Question

Find the amount of $$Rs.12000$$after $$2$$ years compounded annually the rate of interest
being $$5\mathrm{\%}$$pa during the first year and $$6\mathrm{\%}$$pa during the second year also find the compounded
interest?
$$\begin{gathered} \text{A}\text{. Rs}\text{.1356} \\ \text{B}\text{. Rs}\text{.1200} \\ \text{C}\text{. Rs}\text{.1256} \\ \text{D}\text{. None of the above} \end{gathered}$$

Answer 1

Hint-Calculate the amount from the given PRINCIPAL for the first year, which becomes the
PRINCIPAL amount for the second year and then calculate the amount on the PRINCIPAL amount, obtained
from the second year. Compound interest is the sum of the PRINCIPAL and the interest. The PRINCIPAL is
the amount of loan or the deposit on which interest is calculated at a rate for a time period. Compound
interest is generally found by $$A=P{\left(1+{\displaystyle \frac{r}{100}}\right)}^n$$where P is the PRINCIPAL
on which interest is to be calculated for a given rate$$r$$.
The compound can also be calculated half-yearly, quarterly and monthly.
Complete step by step solution:
The given PRINCIPAL is compounded for the $$2$$years, but the rate of interest is different for both the
years hence interest will be calculated yearly, for the first year where
PRINCIPAL$$(P_1)=12000$$
Rate$$\left(r_1\right)=5\mathrm{\%}$$
Time$$\left(n_1\right)=1year$$
Hence the amount $$\left(A\right)$$for the first year will be
$$\begin{gathered} A=P_1{\left(1+{\displaystyle \frac{r_1}{100}}\right)}^{n_1} \\ =12000{\left(1+{\displaystyle \frac{5}{100}}\right)}^1 \\ =12000\times {\displaystyle \frac{105}{100}} \\ =12600 \end{gathered}$$
Now the amount obtained from the first year becomes the PRINCIPAL for the second year
$$\begin{gathered} P_2=12600 \\ {r}_2=6\mathrm{\%} \\ {n}_2=1year \end{gathered}$$
Hence the amount from the second year will be
$$\begin{gathered} A=P_2{\left(1+{\displaystyle \frac{r_2}{100}}\right)}^{n_2} \\ =12600{\left(1+{\displaystyle \frac{6}{100}}\right)}^1 \\ =12600\times {\displaystyle \frac{106}{100}} \\ =13356 \end{gathered}$$
We get the amount for PRINCIPAL$$(P)=12000$$ equals to $$A=Rs.13356$$
We Compound Interest=Amount-Principal
$$\begin{gathered} CI=A-P \\ =13356-12000 \\ =Rs.1356 \end{gathered}$$
Hence compound interest after 2 years will be $$Rs.1356$$

Note: If the interest rates are different for every year, Compound interest can also be calculated by $$A=P\left(1+{\displaystyle \frac{r_1}{100}}\right)\left(1+{\displaystyle \frac{r_2}{100}}\right).......$$for shortcut
method, but interest is being calculated year by year for better understanding.