For city C, if the number of people who will turn 21 years of age next year is…

Concept: When a population is divided in the ratio a : b : c, one part equals the total divided by (a + b + c), so every group size is a whole-number multiple of that one part. Over a year the population changes by cohort flow: a birth adds a person to the youngest group only, while crossing an age boundary transfers a person out of one group and into the next without changing the total. So rebuild each group’s head-count from its own inflows and outflows, then reduce the three new counts by their HCF to read off the new ratio.

Applying it to city C

1. City C has 19.2 lakh = 19,20,000 people in the ratio 1 : 1 : 2. The number of parts is 1 + 1 + 2 = 4, so one part = 19,20,000 ÷ 4 = 4,80,000.

2. Present head-counts: Group I (0 to under 21) = 4,80,000; Group II (21 to under 41) = 4,80,000; Group III (41 and above) = 2 × 4,80,000 = 9,60,000.

3. Group I next year: it loses the 16,000 who turn 21 and gains the 32,000 newborns, giving 4,80,000 − 16,000 + 32,000 = 4,96,000.

4. Group II next year: the same 16,000 who leave Group I cross into it, and no outflow from it is stated, giving 4,80,000 + 16,000 = 4,96,000.

5. Group III next year: the question fixes it, so it stays at 9,60,000.

6. New ratio = 4,96,000 : 4,96,000 : 9,60,000. Dividing all three by their HCF, 16,000, gives 31 : 31 : 60.

Cross-check: The reduced terms total 31 + 31 + 60 = 122 parts, and 122 × 16,000 = 19,52,000. That is exactly the original 19,20,000 plus the 32,000 newborns — the only net addition to the city, since everyone who turns 21 merely moves between groups. The reduction is therefore consistent.

Where this calculation usually goes wrong

• The 16,000 who turn 21 are a transfer, not an arrival. They must be subtracted from Group I as well as added to Group II; counting them on one side only breaks the total.

• The 32,000 newborns join Group I alone. Putting them into Group II instead would make the middle count 4,80,000 + 32,000 = 5,12,000 and change the middle term of the ratio.

• Group III is pinned at 9,60,000, so whatever reduced ratio you write, its third term multiplied by the common part must return 9,60,000 — a quick filter before you commit to an answer.