Question

The difference between the compound interest and simple interest on ₹ 20000 for 2 years is ₹ 32 at the same rate of interest. Find the rate of interest.

Answer 1

Given:

• Principal (P) = ₹ 20,000
• Time (T) = 2 years
• Difference between Compound Interest (C.I.) and Simple Interest (S.I.) = ₹ 32
• Rate of interest = R to be found

Step-wise calculation:

1. Formula for Simple Interest (S.I.):

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

For 2 years,

`S.I. = (20000 xx R xx 2)/100`

`S.I. (40000R)/100`

S.I. = 400R

2. Formula for Compound Interest (C.I.) for 2 years: 

`C.I. = P(1 + R/100)^2 - P`

This can be expanded:

`C.I. = 20000(1 + R/100)^2 - 20000`

3. Difference between C.I. and S.I. is given as ₹ 32: 

C.I. – S.I. = 32

Substitute C.I. and S.I.: 

`20000(1 + R/100)^2 - 20000 - 400R = 32`

4. Simplify:

`20000[(1 + R/100)^2 - 1] = 32 + 400R`

5. Using the identity:

(1 + x)² – 1 = 2x + x² where `x = R/100`:

`20000(2 R/100 + (R/100)^2) = 32 + 400R`

6. Multiply terms inside the bracket:

`20000((2R)/100 + R^2/10000) = 32 + 400R`

Multiply by 20000:

`20000 xx (2R)/100 + 20000 xx R^2/10000 = 32 + 400R`

Simplify:

400R + 2R² = 32 + 400R

7. Cancel 400R on both sides: 2R² = 32

8. Solve for R:

R² = 16

⇒ R = 4

Hence, the rate of interest is 4%.