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# The compound interest on Rs $100000$ at $20\%$ per annum for $2$ years $3$ months compounded annually is:A. Rs $151200$ B. Rs $100000$ C. Rs $51200$ D. Rs $251200$  Hint: In the given question first we will calculate the total time period in years. Then we will apply the formula to find an interest for the given time period. First, we will calculate the interest for the first year and add the original principal to get second-year principal. Then we will calculate the interest for the second year and add it in the second year principal to get the principal for the rest of the time. Then we will find the interest in the remaining time period and make a new amount, finally, we will subtract this amount from the given principal we will get the correct answer.

From the question:
Given that:
$P = 100000 \\ r = 20\% = \dfrac{{20}}{{100}} = 0.2 \\ n = 2{\text{ }}years{\text{ }}3{\text{ }}months \\$
Where $P =$ Principal
$r =$ Rate of interest
$n =$ Time period
Formula for compound interest:
$A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$
Convert the value of n in years.
We get:
$n = 2 + \dfrac{1}{4} = \dfrac{{8 + 1}}{4} \\ \Rightarrow n = \dfrac{9}{4} = 2.25 \\$
Now the time period is $2.25$ years.
Form the question:
Given that:
Interest is compounded annually:
For first year:
Interest$= 100000 \times 0.2 \times 1 = 20000$
Thus for second year ${P_{new}} = 100000 + 20000 = 120000$
Interest for second year $120000 \times 0.2 \times 1 = 24000$
Thus for last year ${P_{new}} = 120000 + 24000 = 144000$
Thus we get the interest for complete two years.
Now interest for rest of $0.25$ years$=$ $144000 \times 0.2 \times 0.25 = 7200$
Now the new amount is $= 14400 + 7200 = 151200$
Thus the interest is $= 151200 - 100000 = 51200$
This is the compound interest on Rs $100000$ at $20\%$ per annum for $2$ year’s $3$ months.
Hence the correct answer is option C.

Note: For the given question first we have to convert the given time in years without this we cannot do this problem. After that, we have to find the interest for the first year as well as for the second year and then for the rest of the time i.e. $0.25$. It means we have to calculate the interest for the complete two years and for the remaining $0.25$years. Thus we get the correct answer.
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