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# As x increases from 209 to 210, which of the following must

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As x increases from 209 to 210, which of the following must [#permalink]

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04 Mar 2012, 10:33
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As x increases from 209 to 210, which of the following must increase?

I. $$3x - 4$$
II. $$\frac{x - 1}{x}$$
III. $$x - x^2$$

(A) I only
(B) III only
(C) I and II
(D) I and III
(E) II and III

How come the answer is C guys?
[Reveal] Spoiler: OA

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Last edited by enigma123 on 04 Mar 2012, 14:25, edited 1 time in total.
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04 Mar 2012, 16:30
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enigma123 wrote:
As x increases from 209 to 210, which of the following must increase?

I. $$3x - 4$$
II. $$\frac{x - 1}{x}$$
III. $$x - x^2$$

(A) I only
(B) III only
(C) I and II
(D) I and III
(E) II and III

How come the answer is C guys?

I. $$3x - 4$$ --> $$x$$ increases from 209 to 210 --> $$3x$$ increases --> $$3x-4$$ increases. Correct.

II. $$\frac{x - 1}{x}$$ --> $$\frac{209-1}{209}=\frac{208}{209}$$ which is less than $$\frac{210-1}{210}=\frac{209}{210}$$, (the same way as 1/2 is less than 2/3). Correct.

III. $$x - x^2$$ --> $$x$$ increases from 209 to 210 --> $$x^2$$ increases more than $$x$$ --> $$x - x^2$$ decreases.

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Re: As x increases from 209 to 210, which of the following must [#permalink]

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25 Jul 2016, 22:18
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When x increases from 209 to 210.

I. 3x-4 will also increase as x is increasing.
II.((x-1)/x) can be written as 1-1/x.
As x increases, 1/x will decrease. That means, lesser value will be subtracted from 1 now. Hence, Increases.
III. As x increases, x^2 increase faster than x. Hence, x-x^2 will decrease.

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Re: As x increases from 209 to 210, which of the following must [#permalink]

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18 Aug 2016, 02:39
What's the best way to compare such large fractions?

Bunuel wrote:
enigma123 wrote:
As x increases from 209 to 210, which of the following must increase?

II. $$\frac{x - 1}{x}$$ --> $$\frac{209-1}{209}=\frac{208}{209}$$ which is less than $$\frac{210-1}{210}=\frac{209}{210}$$, (the same way as 1/2 is less than 2/3). Correct.

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Re: As x increases from 209 to 210, which of the following must [#permalink]

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18 Aug 2016, 03:05
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ameyaprabhu wrote:
What's the best way to compare such large fractions?

Bunuel wrote:
enigma123 wrote:
As x increases from 209 to 210, which of the following must increase?

II. $$\frac{x - 1}{x}$$ --> $$\frac{209-1}{209}=\frac{208}{209}$$ which is less than $$\frac{210-1}{210}=\frac{209}{210}$$, (the same way as 1/2 is less than 2/3). Correct.

Hey ameyaprabhu,

To understand the nature of a fraction, you should first simplify the fraction as far as possible.

So, in this case you can rewrite $$\frac{(x-1)}{x} = \frac{x}{x} – \frac{1}{x} = 1 – \frac{1}{x}.$$

Now, from this simplified version, you can see that x is in the denominator. So, as x increases, the value of $$\frac{1}{x}$$ would decrease. Since $$\frac{1}{x}$$ is being subtracted from 1, the decrease in the value of $$\frac{1}{x}$$ would result in increase in value of the overall fraction.

For example: consider x = 2. So, we have $$1 - \frac{1}{x} = 1 - \frac{1}{2} = \frac{1}{2}$$. Now, when x = 3, we have $$1 - \frac{1}{x} = 1 - \frac{1}{3} = \frac{2}{3}$$. So, as x increases from 2 to 3, the value of the fraction $$1 - \frac{1}{x}$$ also increases from $$\frac{1}{2}$$ to $$\frac{2}{3}$$.

So, you can now confidently say that as x increases from 209 to 210, the value of $$\frac{(x-1)}{x}$$ would increase as well.

Hope this helps.

Regards,
Harsh
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Re: As x increases from 209 to 210, which of the following must [#permalink]

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18 Aug 2016, 06:44
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I. The answer clearly will increase as a negative constant stays the same (-4) whereas the variable with a positive constant(3x) increases.

II.
Any proper and improper fraction that gets a number added to both the numerator and the denominator gets closer to 1.
100/101 is closer to 1 than 99/100.
100/99 is further away from 1 than 101/100.
As the fraction (x-1)/x is less than 1 it will increase. (note how the question would change if II. stated (x+1)/x In that case the fraction would decrease(towards 1).

III.
The positive part (x) gets +1 whereas the negative part -(x^2) receives an exponential increase larger than 1.

I and II must increase. Answer choice: C.
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Re: As x increases from 209 to 210, which of the following must [#permalink]

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11 Oct 2017, 23:06
ameyaprabhu wrote:
What's the best way to compare such large fractions?

Bunuel wrote:
enigma123 wrote:
As x increases from 209 to 210, which of the following must increase?

II. $$\frac{x - 1}{x}$$ --> $$\frac{209-1}{209}=\frac{208}{209}$$ which is less than $$\frac{210-1}{210}=\frac{209}{210}$$, (the same way as 1/2 is less than 2/3). Correct.

One way could be we replace the large number with a smaller one
say X will increase from 3 to 4.
let us first check the extremes.

1- 3x-4=9-4=5 and 12-4=8
8>5-Increase
2. x-1/x=(3-1)/3=2/3=2/3 and (4-1)/4=3/4
2/3>3/4 or 0.6>0.7
3. x-$$x^2$$=3-9=(-6) and 4-16=(-12)
(-6)>(-12)-decrease
hence C
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Re: As x increases from 209 to 210, which of the following must [#permalink]

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16 Oct 2017, 16:54
enigma123 wrote:
As x increases from 209 to 210, which of the following must increase?

I. $$3x - 4$$
II. $$\frac{x - 1}{x}$$
III. $$x - x^2$$

(A) I only
(B) III only
(C) I and II
(D) I and III
(E) II and III

We are being tested on the result when an integer is increased. So, rather than using 209 to 210, let’s use 2 to 3 (since the relationship won’t change).

I.

3(2) - 4 = 2

3(3) - 4 = 5

We see that an increase occurs.

II.

(2 - 1)/2 = 1/2

(3 - 1)/3 = 2/3

We see that an increase occurs.

III.

2 - 2^2 = -2

3 - 3^2 = -6

We see that an increase DOES NOT occur.

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Re: As x increases from 209 to 210, which of the following must   [#permalink] 16 Oct 2017, 16:54
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