CSAT Solved Papers/ 2023/Q58

2023 CSAT — Q58

Quant Data sufficiency 2.5 marks Hard

In a party, $75$ persons took tea, $60$ persons took coffee and $15$ persons took both tea and coffee. No one taking milk takes tea. Each person takes at least one drink.

Question: How many persons attended the party?

Statement-1: $50$ persons took milk.

Statement-2: Number of persons who attended the party is five times the number of persons who took milk only.

Which one of the following is correct in respect of the above Question and the Statements?

A The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone Answer
B The Question can be answered by using either Statement alone
C The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
D The Question cannot be answered even by using both the Statements together

Worked rationale

From the stem (independent of the statements): tea-or-coffee drinkers $= 75 + 60 - 15 = 120$ . Since “no one taking milk takes tea,” a person who took milk only (milk, no tea, no coffee) is disjoint from these $120$ . Every attendee is either in the $120$ or is a milk-only person, so

\text{Total} = 120 + (\text{milk-only}).

Statement-1 alone ( $50$ took milk): “milk” includes those who also took coffee, so milk-only is unknown ( $\le 50$ ). Total $= 120 + \text{milk-only}$ stays undetermined. Insufficient.

Statement-2 alone (Total $= 5 \times \text{milk-only}$ ): combine with Total $= 120 + \text{milk-only}$ :

5M = 120 + M \;\Rightarrow\; 4M = 120 \;\Rightarrow\; M = 30,\quad \text{Total} = 150.

A single value — sufficient.

So one statement (St-2) answers it, the other (St-1) does not.

Answer: (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

Why the other options miss

B

thought a statement was enough when it wasn’t: thinks ” $50$ took milk” pins the total, ignoring that milk-and-coffee overlap leaves milk-only unknown.
C

missed that one statement already closes it: doesn’t notice St-2 alone closes the system via the known $120$ , so wrongly insists both are needed.
D

solved a differently-constrained problem: misses that tea-or-coffee $= 120$ is fixed by the stem, treating the whole thing as under-determined.

Specialist insight

The hinge is the stem fact tea-or-coffee $= 120$ , and that milk-only is disjoint (milk-drinkers never take tea, and “milk only” excludes coffee too). That makes Total $= 120 + M$ a built-in equation, so St-2’s “Total $= 5M$ ” alone solves $M = 30$ , Total $= 150$ . St-1’s ” $50$ took milk” is a decoy — it counts milk-and-coffee people who are already inside the $120$ , so it can’t isolate $M$ . Reading “milk only” vs “milk” precisely is the whole game.

The trap, in one line

Tea-or-coffee $=120$ (stem); Total $=120 + M$ , so St-2's Total $=5M$ gives $M=30$ , Total $=150$ alone — St-1 can't isolate milk-only $\Rightarrow$ (a).

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