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## GMAT

### Course: GMAT > Unit 1

Lesson 1: Problem solving- GMAT: Math 1
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- GMAT: Math 54

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# GMAT: Math 29

148-153, pgs. 172-173. Created by Sal Khan.

## Want to join the conversation?

- For problem #153 at10:40, are we sure the answer is D, 23 years old? I don't have the book so I don't know the other possible answers, but here's how I solved it:

J = B + 14 and J + 10 = 2(J - 14) + 10 because B = J - 14. This turns into J = 2J - 28 which turns into J = 28, so J + 5 = 33 years old. What Sal did was combine the -14 and 10 into -4, but I thought we used order of operations to solve inside the parenthesis first, thus leaving 10 outside the parenthesis to be added later. I also solved the problem a different way and got the same answer, 33.(3 votes)- You can double check your answer thinking about the question logically. If Jack is 33 years old in five years then he is currently 28 years old. The problem begins with the statement: Jack is 14 years older than Bill. This means that Bill would have to be 14 years old (If Jack is 14 years older than Bill and Jack is 28 this means Bill’s age is 14). Already you should see the error here, because Jack is already double Bills age if this condition were true. You can further verify this issue by looking at the second statement in the problem. It states that in ten years Jack will be twice as old as Bill. If you add ten years to both Jack and Bill you get 38 (28 +10) and 24 (14+10) for their ages respectively. Clearly, 38 is not 24 x 2.

You can also verify Sal’s answer with the same logical process. If in five years Jack is 23 years old then it means his current age is 18. Looking at the first statement of the problem, it is understood that if Jack is 18 years old then Bill is 4 years old (J = B +14). The second statement says that Jack will be twice as old as Bill in 10 years. If you add ten years to each individual, Jack is 28 years old and Bill is now 14 years old. Since 28 is twice the amount of 14, you know Sal’s answer of 23 is correct.(0 votes)

- 153) at9:30, Why does Hal set it as J+10 = 2(B+10)? In my head, it should be J+10 = 2B +10.

In my head, Jack in 10 years, is twice as old as Bill is, in 10 years. Jack in 10 years, is not twice as old as Bill is in 20 years...no?(2 votes)- In these problems, you have to think about what time in past or future you are talking about for each clue.

Maybe it would help to look at it this way:

What are their relative ages`now`

?

Jack is some age that is 14 years older than Bill

We call Jack's age`now`

as J and Bill's age`now`

as B

So if Jack is 20, Bill is 6

That is why we say that J = B + 14

OK, so to the heart of your question:

How old will Jack be in 10 years? The actual wording seems to have been, "If`in 10 years`

, Jack will be twice as old as Bill"... Now we have to look ahead to`the future in 10 years`

:

Jack will be J + 10

How old will Bill be in 10 years?

B + 10

So at that point in the future, Jack will be twice Bill's age. If it meant that Jack WILL be twice Bill's age**now**, the wording would have specified**now**.

Now you can set up the equation for`that point in the future`

.

J + 10 = 2 (B + 10)

We need a single variable to solve, so we can use the B = J - 14. No matter what age Jack is, Bill will remain 14 years behind, so you can use that as a constant difference.

which gives us

J + 10 = 2 ( J - 14 + 10)

J + 10 = 2 ( J - 4)

J + 10 = 2J - 8, leading to J = 18

In this problem, there is the additional challenge of asking about a 3rd point in time, which is 5 years from now. So then you have to use Jack's age**now**and add 5 years.

Practice with these age problems helps you remember to look for the time cues.(2 votes)

- q.151 why you multiblied the two percentages ?? i think you should add the additional discount to the first one, so the total discount will be 65% because you said the additional discount is of the original price(1 vote)
- Could you tell me what this is

10=∆+8(1 vote)- I looked through the whole video a second time to find where this was mentioned, but maybe you were just curious.

Δ is usually used to mean change, especially having to do with graphs or even just values. An example is Δy means`change in the y value`

between two points.

So in that case, the meaning of your equation might be

10 = change in a value + 8

So, what number plus 8 equals 10? Δ must equal 2.

Or, you might be doing some pictorial math. Then you would need to find

10 equals Δ + 8

What value, when added to 8 would equal 10. Δ = 2

Or, you can just use algebra:

10 = ∆ + 8; subtract 8 from both sides:

10 -8 = ∆ + 8 - 8

2 = ∆(1 vote)

- 😀😃😄😁😆😅😂🤣☺️😊😇🙂🙃😉😌😍🥰😘😗😙😚😋😛😝😜 show someone but don’t say anything(1 vote)

## Video transcript

We're on problem 148. What is the lowest integer
that is the sum of three different primes, each
greater than 20? So it's the lowest integer, so
we essentially just have to find the three primes
above 20. So let's see, 21 isn't
a prime, 22 isn't a prime, 23 is a prime. 24, no, 25, no, 26,
no, 27, 28, nope. 29, that's prime. 30 isn't, 31 is. 31 plus 29 is 60 plus 23 is
83, and that is choice E. I hope I haven't missed any
primes, no, I don't think I have. 24, 25, 26, 27, 28,
I think I got all the primes in there. What is the lowest integer of
the sum of three different primes, each greater than 20? Yep, that's the answer, E. Problem 149. The average of 6, 8, and 10
equals-- so the average of 6, 8, and 10, so first of all,
what's the average of 6, 8, and 10? Well, let's just do it. 6 plus 8 plus 10. You could immediately see the
average is going to be 8, because you have 8 in the
middle, and both of these numbers are 2 away from it. But that divided by 3 is
equal to 24 divided by 3 is equal to 8. And that's equal to the average
of 7 plus 9 plus x over 3, so that tells us to
multiply both sides of this equation by 3, that says
that 24 is equal to-- what's 7 plus 9? That's 16 plus x. Subtract 16 from both sides,
you get x is equal to 8. Choice C. And you could have said, OK, 7
and 9 or both half-- 8 is in the middle, and they have to
average 8, so you might have just been able to eyeball it and
say 8, but it didn't take that much math, it's not that
much of a shortcut. Problem 150. If x is equal to minus 1, then x
to the fourth minus x to the third plus x squared, all
of that over x minus 1 is equal to what? x to the fourth is 1. x to the third is minus 1, but
then we're going to minus that, so this whole thing is
going to be plus 1, and x squared is plus 1. Negative 1 squared is 1. And you have minus 1 minus 1. So in the numerator you have
1, 2, 3, in the denominator you have minus 2. Let's put the minus out
front, minus 3/2. That's choice A. Problem 151. A toy store regularly
sells all stock at a discount of 20% to 40%. If an additional 25% were
deducted from the discount price during a special sale,
what would be the lowest possible price of a toy costing $16 before any discount? So we want to discount the
$16 as much as possible. So they say they're normally
discounting 20% to 40%, so if you want to discount this as
much as possible, you want to discount this by 40%. So if you discount this by 40%,
that's the same thing as saying it's going to be equal to
60% of its original price. So this would be the price after
being discounted by 40%, and then we would discount
by another 25%. Actually, I think it's easier
to go the other way. Well, let's just
do it this way. So we discount it by
another 25%, so we multiply it by 3/4 again. If you discount by 25%, the
price is going to be 3/4 of its original price. Actually, let's do
it as fractions. Fractions are always easier. 60% percent of your
price is 3/5. If you discount by 2/5, or 40%,
that's the same thing as multiplying by 3/5, and
then we're going to multiply by 3/4. Set 25%. If you're discounting by 25%,
that's the same thing as being worth 75% of your
original price. So what does that get us? 16 divided by 4, 4, you get a 1,
and then you have 4 times 3 times 3, 4 times 9 is 36, over
5, which is equal to-- 5 goes into 36 7 and 1/5. So $7.20. And that's choice B. Problem 152. The shaded portion of the
rectangular lot shown above represents a flowerbed. OK, so let me draw this
rectangular lot. They drew this shaded portion
something like that, and I can even color it in. So they say the shaded portion
of the rectangular lot shown above represents a flowerbed. If the area of the bed is 24
square yards, and x is equal to y plus 2, so they tell us
this side right here is x, which now they're telling
us is equal to y plus 2. And this right here is going
to be equal to y. Then z equals what? z is
this side right there. So if this area is 24, that
tells us that x times y times 1/2 is 24. That's the area of that. So 1/2xy is equal to 24. Or you can even say that
xy is equal to 48. We know what xy is. We can just substitute for
x, so we get y plus 2, that's x, right? X is equal to y plus 2 times
y is equal to 48. And so we get y squared plus
2y is equal to 48. Or y squared plus 2y minus
48 is equal to 0. We could factor this
quadratic. What two numbers when you
multiply by minus 48, and you add you get plus 6? So y plus 8 times y minus 6 is
equal to 0, 8 times minus 6 is minus 48, and then 8 minus 6 is
plus 2, so y could be equal to minus 8 or plus 6, right? Whatever numbers make either
of these equal to zero. Well, we can't have a negative
distance, so y is not equal to minus 8, so y is equal to 6. y is equal to 6, x
is equal to 8. Now, we can just confirm, 6
times 8 is 48 times 1/2 is equal to 24. And now we can use the
Pythagorean theorem to figure out z. So we'll get z squared
is equal to 36. That's y squared plus 64,
which is x squared, so z squared is equal to 100. z is equal to 10. Choice E. Problem 153. Jack is now 14 years older than
Bill, so Jack is equal to Bill plus 14. If in 10 years Jack will be
twice as old as Bill, so 2 times Bill's age, how old
will Jack be in 5 years? So they want to know j plus 5. That's how old Jack will
be in 5 years. So let's just try to
solve for Jack. This top equation tells us that
Bill is equal to Jack minus 14, and then
we can substitute this in for Bill here. So then this equation will
become Jack plus 10 is equal to 2 times Jack minus 14. So Jack plus 10 is equal to
2 times Jack minus 28. If we subtract Jack from both
sides, you get 10 is equal to Jack minus 28. We can add 28 to both
sides, and you get 38 is equal to Jack. So that's how old Jack is now. He's 38 years old. So how old is Jack going
to be in 5 years? 38 plus 5 is equal
to 43 years old. And I've clearly made a mistake,
because that's not one of the options. So let's see where I'm
made an error. So Jack is now 14 years
older than Bill. Fair enough. So in 10 years, Jack will be
twice-- oh, I see my mistake. He's going to be twice
as old as Bill is going to be in 10 years. That was a bone-headed
mistake. So let me clear all of this. That was a bad mistake. I shouldn't have made it. I'm lucky that they didn't have
what I put down as one of the choices, and I would've
gotten it wrong. All right. So we know that in 10 years,
Jack is going to be 2 times what Bill is going to
be in 10 years. That was my mistake. So now let's see
what we can do. We'll use that same information
up there, so Bill is equal to Jack minus 14. Substitute that information
here, so we get Jack plus 10 is equal to 2 times Jack minus
14, instead of Bill, plus 10. So you get Jack plus 10 is
equal to 2 times Jack. Let's see, minus 14 plus 10,
that's minus 4, so Jack plus 10 is equal to 2 times
Jack minus 8. Let's subtract Jack
from both sides. 10 is equal to Jack minus 8. Let's add 8 to both sides. You get 18 is equal to Jack. So that's how old Jack is
right now, he's 18. So in five years, Jack is going
to be 18 plus 5, which is 23 years old, choice D. And I'm out of time. See you in the next video.