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If:
I can make two tools which would serve the same purpose and:
I need 10 times less time to make first tool but work with this tool is a half time slower than work with the second tool, after what period of time I will get advantage by making the tool with which I will work faster by 2 times (the second tool).

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2 Answers 2

up vote 6 down vote accepted

If it takes $x$ hours to make the first tool then it will take $10x$ hours to make the second tool. Also, if a job takes $y$ hours with the first tool it will take $\frac{1}{2}y$ hours with the second tool. So you are asking when

$$ x+y\geq 10x+\frac{1}{2}y. $$

Rearranging this inequality indicates

$$ y\geq 18x. $$

So, once your job requires at least $18x$ hours, the second tool is better.

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HINT $\:$ If the slow tool takes $\rm\:T\:$ hours to build, then the fast tool takes an extra $\rm\: 9\ T\:$ hours to build. For the double-speed tool to make up for this lost time in a job of $\rm\ J\ $ hours, its saved time of $\rm\ J/2\ $ hours must equal the lost build time $\rm\ 9\ T\:,\ $ i.e. $\rm\ J/2\ =\ 9\ T\:.$

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