A bottle is full of antiseptic liquid. One third of it is taken out and then an equal amount of water is poured into the bottle to fill it. This process is repeated three times. If the final ratio of antiseptic liquid and water is \( \displaystyle\frac{p}{q} \) where \( \gcd(p,q)=1 \), then \( (p+q) \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 27
Each time one third of the mixture is taken out and replaced by water, \( \frac{2}{3} \) of the antiseptic present remains. For a repeated replacement,
\( \displaystyle \text{Antiseptic left}=\text{Initial}\times\left(1-\frac{1}{3}\right)^{n} \)
With the bottle of volume 1 and n = 3:
\( \displaystyle \text{Antiseptic}=\left(\frac{2}{3}\right)^3=\frac{8}{27} \)
\( \displaystyle \text{Water}=1-\frac{8}{27}=\frac{19}{27} \)
\( \displaystyle \frac{p}{q}=\frac{8}{19},\ \gcd(8,19)=1 \)
\( \displaystyle p+q=8+19=27 \)