CAT 2023 Slot 1 · Quant · Question 11

Let the jar hold V units and let a units be removed and replaced each time.

Each replacement keeps the fraction \left( 1-\frac{a}{V}\right) of whatever coffee was present, since the removed portion is a uniform mixture.

After two such operations the coffee fraction is {\left( 1-\frac{a}{V}\right)}^{2}.

In mixture Q the ratio of coffee to cocoa is 16:9, so coffee is \frac{16}{25} of the whole.

Therefore {\left( 1-\frac{a}{V}\right)}^{2}=\frac{16}{25}, so 1-\frac{a}{V}=\frac{4}{5} and \frac{a}{V}=\frac{1}{5}.

Take the jar as 25 units for convenience, so 5 units are removed and replaced each time.

Mixture P is formed by one replacement, so its cocoa is exactly the 5 units just added.

Mixture Q has coffee \frac{16}{25}\times 25=16 units, so its cocoa is 25-16=9 units.

The required ratio is cocoa in P to cocoa in Q, that is 5:9 — the third option.

Shortcut. Reading \frac{16}{25} as {\left( \frac{4}{5}\right)}^{2} gives the replacement fraction \frac{1}{5} at sight. Scaling the jar to 25 units then makes both quantities whole numbers — 5 after one step and 9 after two — with no fractions anywhere.