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Imagine you put $ 100 in a savings account with a yearly interest rate of 6 % .
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 + r n ) n t
Example:
Suppose you invest $ 1000 at 9 % interest, compounded monthly. Find the amount you have after 18 months.
Here P = 1000 , r = 0.09 , n = 12 , and t = 1.5 (since 18 months = one and a half years).
Substituting the values, we get:
A = 1000 ( 1 + 0.09 12 ) 12 ( 1.5 ) ≈ 1000 ( 1.143960 ) = 1143.960
Rounding to the nearest cent, you have $ 1143.96 .
What will be the compound interest on Rs 4000 in two years when rate of interest is 5% per annum?
Solution
We know that amount A at the end of n years at the rate of R % per annum is given by A = P\[ \left( 1 + \frac{R}{100} \right)^n . \]
Given:
P = Rs 4, 000
R = 5 % p . a .
n = 2 years
Now,
\[A = 4, 000 \left( 1 + \frac{5}{100} \right)^2 \]
\[ = 4, 000 \left( 1 . 05 \right)^2 \]
= Rs 4,
410
And,
CI = A - P
= Rs 4, 410 - Rs 4, 000
= Rs 410
Answer
Verified
Hint: First we’ll find the compound interest for 2 years by using its formula. Rest of the 2 months’ interest will be simple because it’s given that the compound interest will be applied annually. In the end, we’ll add both the interest and subtract from the principal amount.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.