From a container full of orange juice, 7.5 liters was drawn out and replaced by soda water. This process is repeated 5 more time. The ratio of quantity of orange juice and soda water left in the container is 4 : 5. How much liter of orange juice did the container originally had? (Use $(0.44)^{1/6} = 0.802$)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 37.9 litres
Let the container originally have $V$ liters of orange juice.
Each time, 7.5 liters is removed and replaced by soda. Fraction of orange juice remaining each time:
$\text{Remaining fraction} = 1 - \frac{7.5}{V}$
After 6 repetitions, fraction of orange juice left:
$(1 - \frac{7.5}{V})^6$
Given ratio of orange juice to soda water is $4:5$ → fraction of orange juice:
$\frac{4}{4+5} = \frac{4}{9}$
So, $(1 - \frac{7.5}{V})^6 = \frac{4}{9} \Rightarrow 1 - \frac{7.5}{V} = (\frac{4}{9})^{1/6}$
Given $(0.44)^{1/6} = 0.802$, approximate $4/9 \approx 0.444 \approx 0.44$
$1 - \frac{7.5}{V} = 0.802 \Rightarrow \frac{7.5}{V} = 1-0.802=0.198$
$V = \frac{7.5}{0.198} \approx 37.88 \approx 38$ liters