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.

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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