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## GMAT

### Course: GMAT > Unit 1

Lesson 1: Problem solving- GMAT: Math 1
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# GMAT: Math 16

87-91, pg. 163. Created by Sal Khan.

## Want to join the conversation?

- find the point on the axis that is equidistant from (-2,5) and (4,1)(3 votes)
- or did you mean the point on the "line" that is equidistant => (1,3)?(5 votes)

- i need to know more about the slope method used in problem 89 which math lesson should i watch exactly?(3 votes)
- Well, there are lots of them, depending on what your confusion is.

One of the most important steps here was finding what the slope was according to the equation they gave us.

x = 3y -7

We use`y = mx + b`

as the format that will immediately reveal the slope (m) of the line.

So, the fasted thing to do here, and the best, is to rearrange x = 3y -7 to solve for y.

3y - 7 = x

3y = x + 7

y =`1/3`

x + 7/3

Aha! slope is 1/3 (goes up 1 unit for every 3 units of x, or you can think of it going up 1/3 unit every time c increases by 1.)

OK, so let's look at the coordinates they gave us.(a, b) and (a + 3, b + k)

The definition of slope is`rise over run`

, in other words, the change in**y**divided by the change in**x**.

The workhorse formula for finding slope = (y₂ - y₁)/(x₂ - x₁) That is what Sal used.

So if you have two points, call one Point 1 and the other Point 2. Then take the coordinates and label them for the two points:

Point 1 (x₁ , y₁) and Point 2 (x₂ , y₂)

x₁ = a

y₁ = b

x₂ = a + 3

y₂ = b + k

Plugging in, then (y₂ - y₁)/(x₂ - x₁)

= ((b + k) - b)/((a + 3) - a)

Clearing out the parentheses:

b + k - b is just k

a + 3 - a is just 3

so, the result for the two points is k/3

Wow, that makes it easy, because all we need is the number for**k**that makes

k/3 = 1/3

so k = 1

Actually, I saw that point**a**increased by 3 and point**b**increased by**k**, so I was able to skip most of the steps to the

k/3 = 1/3

so k = 1

And you will too after some practice: put**slope-intercept**into the search box or just**slope**and you will get lots of options for delving into this topic.(4 votes)

- Convert minutes to seconds(0 votes)
- who can do that real quick i dont under stand it(1 vote)
- i always have a problem findind the point equidistance of (4,2) and (5,9)(0 votes)
- The set of all points in the plane equidistant from two non-coincident points would be the line through the midpoint of the segment between the points and perpendicular to it.

In this case that would be the line through (4.5, 5.5) with a slope of -1/7(1 vote)

- At5:43, you spelt minutes wrong(0 votes)

## Video transcript

We're on problem 87. Machine A produces parts twice
as fast as machine B does. So machine A produces 100 parts
twice as fast as machine B, let me just keep reading,
machine B produces 100 parts in 40 minutes. So that's equal to what? 100 divided by 40, that's
2.5 parts per minute. And they told us A produces it
twice as fast, so A is going to be twice that,
so A is going to be 5 parts per minute. If each machine produces parts
at a constant rate, how many parts does machine A produce
in 6 minutes? So 5 parts per minute
times 6 minutes. 6 minutes is equal to what? Minutes cancel out, 6
times 5 is 30 parts. Choice A. 88. A necklace is made by stringing
n individual beads together in the repeating
pattern red bead, green bead, white bead. Let's see. So, red, green, white, blue,
and then yellow, and you keep repeating. If the necklace begins with a
red bead, and ends with a white bead, then n could
be equal to? So let's think about it, it
starts with a red bead, and ends with a white bead. So let's think about this, if
it's exactly a multiple of 5, if n is equal to exactly the
number of beads in this pattern, then it'll always
end with a yellow. So if n is equal to some
multiple of 5, it could be 5, it could be 10, or whatever,
then you're going to end with a yellow bead. That makes sense. If you have 10, you'd just
have the pattern twice. If n is equal to some multiple
of 5 plus 1, then you're always going to complete the
cycle completely, and then your going to have one more,
and you're going to end with a red bead. I think you see the
pattern here. If n is equal to some multiple
of 5, and then it has 2 left over, or when it's divided by
5, it has a remainder of 2, you're going to end
with a green bead. The multiple of 5 parts, you're
going to complete all the way to yellow, and you're
going to have 1, 2 left. And so if n is equal to a
multiple of 5 plus 3, then you're going to end
with a white bead. So a multiple of 5 plus 3 is
essentially any number that ends with either an 8 or 3,
because a multiple of 5 ends with either a 5 or a 0. So let's see what choices
they give us. They give us 16, not a
multiple of 5 plus 3. 32, not a multiple of 5 plus
3, 41 is not, 54 is not. 68, that is a multiple
of 5 plus 3. It's 65 plus 3. 65 is 13 times 5, that's equal
to 13 times 5 plus 3, so the choice is E. Problem 89. In the x-y coordinate system, if
a, b and a plus 3, b plus k are two points on the line
defined by the equation x is equal to 3y minus 7, then
k is equal to what? Well I think an easier thing to
do is to-- well actually we could just stay-- let's just
define y in terms of x. So you get x plus 7 is equal
to 3y, y is equal to 1/3 x plus 7 over 3. OK, let's see what we can do. So the slope of this line is
1/3, if you remember from algebra, this mx plus b form,
but this is how fast the line goes up or down, this 1/3. Every move to the right
in x by 1, you're moving up in y by 1/3. So what's the slope? If you give me two points,
what's the slope? What's the change in y
over the change in x? So I'll call this y1
and this is y2. Or point 1 and point 2, so you
could say change in y is b plus k, the latter y point minus
the first y point, over a plus 3 minus a. And what does this reduce to? b plus k minus k, b's cancel
out, a plus 3 minus a, a's cancel out, I'm left
with k over 3, and that equals the slope. We already know the slope
is equal to 1/3. k over 3 is equal to 1/3,
so k is equal to 1. It's choice D. Problem 90. At the rate of m meters per
second, how many meters does a cyclist travel in x minutes? So remember, this is tricky. They said x meters per second,
how many meters does a cyclist travel in x minutes, so let's
get x in terms of seconds. So x minutes is going to be 60x
seconds, because you have 60 seconds per minute. So 60x seconds. And distance is equal to rate
times time, hopefully that's a little bit of intuition for you
now, but the rate is equal to m meters per second, and
he's traveling for 60x seconds, so you get m times 60x,
which is 60mx in seconds, times meters per second,
that's meters. So 60mx meters. How many meters does a cyclist
travel in x minutes? x minutes is 60x seconds. How many meters does a cyclist
travel-- right, rate times time, at a rate of m
meters per second. I think they made a boo-boo
here, because they have to have the 60m-- oh, I see
the mistake I made. Let me redo it, my bad. They said, at the rate of m
meters per s seconds, so the cyclist's rate is actually m
over s meters per second. That's confusing because they
said that s seconds, and I thought they said, m meters per
second, but it's a rate of m meters per s seconds, how
many meters does a cyclist travel in x minutes, which we
know is 60, if it's x minutes, it's 60x seconds. So you multiply the rate times
the time, and you get 60mx over s, which is choice E. And it saved me that my
original answer wasn't anywhere in any of
the choices. Problem 91. If Sam were twice as old as
he is, he would be 40 years older than Jim. So 2 times Sam, so if Sam were
twice as old as he, so s is Sam's current age. If Sam were twice as old as he
is, he would be 40 years older than Jim, so he would
be Jim plus 40. If Jim is 10 years younger
than Sam, how old is Sam? Jim is 10 years younger than
Sam, so we can say Sam is Jim plus 10. We could've also said Sam
minus 10 is equal to Jim, same thing. Let's see what we can do here,
let's see how best we can solve this equation. They want to know
how old is Sam. We want to solve for s,
so let's do something. Let's subtract this equation
from that equation. So 2s minus s is s, j minus
j is 0, 40 minus 10 is 30. So Sam is 30, and we're done. And there's a lot of ways
you could've done that. You could have solved for j,
and substituted here then solved, it would've taken a
little bit more time, but we just subtracted this equation
from that, and you got s is equal to 30, which
is choice B. And I think I'm about out of
time, so I'll continue into the next video. See you soon.