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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 pq\frac{p}{q}, where gcd⁡(p,q)=1\gcd(p,q)=1, then (p+q)(p+q) is

Solution

✅ Correct Option: 2

Each operation leaves 23\frac23 of the antiseptic present.

After three operations, antiseptic =(23)3=827=\left(\frac23\right)^3=\frac{8}{27} of the bottle, so water =1927=\frac{19}{27}.

Ratio =819=\frac{8}{19} with gcd⁡=1\gcd=1, so p+q=27p+q=27.

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