CAT 2024 Slot 3 · Quant · Question 5

Let the number of girls be g, so the number of boys is 250-g.

The swimmers number 0.5\left( 250-g \right)+0.8g, and the runners number 0.7\left( 250-g \right)+0.6g.

Every student opted for at least one activity, so the union of the two groups is the whole 250.

By the inclusion-exclusion principle, \text{both}=\text{swimmers}+\text{runners}-\text{union}.

Adding the two counts, swimmers plus runners is 1.2\left( 250-g \right)+1.4g=300+0.2g.

Subtracting the union of 250, the number doing both is 0.2g+50.

This is an increasing function of g, so the extremes occur at the extreme values of g.

The percentage of girls runs from 44% to 60%, so g runs from 0.44\times 250=110 to 0.60\times 250=150.

At g=110 the count doing both is 0.2\times 110+50=72.

At g=150 the count doing both is 0.2\times 150+50=80.

So the minimum is 72 and the maximum is 80, the first option.

Shortcut. Since all 250 are covered, the overlap is simply \left( \text{swim}\%+\text{run}\%-100\% \right) applied group by group: boys contribute 50+70-100=20\% and girls contribute 80+60-100=40\%. The overlap is then 0.2\left( 250-g \right)+0.4g=50+0.2g in one line, with no inclusion-exclusion bookkeeping.