Question

On a certain sum and at a certain rate of interest, the simple interest for first year is ₹ 300 and the compound interest for first 2 years is ₹ 630. Find the sum and the rate of interest.

Answer 1

Given:

• Simple Interest (S.I.) for the first year = ₹ 300
• Compound Interest (C.I.) for the first 2 years = ₹ 630

We need to find:

• The principal sum (P)
• The rate of interest (R)

Step 1: Use the formula for simple interest for the first year:

`S.I. = (P xx R xx 1)/100`

S.I. = 300

⇒ P × R = 300 × 100

⇒ P × R = 30000

So, P × R = 30000   ...(Equation 1)

Step 2: Use the compound interest formula for 2 years:

Compound Interest = `P(1 + R/100)^2 - P = 630`

`P((1 + R/100)^2 - 1) = 630`   ...(Equation 2)

Step 3: Divide Equation 2 by Equation 1 to eliminate (P):

`(P((1 + R/100)^2 - 1))/(P xx R) = 630/30000`

`((1 + R/100)^2 - 1)/R = 21/1000`

Simplify `(1 + R/100)^2 - 1`:

`(1 + R/100)^2 - 1 = (1 + (2R)/100 + R^2/10000) - 1`

`(1 + R/100)^2 - 1 = (2R)/100 + R^2/10000`

Now, `((2R)/100 + R^2/10000)/R = 21/1000`

`(2R)/(100R) + R^2/(10000R) = 21/1000`

`2/100 + R/10000 = 21/1000`

Multiply both sides by 10000:

200 + R = 210

R = 10%

Step 4: Substitute R = 10% in Equation 1: 

P × 10 = 30000

⇒ `P = 30000/10`

⇒ P = 3000