A certain sum of money amounts to Rs 1008 in 2 years and to Rs 1164 in 3.5 years. Find the rate of interest.

Given:

Amount (A₁) = ₹1008 in Time (t₁) = 2 years

Amount (A₂) = ₹1164 in Time (t₂) = 3.5 years

Formula Used:

Simple Interest (SI) = Amount (A) - Principal (P)

Simple Interest (SI) = (P × r × t) / 100

Where P = Principal, r = Rate of interest per annum, t = Time in years.

Calculations:

The simple interest earned between 2 years and 3.5 years (i.e., in 3.5 - 2 = 1.5 years) is:

SI for 1.5 years = A₂ - A₁ = ₹1164 - ₹1008 = ₹156

The simple interest earned per year is:

SI for 1 year = ₹156 / 1.5 = ₹104

The simple interest earned in 2 years is:

SI for 2 years = ₹104 × 2 = ₹208

Now, we can find the Principal (P) using the amount after 2 years:

A₁ = P + SI for 2 years

₹1008 = P + ₹208

P = ₹1008 - ₹208 = ₹800

SI for 1 year = (P × r × 1) / 100

₹104 = (₹800 × r × 1) / 100

₹104 = 8r

r = ₹104 / 8 = 13

The rate of interest is 13% per annum.

∴ The rate of interest is 13%.