One can purchase a flat from a house building society for ₹ 55000 cash or on the terms that he should pay ₹ 4275 as cash down payment and the rest in three equal instalments. The society charges interest at the rate of 16% per annum compounded half-yearly. If the flat is purchased under instalment plan, find the value of each instalment.

### Concept & Half-Yearly Compounding Instalments If interest is compounded half-yearly, the annual rate must be halved, and the instalments are paid per half-year. The principal to be compounded is the cash price minus the down payment. $$P = \sum_{t=1}^{n} \frac{x}{\left(1 + \frac{R_{half}}{100}\right)^t}$$ ### Step-by-Step Solution * **Given:** Cash price = ₹ 55000, Down payment = ₹ 4275. * Principal balance $P = 55000 - 4275 = 50725$. * Rate $R = 16\%$ p.a., so half-yearly rate $R_{half} = \frac{16\%}{2} = 8\% = \frac{8}{100} = \frac{2}{25}$. * The discounting factor is $1 + \frac{2}{25} = \frac{27}{25}$. Let each instalment be $x$. * $50725 = \frac{x}{\frac{27}{25}} + \frac{x}{\left(\frac{27}{25}\right)^2} + \frac{x}{\left(\frac{27}{25}\right)^3}$ * $50725 = x \left[ \frac{25}{27} + \frac{625}{729} + \frac{15625}{19683} \right]$ * Take LCM of 27, 729, and 19683, which is 19683. * $50725 = x \left[ \frac{25 \times 729 + 625 \times 27 + 15625}{19683} \right]$ * $50725 = x \left[ \frac{18225 + 16875 + 15625}{19683} \right]$ * $50725 = x \left[ \frac{50725}{19683} \right]$ * The $50725$ on both sides cancels out perfectly! * $x = 19683$. ### Exam Strategy & Shortcut In problems structured for competitive exams, the sum inside the bracket's numerator will frequently perfectly cancel out the initial principal. Notice the denominator $27^3 = 19683$. It is a very strong hint that the answer will just be the denominator itself. ### Common Pitfall Using the full 16% rate as the compounding factor instead of halving it to 8% for half-yearly periods is the most common error in this specific type of question. ### Final Answer Therefore, the correct answer is **₹ 19683**.