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If $20$ $20$ quarts of milk contains $2$ $2$ quarts of butter fat, how much pure water would be need to be added to the milk to make it $4 %$ $4%$ butter fat?

1 Answer

Jun 13, 2017

$30$ $color(red)(30)$ quarts of pure water would need to be added.

Explanation:

Suppose that the required amount of water that would need to be added was $x$ $color(red)x$ quarts.

The total resulting volume would be $20 + x$ $color(blue)(20+color(red)x)$

The ratio of butter fat (given $2$ $color(green)2$ quarts of butter fat) to total volume would be:
$XXX \frac{2}{20 + x}$ $color(white)("XXX")color(green)2/(color(blue)20+color(red)x)$

We are told that this ratio needs to be $4 %$ $color(magenta)(4%)$
and
Noting that $4 % = \frac{4}{100} = \frac{2}{50}$ $color(magenta)(4%)=color(magenta)(4/100)=color(magenta)(2/50)$

We have, as a requirement:
$XXX \frac{2}{20 + x} = \frac{2}{50}$ $color(white)("XXX")color(green)2/(color(blue)20+color(red)x)=color(magenta)(2/50)$

$\Rightarrow 20 + x = 50$ $rArr color(blue)(20+color(red)x)=color(green)50$

$\Rightarrow x = 50 - 20 = 30$ $rArr color(red)x= color(green)50-color(blue)20=color(red)30$

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