Need help! Is there a quick technique?

I encountered the following question. You can brute-force the calculation, but I'm curious to know if there is an easy rule or intuition that I'm missing....

Which is bigger,

100000001/100000002 or
200000001/200000002

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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Comments

They are both slightly less than 1. Which is closer to 1?

Jon Haussmann - 3 years, 8 months ago

That's the question!

Brad Carson - 3 years, 8 months ago

If we want to see which is closer to 1, we should compare $1 - \frac{100000001}{100000002} \quad \text{and} \quad 1 - \frac{200000001}{200000002}.$ What does that give you?

Jon Haussmann - 3 years, 8 months ago

Hint: If $a > b> 0$ , then $\dfrac1a < \dfrac 1b$ , and also $1 - \dfrac1a > 1 - \dfrac 1b$ .

Pi Han Goh - 3 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$