What will be the compound interest on 4000 at 6% per annum for 2 and half years?

After one year, you have 100+6=$106 . After two years, if the interest is simple , you will have 106+6=$112 (adding 6% of the original principal amount each year.) But if it is compound interest , then in the second year you will earn 6% of the new amount:

1.06×$106=$112.36

Yearly Compound Interest Formula

If you put P dollars in a savings account with an annual interest rate r , and the interest is compounded yearly, then the amount A you have after t years is given by the formula:

A=P(1+r)t

Example:

Suppose you invest $4000 at 7% interest, compounded yearly. Find the amount you have after 5 years.

Here, P=4000 , r=0.07 , and t=5 . Substituting the values in the formula, we get:

A=4000(1+0.07)5     ≈4000(1.40255)     =5610.2

Therefore, the amount after 5 years would be about $5610.20 .

General Compound Interest Formula

If interest is compounded more frequently than once a year, you get an even better deal. In this case you have to divide the interest rate by the number of periods of compounding.

If you invest P dollars at an annual interest rate r , compounded n times a year, then the amount A you have after t years is given by the formula:

A=P(1+rn)nt

Example:

Suppose you invest $1000 at 9% interest, compounded monthly. Find the amount you have after 18 months.

Complete step by step solution:
Here, we have given the principal amount (P) as 4000 and interest (r) 10% annually. As we’ll divide the total time duration into two parts because in the time span of 2 years compound interest will be applied and for the rest of the 3 months simple interest will be applied.
The formula for the compound interest is $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}$
Where A is the total amount, P is the principal amount, r is the rate of interest and n is the time duration.
According to our question, $P=4000, r=10\%, n =2 $years. On Calculation A we get,
\[ A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n} \\
   \Rightarrow A = 4000{\left( {1 + \dfrac{{10}}{{100}}} \right)^2} \\
   \Rightarrow A = P{\left( {1 + \dfrac{1}{{10}}} \right)^2} \\
   \Rightarrow A = 4000{\left( {\dfrac{{11}}{{10}}} \right)^2} \\
   \Rightarrow A = 4000 \times \dfrac{{121}}{{100}} \\
   \Rightarrow A = 4840{\text{ Rs}}{\text{.}} \]
Hence the amount after 2 years will be 4840 and it’ll only work as the principal amount for simple interest. The formula for the simple interest is = $\dfrac{{PRT}}{{100}}$, where P is the principal amount, R is the rate of interest, and T is time spam.
On putting the values, we get,
$ \dfrac{{PRT}}{{100}} \\
   = \dfrac{{4840 \times 10 \times 1}}{{100 \times \times 4}} \\
   = \dfrac{{4840 \times 1 \times 1}}{{10 \times \times 4}} \\
   = \dfrac{{484 \times 1 \times 1}}{{1 \times \times 4}} \\
   = 121 $
So, the total amount after 2 years and 3 months will be Rs 4840 + Rs 121 which is equal to Rs 4961.
Total interest will be = total amount after 2 years 3 months- principal amount
That is, $Rs 4961 – Rs 4000$
And, Rs. 961

Hence, option (c) is the correct option.

Note: Students usually make mistakes in such a type of problem where for some time period compound interest is applied and for some time period, simple interest is applied. It’s always recommended from our side to read the question carefully, especially the interest section. Whether the interest is applied monthly or annually also of which kind, simple of the compound.

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

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