Answer
Verified
388.2k+ views
Hint: To calculate compound interest, we have given formula:
$ A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}} $
Where, A = final amount
P = initial amount
r = interest rate
t = number of time periods
Hence, compound interest is the difference of final amount and initial amount.
$ \Rightarrow CI=A-P $
Substitute the values in the formula to calculate compound interest.
Complete step-by-step answer:
As given in the question, the interest is compounded half-yearly, therefore the rate of interest is reduced half times.
That means, interest rate = 10% per annum, so, for compounding half-yearly, the interest rate = 5%.
So, r = 5 %
In the given question, the time period is $ 1\dfrac{1}{2} $ , i.e. three times a six-months interval.
So, t = 3
We have principal = Rs 12000
Using values of P, r and t, we get amount (A) as:
$ \begin{align}
& {{\left( SI \right)}_{3}}=\dfrac{{{A}_{2}}\times r\times t}{100} \\
& =\dfrac{13230\times 5\times 1}{100\times 2} \\
& =Rs.661.50
\end{align} $
Hence, Amount = Rs 13891.50
So, compound interest is:
$ \begin{align}
& CI=A-P \\
& =13891.50-12000 \\
& =1891.50
\end{align} $
Hence, compound interest = Rs 1891.50
Note: The other way to find compound interest compounded half-yearly is applying simple interest for every 6 months for the same interest rate and adding the interest in the initial value to calculate for another 6 months until for the total time period:
As it is given:
P = Rs 12000
r = 5%
t = $ 1\dfrac{1}{2} $ years = 3 $ \times $ 6 months
So, Simple interest for first 6 months is:
$ \begin{align}
& {{\left( SI \right)}_{1}}=\dfrac{P\times r\times t}{100} \\
& =\dfrac{12000\times 5\times 1}{100\times 2} \\
& =Rs.600
\end{align} $
Amount after first 6 months is:
\[\begin{align}
& CI={{\left( SI \right)}_{1}}+{{\left( SI \right)}_{2}}+{{\left( SI \right)}_{3}} \\
& =600+630+661.50 \\
& =Rs.1891.50
\end{align}\]
Now, consider $ {{A}_{1}} $ as principal for another 6 months. So simple interest for another 6 months is:
$ \begin{align}
& {{\left( SI \right)}_{2}}=\dfrac{{{A}_{1}}\times r\times t}{100} \\
& =\dfrac{12600\times 5\times 1}{100\times 2} \\
& =Rs.630
\end{align} $
Amount after another 6 months is:
$ \begin{align}
& {{A}_{2}}={{A}_{1}}+{{\left( SI \right)}_{2}} \\
& =12600+630 \\
& =Rs.13230
\end{align} $
Now, consider $ {{A}_{2}} $ as principal for another 6 months. So simple interest for another 6 months is:
$ \begin{align}
& {{\left( SI \right)}_{3}}=\dfrac{{{A}_{2}}\times r\times t}{100} \\
& =\dfrac{13230\times 5\times 1}{100\times 2} \\
& =Rs.661.50
\end{align} $
Final amount after another 6 months is:
$ \begin{align}
& {{A}_{3}}={{A}_{2}}+{{\left( SI \right)}_{3}} \\
& =13230+661.50 \\
& =Rs.13891.50
\end{align} $
Hence, total interest is
\[\begin{align}
& CI={{\left( SI \right)}_{1}}+{{\left( SI \right)}_{2}}+{{\left( SI \right)}_{3}} \\
& =600+630+661.50 \\
& =Rs.1891.50
\end{align}\]
$ A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}} $
Where, A = final amount
P = initial amount
r = interest rate
t = number of time periods
Hence, compound interest is the difference of final amount and initial amount.
$ \Rightarrow CI=A-P $
Substitute the values in the formula to calculate compound interest.
Complete step-by-step answer:
As given in the question, the interest is compounded half-yearly, therefore the rate of interest is reduced half times.
That means, interest rate = 10% per annum, so, for compounding half-yearly, the interest rate = 5%.
So, r = 5 %
In the given question, the time period is $ 1\dfrac{1}{2} $ , i.e. three times a six-months interval.
So, t = 3
We have principal = Rs 12000
Using values of P, r and t, we get amount (A) as:
$ \begin{align}
& {{\left( SI \right)}_{3}}=\dfrac{{{A}_{2}}\times r\times t}{100} \\
& =\dfrac{13230\times 5\times 1}{100\times 2} \\
& =Rs.661.50
\end{align} $
Hence, Amount = Rs 13891.50
So, compound interest is:
$ \begin{align}
& CI=A-P \\
& =13891.50-12000 \\
& =1891.50
\end{align} $
Hence, compound interest = Rs 1891.50
Note: The other way to find compound interest compounded half-yearly is applying simple interest for every 6 months for the same interest rate and adding the interest in the initial value to calculate for another 6 months until for the total time period:
As it is given:
P = Rs 12000
r = 5%
t = $ 1\dfrac{1}{2} $ years = 3 $ \times $ 6 months
So, Simple interest for first 6 months is:
$ \begin{align}
& {{\left( SI \right)}_{1}}=\dfrac{P\times r\times t}{100} \\
& =\dfrac{12000\times 5\times 1}{100\times 2} \\
& =Rs.600
\end{align} $
Amount after first 6 months is:
\[\begin{align}
& CI={{\left( SI \right)}_{1}}+{{\left( SI \right)}_{2}}+{{\left( SI \right)}_{3}} \\
& =600+630+661.50 \\
& =Rs.1891.50
\end{align}\]
Now, consider $ {{A}_{1}} $ as principal for another 6 months. So simple interest for another 6 months is:
$ \begin{align}
& {{\left( SI \right)}_{2}}=\dfrac{{{A}_{1}}\times r\times t}{100} \\
& =\dfrac{12600\times 5\times 1}{100\times 2} \\
& =Rs.630
\end{align} $
Amount after another 6 months is:
$ \begin{align}
& {{A}_{2}}={{A}_{1}}+{{\left( SI \right)}_{2}} \\
& =12600+630 \\
& =Rs.13230
\end{align} $
Now, consider $ {{A}_{2}} $ as principal for another 6 months. So simple interest for another 6 months is:
$ \begin{align}
& {{\left( SI \right)}_{3}}=\dfrac{{{A}_{2}}\times r\times t}{100} \\
& =\dfrac{13230\times 5\times 1}{100\times 2} \\
& =Rs.661.50
\end{align} $
Final amount after another 6 months is:
$ \begin{align}
& {{A}_{3}}={{A}_{2}}+{{\left( SI \right)}_{3}} \\
& =13230+661.50 \\
& =Rs.13891.50
\end{align} $
Hence, total interest is
\[\begin{align}
& CI={{\left( SI \right)}_{1}}+{{\left( SI \right)}_{2}}+{{\left( SI \right)}_{3}} \\
& =600+630+661.50 \\
& =Rs.1891.50
\end{align}\]
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE