A certain sum of money amounts to Rs. 1,860 in 2 years and to Rs. 2,130 in \(3\frac{1}{2}\) years at simple interest. Find the sum and the rate of interest, respectively.

Given:

A certain sum of money amounts to Rs. 1,860 in 2 years and to Rs. 2,130 in 3¹⁄₂ years at simple interest.

Concept:

Simple Interest (SI) is calculated using the formula:
SI = Principal (P) × Rate (R) × Time (T) / 100

Amount (A) is given by:
A = Principal (P) + Simple Interest (SI)

Calculation:

Let the principal be P and the rate of interest be R%.

For the first condition:

⇒ A₁ = P + (P × R × 2 / 100) = 1860

⇒ P (1 + 2R / 100) = 1860     ...(i)

For the second condition:

⇒ A₂ = P + (P × R × 3.5 / 100) = 2130

⇒ P (1 + 3.5R / 100) = 2130     ...(ii)

From equation (i):

⇒ P (1 + 2R / 100) = 1860

⇒ P = 1860 / (1 + 2R / 100)     ...(iii)

Substitute P from equation (iii) into equation (ii):

⇒ (1860 / (1 + 2R / 100)) × (1 + 3.5R / 100) = 2130

⇒ 1860 (1 + 3.5R / 100) = 2130 (1 + 2R / 100)

⇒ 1860 + 1860 × 3.5R / 100 = 2130 + 2130 × 2R / 100

⇒ 1860 + 65.1R = 2130 + 42.6R

⇒ 22.5R = 270

⇒ R = 270 / 22.5

⇒ R = 12%

Substitute R = 12% back into equation (iii) to find P:

⇒ P = 1860 / (1 + 2 × 12 / 100)

⇒ P = 1860 / 1.24

⇒ P = 1500

∴ The sum is Rs. 1,500 and the rate of interest is 12%.