A certain sum of money amounts to Rs. 6600 in 4 years at a certain rate percent simple interest. If the rate of interest be increased by its 25%, the same sum would amount to Rs. 7000 during the same period. Find the sum.

Given:

Amount (A) = ₹6600 in 4 years at certain rate

Amount (A) = ₹7000 in 4 years with 25% increase in rate

Formula used:

Simple Interest formula: \(A = P + \frac{P \times R \times T}{100}\)

Where P = Principal, R = Rate of interest, T = Time (in years), A = Amount

Calculations:

Let the principal be P and the rate be R.

For the first case:

6600 = P + (P × R × 4) / 100

⇒ 6600 = P(1 + 4R / 100)

⇒ P(1 + 4R / 100) = 6600 ---- (1)

For the second case:

7000 = P + (P × (1.25R) × 4) / 100

⇒ 7000 = P(1 + 5R / 100)

⇒ P(1 + 5R / 100) = 7000 ---- (2)

Now, subtract equation (1) from equation (2):

7000 - 6600 = P(1 + 5R / 100) - P(1 + 4R / 100)

⇒ 400 = P(1 + 5R / 100 - 1 - 4R / 100)

⇒ 400 = P(R / 100)

⇒ P = 400 × 100 / R

Substitute P in equation (1):

6600 = (400 × 100 / R)(1 + 4R / 100)

⇒ 6600 = (400 × 100 / R)(100 + 4R) / 100

⇒ 6600R = 400 × (100 + 4R)

⇒ 6600R - 1600R = 40000

⇒ 5000R = 40000

⇒ R = 8%

Substitute R = 8% into the equation for P:

P = 400 × 100 / 8 = 5000

∴ The sum (principal) is ₹5000.