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Current time:0:00Total duration:11:18

We're on problem 110. Let me scroll this up. Whenever Martin has a restaurant
bill with an amount between $10 and $99 he
calculates the dollar amount of the tip as two times
the tens digit of the amount of his bill. Fair enough. So essentially, if there
is a ten digit, he just multiplies it by 2. If the amount of Martin's most
recent restaurant bill was between $10 and $99, was the
tip calculated by Martin on his bill greater than 15% of
the amount of the bill? That is the question. Statement 1 says, the amount
of the bill was between $15 and $50. So 1 essentially is enough for
this problem, is sufficient, if for every bill, based on
his calculation where you double the tens digit, it's
going to be greater than 15. And I suspect, let's see if we
take the lower end of this, on this he'll pay $2. He'll pay a $2 tip on $15. And what percentage is that? 15 goes into 2.00. 1, 15, 50. 15 goes into 50 3 times. 3 times 50, 45 and it
just keeps going. So that's a 13% tip. Well, it's going to be even
lower than that at $16. Maybe the bill wasn't $15,
the bill was $16. So the tip is, at the lower
end it's 13%, 12%. And at the higher end, if the
bill is $40 exactly and he pays $8 on that. If the bill is $40, then Martin
would pay $8, which would be 20%. So I can pick different numbers
in this range, and based on the way Martin
calculates his tip, he can either pay less than 15%
or more than 15%. So statement 1 alone
is not sufficient. What does statement 2 tell us? The tip calculated
by Martin was $8. Well this is going to be
2 times the tens digit. So that means that the bill
was equal to $40-- I don't know, $40-something. So let's think about it. So the worst case is if the
bill-- if he paid $8 an a $40 bill, that's definitely
more than 15%. That's 20%. That's what I just actually
calculated. Let's see the worst case
is on a $49 bill. That the bill keeps going
up and he just pays $8. So is 8 bigger than 15% of 49? Well yeah. Because 8/50 is equal to what? That's equal to 16%. So if 8/50 is 16%, 8/49, if we
lower the denominator a little bit, that's going to be
greater than 16%. So no matter what range in the
forties the bill was, whether it's $40 or $49 or anything in
between, an $8 tip is going to be more than-- it's actually
going to be more than 16%. Not to speak of even 15%. So statement number 2
alone is sufficient to answer this question. And statement number 1
is fairly useless. Problem 111. The price per share of stock x
increased by 10% over the same time period that the price
per share of stock y decreased by 10%. The reduced price per share of
stock y was what percent of the original price per
share of stock x? Fascinating. So let's do initial and final. So x final is equal to an
increase by 10% from the initial period. OK, so it equals 1.1
times x initial. Fair enough. And then, over the same time
period y decreased by 10%. So y final is equal to 10%
less than y initial. So that's 0.9 times y initial. And what they want to know is
the reduced price per share of stock y-- so that's y f-- was
what percent of the original price per share of stock x? Of x initial? So this is what they
want to figure out. This as a percentage. So let's see if the statements
help us out at all. The increased price per share
of stock x was equal to the original price per
share of stock y. So the increased price
per share of stock x, so that's x f. That's the final. That's the increased
share price. It increased from the initial
to the final. So the increased price per share
of stock x was equal to the original price per
share of stock y. So that equals y initial. So this is interesting. I don't know if it
gets us anywhere. This deals with x final
and x initial. So I think we can-- so if we
could write all of it in terms of y initial. So y final equals-- so
let's rewrite this. This is equal to-- y final
is 0.9 times y initial. That's just from
this equation. 0.9 times y initial. And let's see if we could write
the initial x in terms of the initial y. So x final is equal
to y initial. So that means that y initial
is equal to this. So that equals 1.1 x initial. And that means that we can
divide both sides of this equality by 1.1. And we get x initial is equal to
1 over 1.1 times y initial. So then we have this. 1 over 1.1 times y initial. And then these two
would cancel out. And you would actually
have your answer. So statement 1 alone
is sufficient to answer the question. It wasn't obvious to me at
first, but then you have to realize that, the terminology is
confusing but that you can actually write both of these
in terms of y initial given that information. Given the fact that x final
is equal to y initial. Let's see what statement
2 does for us. The increase in the price per
share of stock x was 10/11 the decrease in the price per
share of stock y. Let me think about that. The increase in the price
per share of stock x. So that means that x final minus
x initial-- that's the increase-- that this was equal
to 10/11 times the decrease in the price per share
of stock y. So what was the decrease? This was y initial
minus y final. Because this was a larger
number and we wanted a positive number here. Because we're just saying
the decrease. We're not saying the
negative increase. So let's see if we can
simplify this at all. So let's see, you get x final
minus x initial is equal to 10/11 y initial minus
10/11 y final. And remember, the whole time
we just want to figure out what y final over
x initial is. So let's think about this. Let's write x final is equal
to 1.1 times x initial. Right So we have 1.1 times x
initial minus x initial is equal to-- actually I should
have done that in the first step. Let's just skip this
right now. Let's just write
the 10 over 11. What do we want? We want y final. So we just want to substitute
for y initial. I am confusing myself. So y initial is going
to be equal to y final divided by 0.9. So 1 over 0.9 y final. I know this is a little
confusing. Minus y final. I just did a substitution
for y initial. And then here, well this is
actually, we know that we can solve this problem. Although it gets quite hairy. Because here, if you think about
it, you're going to get some number, well you're going
to get 0.1 x initial is equal to-- you're going to get some
constant after you do all this math-- times y final. And so you can easily figure out
what y final divided by x initial is, just by dividing
both sides by x initial and then dividing both sides
by a and you would have solved the problem. And I'm not going to do that
because it's actually kind of hairy and I don't have-- 1
divided by 0.9 and then multiplying it by 10/11
is a fairly convoluted way of doing it. But hopefully, you see that
this is solvable. And let me just review that
again because I think I did it in my own head. So the statement itself said the
change, the gain in x-- x final minus x initial-- was
10/11 times the loss in y. So y initial minus y final. Because y final is
the smaller one. x final we can rewrite in terms
of x initial just using our initial, the fact that
it was 10% more. So I just did that here. And y initial, we can
rewrite as y final. You could say y initial
is equal to y final divided by 0.9. And that's what we did there. And then you could see here,
this will simplify to some constant times y final. And then you multiply
that times 10/11. So you get some constant
times y final. And then you have 0.1
times x initial. 1.1 minus 1. And then you can just do some
simple algebra to figure out what y final over
x initial is. So both statements,
independently, are sufficient to solve this problem. I think that was the hardest
one we've done so far. See you in the next video.