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Studying for a test? Prepare with this lesson on Ecuaciones lineales y desigualdades.
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Constructing equations from proportions to solve problems

Video transcript
Rick just finished eating with his family at a restaurant. The total bill for his family was $52. And Rick was pretty impressed with the service. So he's going to leave a 25% tip. He wants to know the total amount he should leave at the table. Select the equations Rick can use and determine the total bill, including the tip. So the unknown thing, the thing that we should figure out, is the total amount he should leave at the table. So let's just call that x. Now, there's a couple of ways we could think about it. We could think about it as the ratio between the total amount he's going to pay and the actual bill. That should be the same thing as the ratio between the respective ratios. So you could view the actual amount of the bill as being 100%. That is 100% of the bill. But he doesn't want to just pay 100%. He wants to pay 100% plus a 25% tip. So we could rewrite everything that I just did here as x over 52 is equal to 100% plus 25%-- I could write that as 125%-- over 100%. Now, if we look at the choices here, this looks exactly like this. If we just swap the left and right-hand side, we get 125% over 100% is equal to x over 52. So we can select this one. And once again, this makes sense. The ratio between what he's going to pay with the tip to 100%, or the percentage that he's going to pay if you include the tip relative to 100% is going to be the actual amount he pays relative to the actual bill. Now, let's see if any of these other ones make sense over here. So here he says it's the ratio of the actual bill at 100% is equal to the ratio you're going to pay at 125%. Now, this makes sense, too. Let me just rewrite it again for emphasis. So here they're constructing a proportion where you have the ratio of the total bill. And you're saying, hey, look, that total bill is 100%. Actually, let me write that in blue. The total bill is 100%. And we want to see this ratio should be the same as the ratio between the actual amount we're paying and 125%. So you increased this by 25%. Well, you're going to have to increase this by 25% as well to get the actual amount that you're paying. So this makes sense as well. Now, what about this? The ratio between 100% and 125% should be equal to x over 52. Now, this does not make sense, because this is the ratio between the full price without paying the tip and the tip. Well, here is the tip and the full price. So if you swapped these two, if you wrote this as 52 over x, then you could check this. But here you're not finding the ratio between the corresponding things. So I would not check this one there. But either way, our last thing to do is actually to figure out how much tip Rick should leave. So we need to actually solve for x. And we could go back to this original equation here. And if we multiply both sides by 52, we get x is equal to 125% divided by 100% is 1.25 times 52. And then I could just multiply that. Let's see. 52. 4 goes into 52 13 times. So this is going to be-- well, actually, let me just divide. Let me just multiply that. I don't want to make a mistake here. So 1.25 times 52. 2 times 5 is 10. 2 times 2 is 4 plus 1 is 5. 2 times 1 is 2. Let's throw a 0 down here, because we're now multiplying by 50. 5 times 5 is 25. 5 times 2 is 10 plus 2 is 12. 5 times 1 is 5 plus 1 is 6. So we get 0. Let's carry that 1. We get a 5. We get a 6. And then we have two numbers behind the decimal point. It is 65. And that's right. That's 52 plus 1/4 of 52, which is 13. 52 plus 13 is 65. So he should leave $65 if we want to include the tip.