Question

Rs 18,000 is kept for $2\dfrac{1}{2}$ at 10 percent compounded annually. What is the interest and final amount received?

Answer 1

Hint: Now we know that principal amount is 18, 000, time is $2\dfrac{1}{2}$ and rate of interest is 10 percent per annum. Now we will first calculate the amount for 2 years with the formula $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$ now once we have the amount we will use simple interest formula $A=\dfrac{P\left( 1+rt \right)}{100}$ with time as $\dfrac{1}{2}$ years and rate of interest 10 percent.

Complete step by step answer:
Now we are given that Rs 18,000 is kept for $2\dfrac{1}{2}$ at 10 percent compounded annually.
Now we have principal amount Rs 18,000, time is $2\dfrac{1}{2}$ and the rate of interest is 10 percent.
Now since time is not an integer we will find the amount received after 2 years by compound interest
Now since the interest is compounded per year, we can use simple interest to find the amount after $\dfrac{1}{2}$ years.
Now we know that amount received by compound interest is given by $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$ .
Hence amount after 2 years will be
$\begin{align}
  & A=18000{{\left( 1+\dfrac{10}{100} \right)}^{2}} \\
 & \Rightarrow A=18000{{\left( \dfrac{100+10}{100} \right)}^{2}} \\
 & \Rightarrow A=18000{{\left( \dfrac{110}{100} \right)}^{2}} \\
 & \Rightarrow A=18000{{\left( \dfrac{11}{10} \right)}^{2}} \\
\end{align}$
$\Rightarrow A=18000\left( \dfrac{121}{100} \right)$
$\Rightarrow A=180\times 121$
$\therefore A=21780$
Hence we have amount received after 2 years is 21780.
Now let us calculate the simple interest in $\dfrac{1}{2}$ years
We know the amount of simple interest is given by $A=P+\dfrac{P\times r\times t}{100}$
The amount after 2 years is our principal amount hence from equation (1) we get
Principal amount is 21780.
Now interest is 10 percent
And time is $\dfrac{1}{2}$ years
Hence the amount received is
\[\begin{align}
  & A=21780+\dfrac{21780\times 10\times \dfrac{1}{2}}{100} \\
 & \Rightarrow A=21780+1089 \\
 & \therefore A=22869 \\
\end{align}\]
Hence the final amount received after $2\dfrac{1}{2}$ years is 22869 Rs.
Now interest is final amount – principal amount
Hence Interest = 22869 – 18000 = 4869

Hence the interest is 4869 Rs.

Note: Note that the formula of compound interest is $P{{\left( 1+\dfrac{r}{100n} \right)}^{nt}}$ where n is number of times interest is compounded in a year. Since we have interest is compounded annually we have taken n = 1. Which is nothing but formula for compound interest compounded per year. Also while substituting always note that rate of interest is per years and time is also in years.