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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.