Raj invests Rs. 60,000 in a scheme that offers 10% annual interest, compounded half-yearly, for 4 years. Rahul invests in the same scheme for 3 years, after which he reinvests the entire amount received for 1 additional year at 8% simple interest. If both Raj and Rahul receive the same amount at the end of the 4^th year, what was the initial amount invested by Rahul?

Using the compound interest formula for half-yearly compounding, Raj's final amount is 60000 times (1.05) raised to the power of 8 (since 4 years have 8 half-year periods), yielding Rs. 88,647.30. Rahul invests in the same scheme for 3 years (6 periods), so his amount after 3 years is P times (1.05) raised to the power of 6, which equals 1.3400956P. He then reinvests this at 8% simple interest for 1 year, making his final amount 1.3400956P times 1.08, which is 1.447303P. Equating this to Raj's amount gives 1.447303P equals 88647.30, so P equals Rs. 61,250.

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