How much will rs 4000 amount to in a year if the interest is compounded quarterly at 8 per annum

A man brought Rs 4000 at 10 percent per annum compounded interest. At the end of each year he has repaid Rs 1000. The amount of money be still incurs after third year is
$\begin{align}
  & \text{a) 2740} \\
 & \text{b) 2104} \\
 & \text{c) 2014} \\
 & \text{d) 3400} \\
\end{align}$

Answer

Hint: Now we are given that a man borrows 4000 Rs at 10 percent interest compounded annually. Hence we will find the total amount after 1 year with the help of formula of amount for compound interest. The formula for compound interest amount for Principal P, rate r, and time t is given by $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$ . Hence we will get the amount for the first year. Now we know the man pays 1000 yearly hence we will subtract 1000 from the amount. Now the obtained value is the principal amount for next year which is second year. We will continue the process to finally find the amount after 3 years.

Complete step by step answer:
Now a man borrowed 4000 Rs at 10 percent per annum
Now here Principal Amount P = 4000 rate of interest is 10 percent.
Now amount in compound interest is given by $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$ where P is principal amount t is time in years and r is rate of interest in percentage
Hence after 1 year the interest will be.
\[\begin{align}
  & A=4000{{\left( 1+\dfrac{10}{100} \right)}^{1}} \\
 & \Rightarrow A=4000\left( 1+0.1 \right) \\
 & \Rightarrow A=4000\times 1.1 \\
 & \Rightarrow A=4400 \\
\end{align}\]
Hence the final amount is 4400 Rs.
Now let the interest is 4400Rs and he pays 1000 each year hence the remaining interest is 3400.
Now for second year principal is 3400 and rate of interest is 10 percent.
Hence we get the amount after 1 year.
$\begin{align}
  & A=3400{{\left( 1+\dfrac{10}{100} \right)}^{1}} \\
 & \Rightarrow A=3400\left( 1.1 \right) \\
 & \Rightarrow A=3740 \\
\end{align}$
Now again he pays 1000 Rs hence the remaining amount is 3740 – 1000 = 2740 Rs.
Now let us calculate the Compound interest for the third year.
Now principal is 2740, Interest rate is 10 percent
Hence after 1 year the amount will be,
$\begin{align}
  & A=2740{{\left( 1+\dfrac{10}{100} \right)}^{1}} \\
 & \Rightarrow A=2740\left( 1.1 \right) \\
 & \Rightarrow A=3014 \\
\end{align}$
Now again he pays 1000 Rs so the amount remaining is 3014 – 1000 = 2014.
Hence after 3 years the amount remaining is 2014.

So, the correct answer is “Option C”.

Note: Here we cannot directly calculate the interest after 3 years since the principal amount changes after each year. Hence for each year we need to find the total amount and subtract monthly installments to get the final amount.

What will be the compound interest on 4000 at 5% per annum in two years.

Answer

Verified

Hint: Here we are given the principal amount of money, the rate of interest and it is to be compounded annually for 2 years. We will find out the compound interest using the formula. Consider all values given in question and try to identify which one to use where.
\[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\]
This will help in understanding the problem.

Complete step-by-step answer:
Here, we have the principal amount given (P) = 4000Rs.
The rate of interest per annum (r) = 5%
The total duration given is 2 years.
As we know, the formula of compound interest is
 \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\]
A is the total amount. R is rate of interest and P is the principal amount
Compounded for two years.
Given in the question, $ P = 4000Rs,r = 5\% ,n = 2years$
\[
  A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n} \\
   \Rightarrow A = 4000{\left( {1 + \dfrac{5}{{100}}} \right)^2} \\
   \Rightarrow A = 4000{\left( {1 + \dfrac{1}{{20}}} \right)^2} \;
 \]
Further, calculating the fraction we get
\[
   \Rightarrow A = 4000{\left( {\dfrac{{21}}{{20}}} \right)^2} \\
  Squaring\,bracket \\
   \Rightarrow A = 4000\left( {\dfrac{{441}}{{400}}} \right) \\
   \Rightarrow A = 4410Rs. \;
 \]
So, the amount after two years will be
 \[A = 4410Rs.\]
The compound interest will be
$
  Interest = Amount - Principal \\
   \Rightarrow Interest = 4410 - 4000 \\
   \Rightarrow Interest = 410\;Rs. \;
 $
So, the correct answer is “410 Rs.”.

Note: 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