The difference between the total number of people in age group II of city A…

Concept

When a total quantity is divided in a given ratio, every term of the ratio stands for one equal share, or “part”. The size of one part is the total divided by the sum of the ratio terms, and the count in any category equals that category's ratio term multiplied by the size of one part.

Each row of such a table carries its own total and its own sum of ratio terms, so one part is worth a different amount in every row. Counts belonging to different rows can therefore be added or compared only after each has been converted into an absolute number of people.

Applying it here

1. City A has 14.4 lakh people in the ratio 1 : 3 : 2, so the terms add to 1 + 3 + 2 = 6 and one part = 14.4 ÷ 6 = 2.4 lakh. Age group II carries 3 parts, giving 3 × 2.4 = 7.2 lakh.

2. City B has 16 lakh people in the ratio 2 : 3 : 3, so the terms add to 2 + 3 + 3 = 8 and one part = 16 ÷ 8 = 2 lakh. Age group II carries 3 parts, giving 3 × 2 = 6 lakh.

3. Age group II for cities A and B together = 7.2 + 6 = 13.2 lakh.

4. City C has 19.2 lakh people in the ratio 1 : 1 : 2, so the terms add to 1 + 1 + 2 = 4 and one part = 19.2 ÷ 4 = 4.8 lakh. Age group III carries 2 parts, giving 2 × 4.8 = 9.6 lakh.

5. City E has 12 lakh people in the ratio 1 : 3 : 1, so the terms add to 1 + 3 + 1 = 5 and one part = 12 ÷ 5 = 2.4 lakh. Age group III carries 1 part, giving 1 × 2.4 = 2.4 lakh.

6. Age group III for cities C and E together = 9.6 + 2.4 = 12 lakh.

7. The required difference = 13.2 − 12 = 1.2 lakh, that is 1,20,000 people.

Cross-check

The four conversions used above are collected below.

Two independent checks confirm these figures. First, the three group counts of a city must add back to its own total: A gives 2.4 + 7.2 + 4.8 = 14.4, B gives 4 + 6 + 6 = 16, C gives 4.8 + 4.8 + 9.6 = 19.2 and E gives 2.4 + 7.2 + 2.4 = 12.

Second, taking each share directly as a fraction of its city's total reaches the same sums without computing one part: (3 ÷ 6) × 14.4 + (3 ÷ 8) × 16 = 7.2 + 6 = 13.2 lakh, and (2 ÷ 4) × 19.2 + (1 ÷ 5) × 12 = 9.6 + 2.4 = 12 lakh. The difference of 1.2 lakh stands.

Where this goes wrong

Two slips account for most wrong values here. One is reading a city's share from the wrong position in its ratio, such as treating the 2 : 3 : 3 of city B as giving age group II two parts rather than three. The other is comparing age group II and age group III within the same pair of cities instead of the pair each group is actually asked about. In both cases every individual division is carried out correctly, yet the final difference changes.