# Lesson: Properties of Fractions - Part I

## Comment on Properties of Fractions - Part I

### I think the last question we

I think the last question we can also apply the fraction property of "if we add the same value to the numerator and denominator then we get closer to 1 and it'll be more bigger value, so the last fraction will be the greatest one.

Perfect!!

### hi Brent,

hi Brent,
Firstly i must say your video courses are really helpful just wanted to say it.

I have a doubt, hoping to gain some insight from you,
Although you have provided with a solution to this problem
I need more clarifications,

https://gmatclub.com/forum/if-mn-0-is-m-n-1-1-m-1-n-2-m-2-n-240815.html

1/n> 1/m
then by the rule that says smaller the denominator greater the value, n becomes the smaller number making m greater which makes the statement sufficient,

-NISHANT

### Thanks for the kind words

Thanks for the kind words about my course, Nishant!

The rule you're citing refers to only POSITIVE fractions.
So, for example, we know that 5/11 > 5/17, since 11 is less than 17

Things are different with NEGATIVE fractions.
For example, -1/2 < -1/5

Since we don't know whether m and n are positive or negative, we can't apply the rule you mentioned.

Cheers,
Brent

### Brent,

Brent,

how come...9/16 is 1/16 away from 1/2, when are looking for the distance from 1/2 ?

### 1/2 = 8/16

1/2 = 8/16
So, 9/16 - 1/2 = 9/16 - 8/16 = 1/6
So, 9/16 is 1/16 greater than 1/2

Cheers,
Brent

### Sir, increase n decrease in

Sir, increase n decrease in numerator n denominator properties are valid only for identical numerators and denominators to know which is bigger fraction? What about for non identical numerators n denominators to know the bigger fraction?

### We can still apply the same

In some cases, we can still apply the same concepts to non-identical numerators and denominators; it just takes an extra step or two.

For example, let's compare 25/99 and 24/100
We know that 24/100 < 24/99
And we know that 24/99 < 25/99
When we combine the results, we get: 24/100 < 24/99 < 25/99, which means 24/100 < 25/99

Cheers,
Brent

### Hi Brent,

Hi Brent,
I am concerned about the question https://gmatclub.com/forum/which-of-the-following-fractions-is-closest-to-220300.html

I did a long division to answer the question however I dont understand your solution. Why did you need to subtract 0.5 from the each answer choice?

Thank you.

We want to determine which fraction is closest to 1/2. So, we need to determine the DISTANCE between 1/2 and each answer choice.

To better understand this approach, let's examine a similar question.
Let's say we want to determine which answer choice is closest to 5, and the answer choices are:
A) 7
B) 6.1
C) 5.9
etc

A) 7 - 5 = 2 (so, 7 is 2 units away from 5)
B) 6.1 - 5 = 1.1 (so, 6.1 is 1.1 units away from 5)
C) 5.9 - 5 = 0.9 (so, 5.9 is 0.9 units away from 5)
etc

Now let's examine my solution to the original question: https://gmatclub.com/forum/which-of-the-following-fractions-is-closest-t...

A) 4/7
We want to find the distance from 4/7 and 1/2
So, we want: 4/7 - 1/2
From here, one approach is to find a common denominator to get: 8/14 - 7/14 = 1/14
However, I chose a slightly faster approach.
I recognized that 1/2 = 3.5/7
So, 4/7 - 1/2 = 4/7 - 3.5/7 = 0.5/7 (which is equivalent to 1/14)

And so on...

Does that help?

### Hi Brent,

Hi Brent,

https://gmatclub.com/forum/if-mn-0-is-m-n-1-1-m-1-n-2-m-2-n-240815.html

How come m is greater than n in statement 1?
Is 1/2 > 1/1?

Isn't it (-3)^2 is equal to 9? in statement 2?

Thank you!

If mn ≠ 0, is m > n?

(1) 1/m < 1/n
(2) m² > n²

Q: How come m is greater than n in statement 1? Is 1/2 > 1/1?
A: No, it's not the case that 1/2 > 1/1
From statement 1, we can't say whether m is greater than n

m = 2 and n = 1 satisfies the condition that 1/m < 1/n. In this case m > n
However, m = -1 and n = 1 also satisfies the condition that 1/m < 1/n. In this case m < n
So, we can see that statement 1 is not sufficient
--------------------------

Q: Isn't it (-3)² is equal to 9? in statement 2?
A: Yes, (-3)² = 9
It's also true that 1² = 1
Since 9 > 1, we can see that the values m = -3 and n = 1 satisfy statement 2.

Does that help?

### Perfect. But in statement 2

Perfect. But in statement 2 what if I test the values where n is greater than m? If n=4 and m=3, in this case, the values does not satisfy the statement and target question. Correct?

### I believe you may be looking

I believe you may be looking at this question the wrong way around

The target question asks "Is m > n?"
This means we don't know whether m is greater than n. In fact, that's what we're trying to determine.
So, it could be the case that m is greater than n, or it could be the case that m and n are equal, or it could be the case that m is less than n.

What we do know is that statements 1 and 2 are true.
So, when it comes to testing values, we must find values that satisfy the statement (NOT the target question).

For example, when it comes to statement 2, we can't test the values n = 4 and m = 3, since they don't satisfy statement 2.
If n = 4 and m = 3, then the inequality m² > n² becomes 3² > 4² which is the same as 9 > 16
This means we can't use the values n = 4 and m = 3 to help us determine whether or not statement 2 is sufficient

Does that help?

### it's clear. Thank you.

it's clear. Thank you.

### Hi Brent,

Hi Brent,

Can you please solve this problem?

List T consists of 30 positive decimals, none of which is an integer, and the sum of the 30 decimals is S. The estimated sum of the 30 decimals, E, is defined as follows. Each decimal in T whose tenths digit is even is rounded up to the nearest integer, and each decimal in T whose tenths digit is odd is rounded down to the nearest integer; E is the sum of the resulting integers. If
1
3
of the decimals in T have a tenths digit that is even, which of the following is a possible value of E − S?
I.−16
II.6
III.10

### Thanks Brent, could you

Thanks Brent, could you please explain why we are doing this?

In order to MAXIMIZE the value of E - S, we must MINIMIZE the value of S
We can do this by making the decimals with an ODD tenths digit 0.1, and by making the decimals with an EVEN tenths digit 0.01

In order to MINIMIZE the value of E - S, we must MAXIMIZE the value of S
We can do this by making the decimals with an ODD tenths digit 0.99999..., and by making the decimals with an EVEN tenths digit 0.89999...
How are we choosing these values?

Could you please also solve the 3 problems I had posted under another post? I cannot track back my post but I assume you would be able to see.

Thank you

### Our goal is to find possible

Our goal is to find possible values of E−S.
So, I first found the SMALLEST possible value of E-S
Then I found the BIGGEST possible value of E-S
Once I found the range of possible values of E-S (-19 to 7.9), it was just a matter of including those numbers that were in the range, and excluding the numbers that were not in the range.

That's odd. I went and answered all three questions, but never actually sent you the links my solutions. You'll find them here: https://www.gmatprepnow.com/module/gmat-integer-properties/video/845

### Yes, thank you Brent. One

Yes, thank you Brent. One last question regarding this one, how did you determine 0.1 for odd and 0.01 for even and not vice versa?

### Our goal is to MINIMIZE the

Our goal is to MINIMIZE the value of S

If the tenths digit must be ODD, then the smallest possible value is 0.1

If the tenths digit must be EVEN, then 0.01 is a nice small value (we could have also gone with 0.001 or 0.0001, etc, but that would have very little effect on the value of S.

Does that help?

### Yes, thank you so much Brent!

Yes, thank you so much Brent! You're amazing!