Question

Find the amount and the compound interest on 2000 at 10% per annum for $$2\dfrac{1}{2}$$ years, compounded annually.

Answer 1

Hint:So to solve this question we first need to find the compound interest for the first two years and after that we have to find the simple interest for the last $$\dfrac{1}{2}$$ year where principal amount will be the amount after 2 years and the summation of the all amount will be the final amount after compounded.
So for this we need to know
Compounded Amount (A) = $$p\left( 1+\dfrac{r}{100} \right)^{nt} $$.....(1)
And simple Interest = $$\dfrac{p\times r\times t}{100}$$.......(2)
Where, p = principle amount .
r = interest rate.
t = number of time periods elapsed.
n = number of times interest applied per time period.

Complete step by step answer:
Here first we will find the compounded amount for first two years,
So for this,
p = 2000, r = 10, t = 1 year and n = 2 times.
Therefore by formula (1) we can write,
A = $$p\left( 1+\dfrac{r}{100} \right)^{nt} $$
   = $$2000\left( 1+\dfrac{10}{100} \right)^{2\times 1} $$
   = $$2000\left( 1+\dfrac{1}{10} \right)^{2} $$
   = $$2000\left( \dfrac{11}{10} \right)^{2} $$
   = $$2000\times \dfrac{11^{2}}{10^{2}}$$
   = $$2000\times \dfrac{121}{100}$$
   = $$20\times 121$$
   = 2420
Now this amount will be used as the principal amount for the next half year.
Therefore the principle amount, p = 2420, r = 10 and t = $$\dfrac{1}{2} =0.5$$
Therefore by formula (2) we can write,
SI = $$\dfrac{p\times r\times t}{100}$$
    = $$\dfrac{2420\times 10\times 0.5}{100}$$
    = $$242\times 0.5$$
    = 121
Therefore after 2.5 year the compounded amount will be (2420 + 121) = 2541
Therefore compound interest = (compounded amount - principal amount)
                                               = 2541 - 2000 = 541
Hence the amount is Rs 2541 and the compound interest is Rs 541.

Note:
To solve this we need to know that Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously accumulated interest.
The simple annual interest rate is the interest amount per period, multiplied by the number of periods per year. The simple annual interest rate is also known as the nominal interest rate.