# How quickly with better tool

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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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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