Multiplying fractions word problem: pumpkin pie

Ken and Isaiah are eating pumpkin pie with their friends. They want to figure out how much pie they have eaten so far. There are 7 of them in total and each of them has eaten 2/5 of a pie. Ken said you can solve this problem by multiplying 2/5 times 7. And that makes sense. Each of them ate 2/5, and there are 7 folks. I would multiply 2/5 times 7. So he multiplied and got 14/35. Now this is a little bit suspicious. And here's the work that he did. He said 2/5 times 7 is equal to 2 times 7 over 5 times 7, which equals 14/35. So this is starting to smell real fishy right now. Isaiah says that 14/35 cannot be correct because 14/35 is less than one whole pie. So this is definitely true, that 14/35 is less than one whole pie. Which of the following best explains this situation? So before even looking at the explanations, let's see what's fishy about this. So let me get my little notepad and write it out. So he literally took 2/5, and he attempted to multiply it by 7. And he said that this is the same thing as taking the 2 times the 7, over the 5 times the 7. And that's how he got the 14/35. And I encourage you to pause this and try to figure out why this doesn't make sense yourself. Now let's think about what actually went on here. If you multiply the numerator times 7 and the denominator by 7, you're actually not changing the value of the fraction. This is equivalent to saying 2/5 times 7/7, which is equivalent to 2/5 times 1, which is equivalent to 2/5. 14/35 is just another way, it's an equivalent representation of 2/5. This is just how much pie only one person should have eaten. So how should he have thought about this? Well, there's a couple of ways you could think of it. 2/5 times 7 could literally mean seven 2/5. It literally could mean, so 2/5 times 7, literally means one 2/5, plus another 2/5, plus another 2/5. So we do this seven times. So that's four 2/5. That's five 2/5. That's six 2/5. And that's seven 2/5. Sorry, my brain isn't working. And that's seven 2/5. And if you were to add all of these together, how many fifths do you have? How many fifths do you have? Well, you have 2 plus 2 plus 2 plus 2 plus 2. Let's see, that's five. Plus 2, plus 2. You have 2, seven times, fifths. Or this is another way of saying you have 7 times-- let me write it this way-- you have 2 times 7 fifths. Or another way of saying it is this is equal to 14/5. This is well over more than 1. 5 goes into 14 two times, and you have a remainder of 4. So it's 2 and 4/5 pie. So this is what he should have done. But you might be saying, well, how would I just multiply this if I didn't even have to think it through, adding all these 2/5 together? Well, one way to think about it is that 7 is the same thing as 7/1. So he could have just said 2/5 times 7/1. 7/1 is the exact same thing as 7. And that would be equal to 2 times 7 in the numerator, which is 14. And the denominator would be 5 times 1, which is equal to 5. And you would get the same answer. So now let's actually go back and select the right choice. I forgot that we actually had to say which explanation is the right explanation. So explanation 1 is that Ken didn't multiply correctly. Multiplying 2/5 times 7 is the same as adding 2/5 seven times. This is exactly right. This is exactly what we just went through. The correct answer is 14/5 or 2 and 4/5. So explanation A seems to be the right one, but we'll just read the other ones just to see if there's some flaws in them. Explanation B, Ken multiplied correctly but forgot to cancel out the 7's in the fractions. Since 7/7 is equal to 1, 2/5 times 7 is equal to 2/5 times 1. Well, obviously, 2/5 times 7 is not the same thing as 2/5 times 1. So this is kind of nutty. Explanation C, since Ken must add up all the pie for 7 of them, he should have added 7 and 2/5 instead of multiplying. No, that makes no sense. He should be multiplying. Ken is correct. His friends just didn't each that much pie. No, that doesn't make sense either. It's explanation A.