big slices which uses up four pies (3×4), and then they each get one of the six slices from the fifth pie.

Mush everything into a big pizza ball, and divide that by 6.

Tell #6 to not be a mooch.
Have him order his pizza.
He could eat two pieces from the existing pizzas while waiting, then replace them when his pizza arrives.

In most pizzerias, the slicing was done in the kitchen so the clever post-hoc slice approach wouldn’t work.

So I see that you are not asking us to solve the math problem, much less to solve it in practical real-world terms. If what you want is help thinking up a (presumably better) word problem, why don’t you tell us (a) the answer and (b) what’s wrong with your story?

But anyway—coming back an hour later: take three pizzas and cut each in half (3/6). Everyone gets half a pizza, Take the remaining two and cut each in thirds (2/6). Everyone gets one-third. Now everyone has an identically configured 5/6 of a pizza.

@Jeruba , That is the answer I was looking for. So you think the story behind the problem is okay?

I don’t like cold pizza.
Order the sixth pizza and start eating. By the time you eat your slices from the 5 pies, the sixth one will be done.

Why do you say the pizza is cold? It should only take about a minute or so to cut them.
If you don’t like the pizza story, can you come up with a better one?

There is a fundamental difficulty with this word problem. One does not cut pizza on a radius, but on a diameter.

Try cutting a pizza into thirds and someone is going to have a legitimate gripe.

Ideally the best way to order pizza so it’s not cold is to order it a pie at a time, or if there are not enough slices foe everyone to have one, two pies at a time. That way it will be hot or almost hot. If you ordered all the pies at once, some would be cold.

One final point is to generalize the problem. The reason this method works for splitting 5 among 6 is that we can find two numbers, 2 and 3 such that 5=2+3 and 6=2×3. Alternatively, we can start with 2 and 3 to generate 5 and 6. If we use 3 and 4 in place of 2 and 3 then we get 3+4=7 and 3×4=12. We could split 7 pizzas among 12 people by giving everyone ⅓ and ¼.

@LostInParadise, it is in fact a very silly story, but at least it’s better than the pies and cookies of the old days. The use of it is clear enough for the purpose, and harmless, as long as your students don’t have romping attacks of realism such as worrying about the temperature of the pizza or how it’s usually cut. I can’t see that going to elaborate lengths to make it realistic is going to improve solving ability.

As a sad veteran of many word problems, though, I can see that it might be helpful if the story gave some hint of why anyone would really ever want to do this. And if it’s never meant for any practical application but just an exercise to explore the concept, then I’d say make it as fanciful as possible so no one is looking for realism and instead sees the abstraction for what it is.

I appreciate what you have to say, but it just seems that there should be a practical application to being able to share something by splitting things apart while maximizing the sizes of the pieces.

In which case: aren’t they going to cut them into smaller slices anyway, rather than chomping on half a pizza at once, even a small one?

I was assuming that a sixth of a small pizza would be rather small. Without that, the story behind the problem does not make any sense. Instead of small pizzas, we could talk about individual slices. 1/6 of an individual slice is rather puny.

The thing is, if you want to be realistic, no one would do this. In that case, I happen to think @filmfann‘s solution makes the most sense.

Realistically, if they’re ordering from one of those fast-pizza places that make the small single-serve pizzas, tardy Tom should just go ahead and order, and his pizza will be along in a few minutes. The others can slow down a little so theirs won’t be all gone before his arrives.

Otherwise you could add Alicia, the neurotic perfectionist, who is the birthday girl and who wants all the portions to be identical and conform to an aesthetic standard. She won’t tolerate all those irregularities and will fish a protractor out of her purse to guide the cutting in thirds.

Now waiting for the next logical question, which is, what about the toppings?

This is my argument for making it as fanciful as possible so no one is tempted to seek realism in it.

Yes in real life, people are usually casual about this stuff and just share and make it work.

Then I return to my original question. Is there a case where you might switch from sharing among five people to sharing among six people where you would want the size of the portions to be as large as possible?

@LostInParadise ^^^^ it is easy if one shares food that is not apportioned geometrically in two dimensions, but uses volume instead of acreage. If one is serving mac and cheese (everybody’s favorite side!) then one just uses smaller spoon fills to mete out six portions rather than five.

Here is a story that does not involve food. An army has 5 equal size divisions. They want to regroup into 6 divisions. How can they do that so that everyone is with a lot of members of their old division? In line with what was done for the pizzas, we have each of the new divisions containing half of one of the old divisions and ⅓ of another one of the old divisions. In the worst case, for the people from the ⅓ divisions, (⅓)/(5/6) or 40% of the members of the new division are members of their old division.