A tank contains $120$ liters of pure juice. $15$ liters are drawn out and replaced by water twice. What is the ratio of juice to water in the final mixture?

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Solution:
A tank contains:
$$ 120 	ext{ liters} $$
Amount drawn out each time:
$$ 15 	ext{ liters} $$
So, the fraction of juice left after each replacement is:
$$ 1 - \frac{15}{120} $$
$$ = 1 - \frac{1}{8} $$
$$ = \frac{7}{8} $$
Since the process is done twice, the amount of juice left is:
$$ 120 	imes \left(\frac{7}{8}ight)^2 $$
$$ = 120 	imes \frac{49}{64} $$
$$ = \frac{735}{8} $$
$$ = 91.875 	ext{ liters} $$
Now, total mixture is still:
$$ 120 	ext{ liters} $$
So, amount of water in the final mixture is:
$$ 120 - 91.875 $$
$$ = 28.125 	ext{ liters} $$
Now, the ratio of juice to water is:
$$ 91.875 : 28.125 $$
Multiply both by 8 to remove decimals:
$$ 735 : 225 $$
Divide by 15:
$$ 49 : 15 $$
Therefore, the ratio of juice to water in the final mixture is:
$$ \boxed{49 : 15} $$
Final Answer: **49 : 15**

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