Lesson: Properties of Fractions - Part II

Comment on Properties of Fractions - Part II

Hi Brent,
For this question https://gmatclub.com/forum/if-0-y-x-then-which-of-the-following-is-233173.html
can we use the approach of testing some values?
Since x > y > 0 ,
let x=2 and y=1
27x+23y / 3x+2y = 54 + 23 / 6 + 2 = 77 / 8 =9.625
Let x = 3 and y =2
81+46 / 9+4 = 127/13=9.76
let x= 5 and y =4, result 9.86
let x = 13 and y =2, result 9.23
I just tested these values, but not sure if at some values the result will deviate from 9.
Thanks for your help.
Aladdin
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-0-y-x-then-which-of-the-following-is-23317...

Hi Aladdin,
Your approach is a good idea. The only problem is that this strategy only gives us the FEELING that the expression COULD equal 9.2

However, it's hard to have any certainty about whether or not it's possible to get the other two values (8.7 and 10.8)

That said, if you don't spot any other approach, this will at least help give you an idea of what the correct answer might be.

Cheers,
Brent

hi brent,

SHOW TIMER STATISTICS
For which of the following values of n is (100+n)/n NOT an integer?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

In this question, I understand the answer is 3, used divisibility to ans the question. But I can't understand why it is incorrect, can't integer be a decimal?
gmat-admin's picture

An integer is a whole number: . . . -3, -2, -1, 0, 1, 2, 3, . . .
So, for example, the decimal 3.2 is not an integer.

Here's my full solution: https://gmatclub.com/forum/for-which-of-the-following-values-of-n-is-100...

Cheers,
Brent

For all even integers n, h(n) is defined to be the sum of the even integers between 2 and n, inclusive. What is the value of h(18)/h(10) ?

(A) 1.8
(B) 3
(C) 6
(D) 18
(E) 60

Hi Brent, can you please share step by step method to solve this?
gmat-admin's picture

I'm happy to help.
Here's my step-by-step solution: https://gmatclub.com/forum/for-all-even-integers-n-h-n-is-defined-to-be-...

Cheers,
Brent

https://gmatclub.com/forum/the-expression-5x-2-x-3-is-equivalent-to-which-of-the-following-222362.html

Hi Brent,

The expression (5x-2)/(x+3) is equivalent to which of the following?

A) (5-2)/3

B) 5 – (2/3)

C) 5 – (x)/(x+3)

D) 5 – (17)/(x+3)

E) 5 + (17)/(x+3)

This solution to this question seems quite rhetorical to me and I haven't come across this being taught in any of your videos. How would you know to suddenly transform (5x-2) into (5x-15+17). I would never have thought of this given my basic number sense.

Could you please elaborate further in terms of how this solution is warranted. And, what scenarios present the opportunity to apply a similar approach?

Thank you!
gmat-admin's picture

Question link: https://gmatclub.com/forum/the-expression-5x-2-x-3-is-equivalent-to-whic...

We have a video on simplifying rational expressions here: https://www.gmatprepnow.com/module/gmat-algebra-and-equation-solving/vid...

Of course that video lesson appears after the above video, so you may not have covered it yet.

That said, here's how we might tackle this question.
Upon scanning the answer choices, we see that part of this expression might simplify to be 5.

So, we should ask ourselves, "How do we get a 5 from (5x-2)/(x+3)?"
Well, we can see that 5x/x = 5, so that's a start.
However, we need to deal with the rest of the expression.

So, let's put it this way: If SOMETHING/(x + 3) = 5, what is the value of SOMETHING?
Well, we know that SOMETHING must equal 5x + 15
That is (5x + 15)/(x + 3) = 5

So, from there, it's a matter of making the numerator 5x-2 look more like 5x + 15
Notice that we can rewrite 5x - 2 as 5x + 15 - 17
So, we get: (5x-2)/(x+3) = (5x + 15 - 17)/(x + 3)
= (5x + 15))/(x + 3) - 17/(x + 3)
= 5 - 17/(x + 3)

Question: Could you please elaborate further in terms of how this solution is warranted. And, what scenarios present the opportunity to apply a similar approach?
I'd say that this is a pretty advanced approach, so it's likely a student may never need to use it on test day.
However, the approach is useful when we want to make an expression LOOK LIKE a different expression.
So, for the above question I wanted the numerator to be 5x + 15, because (5x + 15)/(x + 3) = 5
I accomplished this by essentially adding 0 to the numerator.
As you can imagine adding zero to an expression does not alter its meaning.
So, I took 5x - 2 and added zero to it in a unique way: 5x - 2 + (15 - 15)
Then after some simplifying and rearranging, I got to: 5x + 15 - 17

Does that help?

Cheers,
Brent

Thanks Brent, I still have to watch the algebra module. This answer definitely clarifies the approach you took for this question.
Yulia's picture

Hi Brent,

(5x-2)/(x+3) = (5x + 15 - 17)/(x + 3)
= (5x + 15))/(x + 3) - 17/(x + 3)
= 5 - 17/(x + 3)
Could you please tell me how did we get rid of +15 to get 5 - 17/(x + 3)?
gmat-admin's picture

I'm happy to help.

General property: If k ≠ 0, then kx/ky = x/y

Take: (5x + 15)/(x + 3)
Factor the numerator and denominator to get: 5(x+3)/1(x+3)
Apply general property (above) to get: 5/1 (aka 5)

Does that help?

Hi Brent. Can you explain how you come across this in your latest comment?:

"So, let's put it this way: If SOMETHING/(x + 3) = 5, what is the value of SOMETHING?
Well, we know that SOMETHING must equal 5x + 15
That is (5x + 15)/(x + 3) = 5"

I do not see how that SOMETHING must equal 5x+15?
gmat-admin's picture

I'm happy to help.

Let's start with two analogous questions:
1) If SOMETHING/2 = 3, what is the value of SOMETHING?
2) If SOMETHING/237 = 11, what is the value of SOMETHING?

For question #1, we can see that SOMETHING = 6.
Notice that we can calculate that value multiplying 2 and 3 to get 6

For question #2, SOMETHING = 2607.
Here, we can calculate the value multiplying 237 and 11 to get 2607

In general, we can say that, if SOMETHING/x = y, then SOMETHING = xy

So, if SOMETHING/(x + 3) = 5, then SOMETHING = (5)(x + 3) = 5x + 15

Does that help?

Cheers,
Brent

Yes thank you

Hi Brent,

Do you have a solution for this question https://www.beatthegmat.com/mba/2010/10/18/beat-the-gmat-math-challenge-question-october-18-2010

I do not quite understand why the sum can be expressed as (1/1-1/2)+(1/2-1/3)+(1/3-1/4). Please help.
gmat-admin's picture

Hi Olga,

Funny coincidence! That's MY question and my solution in the video (https://www.youtube.com/watch?v=TkTs4mEYPDg)

This is a CRAZY HARD question, since most students will never recognize that 1/(k)(k+1) = 1/k - 1/(k + 1)

To see how these expressions are equal, let's work backwards.

Take: 1/k - 1/(k + 1)

Find common denominators: (1)(k + 1)/(k)(k + 1) - (1)(k)/(k)(k + 1)

Simplify: (k + 1)/(k² + k) - (k)/(k² + k)

Simplify: 1/(k² + k)

Factor denominator: 1/(k)(k + 1)

Once we've recognized that 1/(k)(k+1) = 1/k - 1/(k + 1), we can take each fraction and rewrite it.

1/(1)(2) = 1/1 - 1/2

1/(2)(3) = 1/2 - 1/3

1/(3)(4) = 1/3 - 1/4

etc.

Does that help?

Cheers,
Brent

https://gmatclub.com/forum/if-n-is-an-integer-is-n-positive-208378.html
Im unable to undertand the second statement. He has directly given that n is negative.
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-n-is-an-integer-is-n-positive-208378.html

Many people will see -n and conclude that this must be a negative number. However, this is not necessarily true.
If n = 1, then -n = -1, which is negative.
If n = -1, then -n = -(-1) = 1, which is positive.

To better understand the value of n, we must solve the equation for n.
Take: n=−n
Add n to both sides to get: 2n = 0
Divide both sides by 2 to get: n = 0

So, it turns out that n is neither positive not negative.
n = zero

Does that help?

Cheers.
Brent

Hi Brent,

Do you have a solution to this question?
https://www.beatthegmat.com/mba/2010/10/18/beat-the-gmat-math-challenge-question-october-18-2010
gmat-admin's picture

That's a question I created.
The solution is explained in the following video: https://www.youtube.com/watch?v=TkTs4mEYPDg

Cheers,
Brent

Hi Brent, is there a simpler way to solve https://gmatclub.com/forum/if-0-y-x-then-which-of-the-following-is-233173.html ?
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-0-y-x-then-which-of-the-following-is-23317...

There are 2 nice solutions above mine.
This one is my favorite: https://gmatclub.com/forum/if-0-y-x-then-which-of-the-following-is-23317...

NOTE: As I mention in my post, my solution is an alternate (much slower) solution. It's a total time killer.

By the way, here's my full (faster) solution on a different forum: https://greprepclub.com/forum/if-0-y-x-then-which-of-the-following-3202....

Cheers,
Brebt

https://gmatclub.com/forum/if-0-y-x-then-which-of-the-following-is-233173.html

I let y = 1 and x = 2, found out that it's 77/8, which is 9.something, so I chose the correct answer, however, I'm not heart-easy with this...
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-0-y-x-then-which-of-the-following-is-23317...

Testing values is a great approach (if you don't spot an algebraic solution).

The main problem with your solution is that you only tested one pair of values (y = 1 and x = 2).
So, this one result only tells us that the expression could evaluate to 77/8 (or 9.265). This doesn't necessarily mean the expression could evaluate to 9.2.

But even if we were able to find values of x and y such that the expression evaluates to exactly 9.2, this just tells us that statement II is possible, which means the correct answer is B, D or E.

We still need to determine whether the other values (8.7 and 10.8) are possible values of the expression.

By testing several different pairs of values (for x and y), we'll get a better idea of which values are possible and which aren't.

I see the point, hmm, actually, algebraic solution makes sense for the first answer post, i should be flexible enough

https://gmatclub.com/forum/if-n-is-an-integer-is-n-positive-1-2n-1-n-1-is-an-integer-208378.html

I mean, honestly... I got the idea, but, still my brain could function against me, I may not notice two statements are identical here, is it just because I'm not smart enough? so ACE could be eliminated and then I chose B eventually, but it is wrong unfortunately
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-n-is-an-integer-is-n-positive-1-2n-1-n-1-i...
This is a super tough question. Only 50% correctly answered it, and that number is inflated!

The two statements here are not identical. One statement essentially tells us that n/(n+1) is an integer, while the other statement tells us that n = -n.

We can say that both statements are sufficient, but we can't say they're identical.

Now that you've seen and understood the solution, you have an additional tool for your mathematical toolbox on test day.

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