CAT 2022 Slot 1 · Quant · Question 14

Let the juice cost j per kg, so the syrup, being 20% cheaper, costs 0.8j per kg.

His total outlay is 110\times 0.8j+120\times j.

That is 88j+120j=208j.

He sells 10 kg of syrup at a 10% profit, fetching 10\times 1.1\times 0.8j=8.8j.

He sells 20 kg of juice at a 20% profit, fetching 20\times 1.2j=24j.

That leaves 110-10=100 kg of syrup and 120-20=100 kg of juice, a 200 kg mixture.

The mixture sells at ₹308.32 per kg, fetching 200\times 308.32=61664.

His total revenue is 8.8j+24j+61664=32.8j+61664.

An overall profit of 64% means the revenue is 1.64 times the outlay: 1.64\times 208j=341.12j.

Equating, 32.8j+61664=341.12j.

So 61664=308.32j, giving j=200.

The syrup costs 0.8\times 200=160, that is ₹160 per kg.

Shortcut. Notice that 308.32 appears on both sides in disguise: the mixture revenue is 200\times 308.32 and the surviving coefficient is 341.12-32.8=308.32. So j is simply \frac{200\times 308.32}{308.32}=200, and the syrup is four-fifths of that.