# The compound Interest on a certain sum of money in 2 years is Rs. 920.25 and the simple interest is Rs. 900. Find the yearly interest rate r and the principal amount P.

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If the compound interest is Rs 920.25 and the simple interest is Rs 900, then 920.25-900 = Rs 20.25 is the interest made on the gathering interest *only.*

The simple interest made on a principle P in 2 years (2 periods) is given by the formula

`I_s = 2rP `

where `r ` is the period (yearly here) interest rate.

Since we are given that `I_s = 900 ` then we have the relation

`rP = 450 `

The compound interest made on a P in 2 years is contrastingly given by the formula

`I_c = P(1+r)^2 - P = 2rP + r^2P `

The final term is the pure interest in the second period made on the pure interest from the first period. This pure interest term is not included when calculating the simple interest, which consists of the first term only (2rP).

Since we are given that `I_c = 920.25 ` then we have that the pure interest term

`r^2P = 20.25 `

We now, then, have two formulae in the interest rate r and the principal P, meaning we can solve for both:

`rP = 450 qquad (1)`

` r^2P = 20.25 qquad (2)`

Dividing (2) by (1) gives us `r = 20.25/450 = 0.045 `

and then plugging this into equation (1) we obtain `P = 450/0.045 = 10,000 `

Check that equation (2) adds up: `r^2P = (0.045^2)10000 = 20.25 ` True.

**Therefore the principal was `P = 10,000 ` Rs and the period interest rate is `r =0.045 ` , that is 4.5%.**