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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?

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.

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

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25 Jul 2016, 21:18

1

This post was BOOKMARKED

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.

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

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18 Aug 2016, 01:39

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.

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.

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.

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

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18 Aug 2016, 05:44

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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, 22:06

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.

Answer: C.

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

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.

Answer: C
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