Question

The simple interest on a certain sum of money at a certain rate is ₹ 900 in 2 years while the compound interest is ₹ 927. Find the sum and the rate of interest.

Answer 1

Given:

• Simple Interest (S.I.) = ₹ 900 in 2 years
• Compound Interest (C.I.) = ₹ 927 in 2 years
• Find the Principal (sum) and the Rate of Interest (r)

Step 1: Express Simple Interest formula:

`S.I. = (P xx r xx t)/100`

Given S.I. = 900,

t = 2 years,

So, `900 = (P xx r xx 2)/100`

⇒ `900 = (2Pr)/100`

⇒ 900 × 100 = 2Pr

⇒ 90000 = 2Pr

⇒ 2Pr = 90000

or

⇒ Pr = 45000   ...(Equation 1)

Step 2: Express Compound Interest formula:

`C.I. = P(1 + r/100)^t - P`

Given C.I. = 927,

t = 2 years,

`927 = P((1 + r/100)^2 - 1)`

Let `x = 1 + r/100`,

So, 927 = P(x² – 1).

From Equation 1,

`P = 45000/r`

Substitute:

`927 = 45000/r (x^2 - 1)`

Step 3: Rewrite x in terms of r : 

`x = 1 + r/100`

So, `x^2 - 1 = (1 + r/100)^2 - 1`

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

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

Convert to a common denominator:

= `(200r + r^2)/10000`

= `(r(200 + r))/10000`

Plug this into the C.I. equation: 

`927 = 45000/r xx (r(200 + r))/10000`

Simplify (r) in the numerator and denominator:

`927 = (45000(200 + r))/10000`

Multiply both sides by 10000:

927 × 10000 = 45000(200 + r)

9270000 = 45000(200 + r)

Divide both sides by 45000:

`9270000/45000 = 200 + r`

206 = 200 + r

Solve for r:

r = 206 – 200

r = 6 

So the rate of interest is 6%.

Step 4: Find the principal using Equation 1:

Pr = 45000

⇒ P × 6 = 45000

⇒ `P = 45000/6`

⇒ P = 7500