Please can you explain the reasoning behind this?

[shibotta23]

As x increases from 165 to 166, which of the following must increase?

I. 2x-5

II. 1-1/x

III. 1/x squared - x

a) I only
b)III only
c)I and II
d)I and III
e)II and III

[Stuart@KaplanGMAT]

As x increases from 165 to 166, which of the following must increase?

I. 2x-5

II. 1-1/x

III. 1/x squared - x

a) I only
b)III only
c)I and II
d)I and III
e)II and III

Hi!

We're told that x is positive and increasing and we're asked to determine which of 3 expressions will increase along with x.

Here's something to remember: the GMAT is not trying to be your friend, but it is trying to see how clever you are. So, we're always using our cleverness to outwit the test and turn complex questions into simpler ones.

How can we do so on this question? By recognizing that "165 to 166" is an arbitrary range. We can make our lives much easier by saying that x is increasing from 5 to 6 instead, then using those numbers to test the choices.

(I) looks simplest, so let's start there.

If x=5, then 2x-5 = 10-5=5
If x=6, then 2x-5 = 12-5=7

Did (I) increase? YES. Eliminate (b) and (e), since they don't include (I).

(As an aside: the "-5" is irrelevant, since it's a constant and has no impact on whether that expression grows, so we could have just tested "2x" instead.)

Our 3 remaining choices are I only, I and II and I and III. Not much help there, so let's test II next, since it seems simpler than III.

(II) 1 - 1/x

If x = 5, then we get 1 - 1/5 = 4/5
If x = 6, then we get 1 - 1/6 = 5/6

Is 5/6 > 4/5? YES. So, x is increasing once again, which means that (II) is also a winner.

(As another aside, 1 is constant so we could have ignored it and only tested "-1/x"... but you have to remember to include the negative sign, so this one may be easier to test by leaving the 1 in there.)

Only one choice includes both I and II... choose (C)!

[Scott@TargetTestPrep]

As x increases from 165 to 166, which of the following must increase?

I. 2x-5

II. 1-1/x

III. 1/x squared - x

a) I only
b)III only
c)I and II
d)I and III
e)II and III

Solution:

Before jumping into this problem using the numbers provided, we must analyze what is really being tested here. We are really being asked what happens when we increase a positive integer by a value of 1. Using 165 and 166 is cumbersome, so we will use two much smaller numbers, 2 and 3. We can now rephrase the question:

As x increases from 2 to 3, which of the following must increase?

Let's test the new numbers for each Roman numeral.

I. 2x - 5

When x = 2:

2(2) - 5 = -1

When x = 3:

2(3) - 5 = 1

This DOES increase. Roman numeral I is correct. We can eliminate answer choices B and E.

II. 1 - 1/x

When x = 2:

1 - 1/2 = 1/2

When x = 3:

1 - 1/3 = 2/3

This DOES increase. Roman numeral II is also correct. We can eliminate answer choices A and D. Although we know that the answer must be C, let's check Roman number III anyways.

III. 1/(x^2 - x)

When x = 2:

1/(2^2 - 2)

1/(4 - 2) = 1/2

When x = 3:

1/(3^2 - 3)

1/(9 - 3) = 1/6

This DOES NOT increase. Roman numeral III is not correct.

Thus, only I and II are correct.

Answer: C

[[email protected]]

Hi All,

Roman Numeral questions are relatively rare on Test Day (you'll likely see just 1 in the Quant section), but they are almost always designed around Number Properties and logic 'shortcuts'...

Here, we're told that X increase from 165 to 166. We're asked which of the following MUST increase. While this prompt might appear 'calculation heavy', you can actually solve it WITHOUT doing calculations (and paying attention to the answer choices).

I. 2X - 5

The "-5" has the same 'math effect', regardless of what X is: the '-5' reduces the value by 5. So as X goes from 165 to 166, 2X gets BIGGER. Since the '-5' has the same effect, 2X-5 definitely gets bigger as X increases from 165 to 166.
Roman Numeral 1 INCREASES
Eliminate Answers B and E.

II. 1 - 1/X

As the denominator of a fraction gets BIGGER, the fraction gets SMALLER. Thus, 1/165 > 1/166. The "1" in this calculation is constant, so subtracting a SMALLER fraction from it will lead to a BIGGER result. As X increases from 165 to 166, 1/X gets SMALLER, so 1-(1/X) definitely gets bigger.
Roman Numeral 2 INCREASES
Eliminate Answers A and D.

There's only one answer remaining (and we don't even have to deal with Roman Numeral 3)...

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

[nikhilgmat31]

Answer is C by substituting values & subtract 2 values.
It answer is positive than second one is greater

It answer is negative than second one is fewer.