Question

When the interest is compounded annually
Let the principal = 2P, rate = R% per annum and the time = n years. Then, the amount(A) is given by the formula.

Answer 1

Hint: To solve the given question, we should know the definition of compound interest, and some formulas to calculate the terms associated with compound interest. Compound interest is the addition of interest to the principal sum of a loan, 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.

Complete step-by-step solution:
For this question, we will need the formula to calculate the amount gain in compound interest. The formula is \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\]. Here A is the amount gain, P is the principal amount, r is the interest in, n is the number of times interest is compounded in a time period, t is the time period.
For the given problem, we have principal = 2P, rate of interest as R% so \[r=\dfrac{R}{100}\], time period is n years, as we are compounding annually the value of n is 1.
Substituting these values in the above formula, we get
\[\begin{align}
  & A=2P{{\left( 1+\dfrac{\dfrac{R}{100}}{1} \right)}^{n}} \\
 & A=2P{{\left( 1+\dfrac{R}{100} \right)}^{n}} \\
\end{align}\]
Hence proved.

Note: We should also know the basic difference between the simple interest and compound interest. Simple interest is calculated on the principal, or original, amount of a loan. Compound interest is calculated on the principal amount and also on the accumulated interest of previous periods, and can thus be regarded as interest on interest.