Rishi has ₹1,612 with him. He divided it amongst his sons Shan and Piyush and asked them to invest it at 8% rate of interest per annum, compounded annually. It was seen that Shan and Piyush got the same amount after 19 and 20 years, respectively. How much (in ₹) did Rishi give to Shan?

Given:

Total money (₹) = ₹1,612

Rate of interest (r) = 8% per annum

Time for Shan (t₁) = 19 years

Time for Piyush (t₂) = 20 years

Amount received by both = Equal

Formula used:

A = P × (1 + r/100)^t

Calculation:

Let the amount given to Shan be ₹x. Then, the amount given to Piyush will be ₹(1612 - x).

Using the formula for compound interest:

P × (1 + r/100)^t₁ = P × (1 + r/100)^t₂

⇒ x × (1 + 8/100)¹⁹ = (1612 - x) × (1 + 8/100)²⁰

⇒ x × (1.08)¹⁹ = (1612 - x) × (1.08)²⁰

⇒ x × (1.08)¹⁹ = (1612 - x) × 1.08 × (1.08)¹⁹

Cancel out (1.08)¹⁹:

⇒ x = (1612 - x) × 1.08

⇒ x = 1612 × 1.08 - 1.08x

⇒ x + 1.08x = 1612 × 1.08

⇒ 2.08x = 1740.96

⇒ x = 1740.96 / 2.08

⇒ x = 837

∴ The correct answer is option 2 (₹837).