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?
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?
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.
Thank you, Mathmate! I don't know where my brain was at.
To answer a question like this, you can approach it using a systematic method, such as finding a common denominator, making calculations, and considering all possible scenarios.
First, let's review your initial reasoning. You mentioned that if one slice had all three toppings, there would be at least one slice without any toppings, which would not be allowed. While this reasoning seems logical, it may not necessarily hold true. It's important to consider all possible combinations and see if any scenario satisfies the given conditions.
Now, let's approach the problem systematically. We need to find the number of slices that can have all three toppings, given that every slice must have at least one topping.
To calculate the number of slices, we can first find a common denominator for the fractions representing the toppings. In this case, the common denominator is 16.
Now, let's calculate the number of slices with each topping:
- Bacon: 3/4 of the slices = (3/4) * 16 = 12 slices
- Mushroom: 5/8 of the slices = (5/8) * 16 = 10 slices
- Pepperoni: 4/16 of the slices = (4/16) * 16 = 4 slices
To find the maximum number of slices that can have all three toppings, we need to consider the minimum value among the three toppings. In this case, the pepperoni has the fewest slices, so we know that a maximum of 4 slices can have all three toppings.
Now, to distribute the toppings, we subtract the number of slices with all three toppings from the total slices:
Total slices - Slices with all three toppings = Number of slices remaining
16 slices - 4 slices = 12 slices remaining
Out of the 12 remaining slices, we have to distribute them among the other two toppings, mushroom and bacon, keeping in mind that every slice must have at least one topping.
Let's consider a scenario where all 12 remaining slices have bacon. In this case, there would be no slice with mushroom or pepperoni, contravening the requirement that every slice must have at least one topping.
Hence, we need to distribute the remaining slices among mushroom and bacon. We can have some slices with both mushroom and bacon, but there should be slices with only mushroom or only bacon as well. This allows us to fulfill the requirement that every slice must have at least one topping.
From your revised answer, you suggested having 3 slices with all three toppings. If we subtract these 3 slices from the 12 remaining slices, we have 9 slices left.
Now, distribute these remaining slices among the two toppings, bacon and mushroom. Let's consider a scenario where 9 slices have bacon, leaving none with mushroom. In this case, we would not be able to satisfy the requirement of having at least one topping on every slice.
So, we need to distribute the remaining slices between bacon and mushroom. We can have some slices with both bacon and mushroom, and the remaining slices with only one of these toppings.
As there are 9 slices remaining, we can have a maximum of 9 slices with both toppings (bacon and mushroom). This would also mean that none of these 9 slices would have just one topping.
Therefore, the correct answer is that a maximum of 9 slices can have all three toppings (bacon, mushroom, and pepperoni), while still satisfying the condition that each slice must have at least one topping.
So, to summarize, the maximum number of slices with all three toppings is 9, and this distribution satisfies the requirements given in the question.