In a family, there are ‘n’ persons. The expenditure of rice per month is directly proportional to 4 times the square of the number of persons of the family. If one of them left the family there was a decrease in consumption of 28 kg rice per month. Then initially how many persons were in the family?

TITA type i.e. Type In The Answer type

4

Initial consumption = \(k \times 4{n}^{2}\)

Final consumption = \(k \times 4 {(n -1)} ^{2}\)

It is given that: \(k \times \left(4{n}^{2}- 4\left({n}^{2}\; – 2n + 1\right) \right) = 28\)

⇒ k × 4 × (2n – 1) = 28

⇒ k × (2n – 1) = 7

7 is a prime number and (2n – 1) is a whole number.

And 7 can be factorised as 1 × 7 or 7 × 1.

If (2n – 1) = 1, then n = 1 and after 1 person reduces, then the number of members in the family will become 0 and this case will become a trivial solution.

Thus, (2n – 1) = 7 and we get n = 4.