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I want to compare two following fractions $$ a = \frac{1}{411}\cdot\frac{1}{412}\cdot\frac{1}{413}; b = \frac{1}{63990006} $$

I want to compare these without any calculations, but simply manipulating, something like this:

$$ a=\frac{791}{993}; b=\frac{792}{991} $$

$$ a=\frac{791}{993}\cdot\frac{991}{991}=\frac{(792-1)\cdot(993-2)}{993\cdot991}=\frac{792\cdot993-2\cdot792-993+2}{993\cdot991}=\frac{792\cdot993-(2\cdot792+991)}{993\cdot993} $$

$$ b=\frac{792\cdot993}{991\cdot993} $$ so as the numerator of a is less than b's then $a < b$.

I am not sure if such an approach is applicable at all.

P.S. I see that the example provided does not need any manipulations, it's just for demonstration.

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MisterAlejandro
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1 Answers1

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Note that

$$400^3=64\,000\,000$$

therefore

$$a = \frac{1}{411}\cdot\frac{1}{412}\cdot\frac{1}{413}< b = \frac{1}{63\,990\,006}$$

gen-ℤ ready to perish
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user
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  • It seems that you did not check my example out properly, as a result did not get what I asked about. My question is about practicing juggling numbers. Process sometimes is more important that result. Don't understand why so many up votes. – MisterAlejandro Jun 13 '18 at 19:38
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    @MisterAlejandro As in your example here we have that denominator for $a$ is greater then denominator for $b$ and thus $a<b$. For that comparison of course we can manipulate a little bit the expression as for example $(411\cdot 412 \cdot 413)=(400+11)\cdot (400+12) \cdot (400+13)>400^3>63,990,006$ or in others similar ways. – user Jun 13 '18 at 19:49