# Simple interest on a certain sum of money at 9% is Rs 450 in 2 years. Find the compound interest on the same sum at the same rate for 1 year if the interest is reckoned half-yearly.

Simple interest refers to the interest charged on a certain sum at a certain rate of interest per year.

Compound interest refers to the addition of the interest to the principal and then compound interest is charged on the whole sum.

The main difference between simple interest and compound interest is that for simple interest, the interest is the same for all the years whereas the compound interest the interest for every year increases.

## Answer: For a certain principal amount whose simple interest at 9% is Rs 450 in 2 years, the compound interest on the same sum at the same rate for 1 year if the interest is reckoned half-yearly is Rs 230.0625.

Simple interest for a given principal (P) sum of money, for some time (T) in years at a certain rate (R) percent per year is given as:

SI = (P × R × T)/100

Compound interest for the same principal sum for the same time and for a certain period of time is given as:

Compound interest = Amount - Principal, where Amount, A = P(1 + R/100)^{T}.

**Explanation:**

Let the principal sum be Rs 'P'. The given values are as follows:

Simple interest = Rs 450

Rate% = 9%

Time = 2 years

Now using the simple interest formula to find the principal amount:

SI = (P × R × T)/100

or, 450 = (P × 9 × 2)/100

or, 18P/100 = 450

or, P = (450 × 100)/18

or, P = 45000/18

or, P = Rs 2500

Now let's calculate the compound interest for the available values:

Principal, P = Rs 2500

Since the interest is compounded half-yearly, there are 2 half years in one year.

Time = 1 year = 2 half-years

Rate%, = 9% per annum = 9/2% per half year

Now Amount, A = P(1 + R/100)^{T}

or, A = 2500 (1 + 9/200)^{2}

or, A = 2500 × (209/200)^{2}

or, A = 2500 × (209/200) × (209/200)

or, A = (2500 × 209 × 209)/(200 × 200)

or, A = 2730.o625

Since the amount is Rs 2730.0625 and the principal is Rs 2500, therefore, compound interest is calculated as:

Compound Interest = Amount - Principal

⇒ CI = A - P

⇒ CI = 2730.0625 - 2500 = Rs 230.0625