A housing society collected money for a garden. Each family gave an amount (in Rs) equal to four times the number of families, and the builder gave three times the number of families. Had each family given three times the number of families and the builder five times the number of families, Rs 528 less would have been collected. How many families are in the society?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 24
Write both totals in terms of the number of families \(n\); their difference is Rs 528.
Actual collection \(=n\times 4n+3n=4n^{2}+3n\)
Other plan \(=n\times 3n+5n=3n^{2}+5n\)
Difference \(=n^{2}-2n=528\)
\(n=24\) gives \(24^{2}-2\times24=528\), and the difference grows as \(n\) grows, so no other positive whole number works
The number of families is 24.
Why the other options are incorrect
Option (2) 23 — Incorrect. With 23 families the difference is \(23^{2}-2\times23=483\), not Rs 528.
Option (3) 25 — Incorrect. With 25 families the difference is \(25^{2}-2\times25=575\), not Rs 528.
Option (4) 28 — Incorrect. With 28 families the difference is \(28^{2}-2\times28=728\), not Rs 528.