Question

Find the compound interest on Rs. $ 4000 $ for $ 2\dfrac{1}{2} $ years at $ 10\% $ per annum compounded annually.

Answer 1

Hint: Interest is the amount of money paid for using someone else’s money. There are two types of interest. $ 1) $ Simple Interest and $ 2) $ Compound interest. Here we will use both the concept of interest. First will find the Amount paid for two years and simple interest for half year and then sum both the values for the amount after $ 2\dfrac{1}{2} $ year. Use formula – $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $ Where A is the amount, P is the Principal amount and R is the rate of interest and $ I = \dfrac{{PRT}}{{100}} $ for simple interest.

Complete step-by-step answer:
Sum (P) $ = 4,000 $
Rate of Interest, $ = 10\% $ per annum
Time $ = 2\dfrac{1}{2}{\text{year}} $
Split the given term period as, $ 2\dfrac{1}{2} = 2 + \dfrac{1}{2} $
First take $ n = 2 $ in the below formula,
Amount, $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n} $
Place values in the above equation –
 $ \Rightarrow A = 4000{\left( {1 + \dfrac{{10}}{{100}}} \right)^2} $
Simplify the above equation –
 $ \Rightarrow A = 4000{\left( {1 + \dfrac{1}{{10}}} \right)^2} $
Take LCM (Least common multiple) and simplify
 $ \Rightarrow A = 4000{\left( {\dfrac{{11}}{{10}}} \right)^2} $
Further multiplication and division implies –
 $ \Rightarrow A = 4840{\text{ Rs}}{\text{.}}\;{\text{ }}.....{\text{ (a)}} $
Now, Simple interest for last half year is –
 $ I = \dfrac{{PRT}}{{100}} $
Place values in the above equation, where $ T = \dfrac{1}{2}year $ and $ P = 4840\;{\text{Rs}}{\text{.}} $
 $ I = \dfrac{{4840 \times 10 \times \dfrac{1}{2}}}{{100}} $
Numerator’s denominator goes to denominator-
 $ I = \dfrac{{4840 \times 10 \times 1}}{{100 \times 2}} $
Simplify the above equation –
 $ \Rightarrow I = 242{\text{ Rs}}{\text{.}}\;{\text{ }}....{\text{ (b)}} $
By using equations (a) and (b)-
The Net amount after \[2\dfrac{1}{2}{\text{year is}}\]
 $
  {A_N} = 4840 + 242 \\
  {A_N} = 5082{\text{ Rs}}{\text{.}} \;
  $
Now, the compound Interest $ = Amoun{t_{Net}}{\text{ - Principal (Sum)}} $
Place the values
 $
  C.I. = 5082 - 4000 \\
  C.I. = Rs.{\text{ 1082}} \;
  $
Hence, the required answer is – the compound interest is $ Rs.\;{\text{1082}} $

Note: Always convert the percentage rate of interest in the form of fraction or the decimals and then substitute further for the required solutions. Remember the difference between simple interest and compound interest and apply its concept wisely. Compound interest is the interest paid for the interest earned in the previous year. Be good in multiples and do simplification carefully.