IPMAT Indore 2023 · Quantitative Ability · Question 21

Step 1 — lay out the two partitions. The 120 students are split two independent ways:

By stream: Science 80, Commerce 120-80=40

By team: Mumbai Indians 70, Chennai Super Kings 120-70=50

Nothing links the two, so the overlap is not determined — only bounded.

Step 2 — find the minimum overlap. To make Science ∩ MI as small as possible, push as many MI supporters as possible into Commerce. Commerce has only 40 students, so at most 40 of the 70 MI supporters can be from Commerce:

\text{Science}\cap \text{MI}\ge 70-40=30

Step 3 — find the maximum overlap. All 70 MI supporters could be Science students (Science has 80 places, which is more than enough):

\text{Science}\cap \text{MI}\le 70

Step 4 — state the range.

30\le \text{Science}\cap \text{MI}\le 70

Among the options, only “30 or more” is consistent with this. The others assert a cap of 20 or 25, neither of which the data supports.

Answer: Option 1 — 30 or more

Shortcut. The minimum overlap of two sets is the standard formula

n\left( A\cap B \right) \ge n\left( A \right)+n\left( B \right)-n\left( U \right)=80+70-120=30

and the maximum is \min \left( 80,70 \right)=70. So the answer is the interval \left[ 30,70 \right].

Reading the options is the real skill here. Three of them claim an upper bound of 20 or 25, but the overlap can be as high as 70, so they are all impossible. Only the open-ended “30 or more” survives.

Recognise the question type: when a problem gives two independent classifications and no link between them, it is asking for a bound, not a value — and the answer will be phrased as a range or an inequality.