Proportion word problem: hot dogs

- We're told that Mika can eat 21 hot dogs in 66 minutes. She wants to know how many minutes, m, would it take her to eat 35 hot dogs, if she can keep up the same pace. So, big clue is the same pace, I have to remove a hair from my tongue, alright. (laughs) A big clue is the same pace. That means that the hot dogs, hot dogs per minute, per minute, is going to be constant, is always going to be the same, always the same. Always the same, because this is essentially the pace. Her hot dogs per minute are going to stay the same. She's gonna stay at the same pace. So, we it tells us that she can eat 21 hot dogs in 66 minutes. So, her hot dogs per minute, at least up here, is 21 hot dogs in 66 minutes, So, it's 21 hot dogs in 66 minutes. Well, if her pace is always going to be the same, well it's gonna take her this ratio over here, is going to be the ratio between 35 hot dogs and however long it takes her to eat 35 hot dogs. So, once again hot dogs per minute are going, has to be a constant because it's gonna be the same pace. Hot dogs per minute. If it takes, 21 hot dogs takes 66 minutes, 35 hot dogs take m minutes, these two ratios are going to be the same. We're dealing with a proportional relationship. It's going to be happening at the same rate. And then, we're left with a situation where we just have to solve for m, and there is a bunch of different ways you could tackle this. The easiest way, that I can think of doing it is, I like, I don't like this m sitting here in the denominator, so let's multiply both sides by m. Let me do that in a different color. So, if I multiply that side by m and then this side by m, and so what do we get? On the left hand side, we have 21 over 66m. 21 over 66 times m, times m, is equal to, is equal to, well, you divide by m and multiply by m. Those are gonna cancel out and you're just gonna have 35. And now, you just have to solve for m and the best way I can think of doing that, is multiply both sides times the reciprocal on, both sides times the reciprocal of the coefficient on the m. So, let's multiply both sides by, let's multiply both side by 66 over 21. Once again, I've just swapped the numerator and the denominator here to get the reciprocal, but I can't just do it to one side of the equation, I have to do it to both sides, otherwise, it's not going to be an equa, it's not gonna be equal anymore. So, times 66 over 21, this is just going to be one. You multiply something times it's reciprocal, you're just going to end up with one. So, you're gonna be left with, m is equal to. Now, 35 times 66 divided by 21. Well, 35 is the same thing as, 35 is five times seven and 21 is three times seven. So, you're multiplying by seven up here and here, you have a seven in the denominator, you're dividing by seven, so they're going to cancel out. So, this is going to simplify to five times 66 over 3, and then we could simplify it even more, because 66 is the same thing as three times 22. Three times 22 and so, you have a three in the numerator, you're multiplying by three and three in the denominator, dividing by three. Three divided by three is one, so you're left with five times 22, which is 110. So, it would take her m minutes to eat 35 hot dogs at the same pace. Now, when some of you might have tackled it, you might have had a different equation set up here. Instead of thinking of hot dogs per minute, you might have thought about minutes per hot dog. And so, in that situation, if you thought in terms of minutes per hot dog, you might have said, ok look, it took Mika 66 minutes to eat 20, to eat 21 hot dogs,and it's gonna take her m minutes to eat 35 hot dogs and if it's the same pace, then these two rates are going to be equal. They have to be the same pace. And so, then you can solve for m and actually, this one's easier to solve for m, you just multiply both sides by 35. Multiply both sides by 35 and you're left with, on the right hand side you're left with just an m, and on the left hand side, same, same idea. You're taking 35, you have 35 times 66/21, which we already figured out is 110. So, 110 is equal, is equal to m. So, once again, multiple ways to tackle it, but it's important that we got the same answer.