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Problem 77. If Miss Smith's income was 20%
more for 1991 than it was for 1990, how much was her
income in 1991? Let's call it income 1991,
is 20% more than 1990. So that means 1.2 times
income in 1990. I hope you get that. If something is 20% more than
something else, it's going to be 1.2 times that. 12 is 20% more than 10. And 12 is 1.2 times 10. Or you could view it as income
plus 20% of income, which is 1.2 times income. Either way. So let's see what
they tell us. Well, what do we have
to figure out? OK, we were trying to figure
out this, income in 1991. Statement number one. Miss Smith's income for the
first six months of 1990 was $17,500, and the income for the
last six months of 1990 was $20,000. So they're telling that she made
$17,500 in the first six months of 1990, and her income
for the last six months of 1990 was $20,000. Well, they're essentially
telling us the total income for 1990. The first six months and
the last six months. There are 12 months in a year. So her total income for
1990 was $37,500. That equals income for 1990. So clearly, if we know this is
this, we just multiply that times 1.2, and we get
the income for 1991. So this statement alone
is sufficient. Let's see what they give us
for statement number two. Miss Smith's income for
1991 was $7,500 greater than for 1990. So they say income of 1991 was
$7,500, so it equals income for 1990 plus $7,500. Well, this alone does help
us, because they've already given us this. So we have two linear
equations, right? This is one linear equation
and two unknowns. This is another linear equation
and two unknowns. So we have two linear equations
and two unknowns. We can solve this. Probably the easiest way is just
to substitute, depending on what you want to solve for. But we've done that
multiple times. You could substitute 1.2
times 1990 here, and then solve for it. Or you could do the other way. You could divide by 1.2 here and
then substitute it there. But either way, this is trivial
algebra, hopefully, by this point, to solve. But this and this is definitely
enough information to solve the problem. So two equations with
two unknowns. And you can do that in your
spare time if you don't believe me. So both statements alone are
sufficient for this one. 78. In the figure above-- so
I think I have to draw. This is the y-axis, and
that's the x-axis. And then they have a line. Let's see what I can do. The line looks something
like that. And then they tell us--
what do they tell us? This, of course, is y. This is x. And then they say this is P. And this right here is Q. And then they draw this. They call this point
right here R. And they draw-- that's
like that. And this Q is at point c, d. And P is at point a, b. And then they say, in the figure
above, segments PR and QR-- so let me draw that out
a little bit better. --are each parallel
to one of the rectangular coordinate axes. OK, fair enough. This is parallel to
the y-axis, PR is parallel to the x-axis. Fair enough. Is the ratio of the length
of QR:PR equal to 1? So is the ratio of
QR:PR equal to 1? So they want to know QR/PR,
is that equal to 1? And immediately this should
trigger something from Algebra 1. They're asking you, essentially,
is the slope of this line equal to 1? Right? Is the ratio of QR:PR, so rise
over run, is the slope of this line equal to 1? So let's see what we can do. And slope is just change
in y over change in x. So what's this point,
first of all? You actually don't have to know
anything about slope. I don't want to make you
feel like you have to memorize some formulas. What's this point going to be? So it's going to be-- actually, let's do it even better. What is the length of
QR going to be? I haven't looked at any of the
data points right now. What is length of QR? Well, it's going to
be this height. So it's this y, which
is d, minus this y. This y is going to
be b, right? Because all of this is y is
equal to b, right here. So QR is going to be
equal to d minus b. And PR is the length
on the x-axis. It's going to be this x. What is this x? Well, this x is right here, c. x equal to c. It's going to be this
x minus this x. Well, here, x is equal to a. And so the ratio is equal to
d minus b, over c minus a. Which is, if you remember,
the formula for the slope of a line. You just take the y1 minus
y2, over x1 minus x2. But we didn't have
to memorize that. It's intuition. See the x-coordinate is c and
the y-coordinate is b. So hopefully that gives
you intuition. Now let's look at
the statements. You wouldn't have to do
that on the real GMAT. That would all be
a waste of time. Statement number one tells
us, c is equal to 3 and d is equal to 4. So that by itself, that
just gives us the first part of this. That doesn't help us figure
out this entire ratio. So this by itself isn't
that useful. Maybe in conjunction with
what else they give us. Statement two. a is equal to minus 2, and
b is equal to minus 1. Well, if you used both of these
statements together, then we have everything here. We have d, we have b, we
have c, and we have a. So we can solve it. So both statements together
are sufficient for knowing whether the ratio
of QR:PR is 1. Or essentially, is the slope
of this line equal to 1? Next problem, 79. While on a straight road, car x
and car y are travelling at different constant rates. If car x is now 1 mile ahead
of car y, how many minutes from now will car x be 2
miles ahead of car y? So x is here. y is here. And they're going at
constant rates. 1 mile. So they've been traveling for
some amount of time and x is 1 mile ahead. And they're saying how long is
it going to be before x is-- how many minutes before
x is 2 miles ahead? And they're at constant rates. So if they started off-- and
let's just think about it-- if they started off at the same
point and it took 10 minutes for x to get 1 mile ahead, it
would take another 10 minutes for it to get 2 miles. Well, that's how I'm
thinking about. Let's see what they give
us for the statements. Statement number one. Car x is traveling at 50 miles
per hour, and car y is travelling at 40
miles per hour. Well, that seems to be pretty
good information. So essentially, car
y is moving away from car x at what? It's moving away 50 minus
40 miles per hour. So from car y's point of view,
car x is always pulling away at 10 miles per hour. Right? Does that make sense? If car x was going at 40 miles
per hour, you wouldn't be pulling away at all. If it was going at 41 miles per
hour, it'd be pulling away at a increment of
1 mile per hour. And as long as we're not
approaching the speed of light, we can assume Newtonian
classical physics, and we could just take the difference
between the two. So how long does it take for it
to pull away another mile? Well, if you're going 10 miles
per hour, relative to something else, how many
minutes does it take to go a mile? Well, one, you know you
can figure that out. But let me figure that
out for you. So you know distance is equal
to rate times time. So if your distance you want
to know is 1 mile, and your rate is equal to 10 miles per
hour, times time, what's the time going to be equal to? Time is going to be equal to
1/10 of an hour, or 6 minutes. So that's the answer
to number one. One alone is sufficient. Or the answer to number 79,
one alone is sufficient. Let's see what they give
us for number two. Statement two. 3 minutes ago, car x was 1/2
mile ahead of car y. OK. So 3 minutes ago, the state of
affairs was this: y was here, x was here. And it was a 1/2 mile
difference. So what does that tell us? That's actually pretty
good information too. 3 minutes ago, car x was 1/2
mile ahead of car y. Now car x is 1 mile ahead. So in 3 minutes, x pulls away
by 1/2 a mile, right? And they're going at constant
velocities. So the relative velocities
between the two don't change. So if it takes 3 minutes for x
to pull away by 1/2 a mile, it would take 6 minutes for x to
pull away by a mile, right? You just multiply them by 2. They're all going at the
same constant velocity. So in 6 minutes, x pulls
away by 1 mile. And that's actually what
they're asking. Because they say how many more
minutes does it take x to pull away by another mile? They've probably been traveling
for 6 minutes already, and then in another 6
minutes x would pull away by another mile. So two alone is also
sufficient. So each of them independently
are good enough to answer this question. See you in the next video.