In a city, 45% of people read both English and Hindi newspapers, 15% read only…

Concept: For three overlapping sets, split the total population into every mutually exclusive region: three only-regions, three pairwise-overlap-only regions, and the triple overlap. When a problem forces the triple overlap to zero, a given pairwise overlap becomes one clean, disjoint block, and any two independent overlap variables can be pushed to their extreme values to find the maximum or minimum size of an unknown group.

Application:

1. Let the total population be 100%. Since nobody reads all three newspapers, the triple-overlap region is 0%, so the English-and-Hindi readers (45%) form one clean block containing no Urdu readers.

2. Split Hindi readers into three disjoint pieces: only-Hindi (b), Hindi-and-English (45%, given), and Hindi-and-Urdu (f). Since total Hindi readers are 75%, b + f = 75 minus 45 = 30%.

3. The whole population splits into: only-English (a) + only-Hindi (b) + only-Urdu (15%, given) + Hindi-and-English (45%) + English-and-Urdu (e) + Hindi-and-Urdu (f) = 100%. Substituting the known values gives a + b + e + f = 100 minus 15 minus 45 = 40%.

4. Since b + f = 30%, this reduces to a + e = 10%.

5. Total Urdu readers = only-Urdu + English-and-Urdu + Hindi-and-Urdu = 15 + e + f = 15 + e + (30 minus b) = 45 + (e minus b).

6. e can range from 0 to 10 (because a = 10 minus e must stay at least 0), and b can range from 0 to 30 (because f = 30 minus b must stay at least 0), independently of each other. To maximise Urdu readers, push e to its largest value (10) and b to its smallest (0): maximum Urdu readers = 45 + 10 minus 0 = 55%.

Cross-check: Check by filling every region with e = 10 and b = 0 (so a = 0, f = 30): only-English 0% + only-Hindi 0% + only-Urdu 15% + Hindi-and-English 45% + English-and-Urdu 10% + Hindi-and-Urdu 30% + all-three 0% totals 100%, and Hindi readers = 0 + 45 + 30 = 75% -- both match the given data, and total Urdu readers = 15 + 10 + 30 = 55%.

So the correct reading is Max. 55%.