Answer

Verified

481.8k+ views

Hint: Use compound interest formula for the calculation of amount $A$, given by: \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\]. From this calculate amount $A$ after \[t=2\text{ years}\] and \[t=3\text{ years}\] and then take the difference for the calculation of interest obtained in\[3\text{rd year}\].

Complete step-by-step answer:

Compound interest is the addition of interest to the principal sum of a loan or deposit. It is the result of reinvesting interest, rather than paying it out, so the interest in the next period is then earned on the principal sum plus previously accumulated interest.

The total accumulated amount $A$ , on the principal sum \[P\] plus compound interest $I$ is given by the formula \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\].

Here, \[A\] is the amount obtained, $t$ is the number of years, \[r\] is the rate, $P$ is the principal and \[n\] is the number of times the interest is given in a year.

The total compound interest generated is given by: $I=A-P$.

Now, we have been given that:

$P=\text{Rs 8000}$, $r=5%=\dfrac{5}{100}=0.05$, $n=1$.

(i) The amount credited at the end of the second year i.e. $t=2\text{ years}$ can be calculated as:

\[\begin{align}

& \therefore {{A}_{2}}=8000{{\left( 1+\dfrac{0.05}{1} \right)}^{1\times 2}} \\

& \text{ }=8000{{\left( 1+\dfrac{5}{100} \right)}^{2}} \\

& \text{ }=8000{{\left( \dfrac{100+5}{100} \right)}^{2}} \\

& \text{ }=8000{{\left( \dfrac{105}{100} \right)}^{2}} \\

& \text{ }=8000\times \dfrac{105}{100}\times \dfrac{105}{100} \\

& \therefore {{A}_{2}}=\text{Rs }8820. \\

\end{align}\]

Hence, the credited amount after two years is \[\text{Rs }8820\].

(ii) Now, to calculate the interest for the third year we need to subtract the amount ${{A}_{2}}$ obtained after two years form the amount ${{A}_{3}}$ obtained after three years.

\[\begin{align}

& \therefore {{A}_{3}}=8000{{\left( 1+\dfrac{0.05}{1} \right)}^{1\times 3}} \\

& \text{ }=8000{{\left( 1+\dfrac{5}{100} \right)}^{3}} \\

& \text{ }=8000{{\left( \dfrac{100+5}{100} \right)}^{3}} \\

& \text{ }=8000{{\left( \dfrac{105}{100} \right)}^{3}} \\

& \text{ }=8000\times \dfrac{105}{100}\times \dfrac{105}{100}\times \dfrac{105}{100} \\

& \therefore {{A}_{3}}=\text{Rs 9261}. \\

\end{align}\]

The interest $I$ for the third year is given by:

$\begin{align}

& I={{A}_{3}}-{{A}_{2}} \\

& \text{ }=9261-8820 \\

& \text{ }=441. \\

\end{align}$

Hence, the interest for the third year is \[\text{Rs }441\].

Note: Here, the value of $n$ must be substituted carefully. We have to read the question carefully as it is given that the rate is compounded annually, therefore, $n=1$ is substituted. We must divide the given rate by 100 and then substitute in the equation.

Complete step-by-step answer:

Compound interest is the addition of interest to the principal sum of a loan or deposit. It is the result of reinvesting interest, rather than paying it out, so the interest in the next period is then earned on the principal sum plus previously accumulated interest.

The total accumulated amount $A$ , on the principal sum \[P\] plus compound interest $I$ is given by the formula \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\].

Here, \[A\] is the amount obtained, $t$ is the number of years, \[r\] is the rate, $P$ is the principal and \[n\] is the number of times the interest is given in a year.

The total compound interest generated is given by: $I=A-P$.

Now, we have been given that:

$P=\text{Rs 8000}$, $r=5%=\dfrac{5}{100}=0.05$, $n=1$.

(i) The amount credited at the end of the second year i.e. $t=2\text{ years}$ can be calculated as:

\[\begin{align}

& \therefore {{A}_{2}}=8000{{\left( 1+\dfrac{0.05}{1} \right)}^{1\times 2}} \\

& \text{ }=8000{{\left( 1+\dfrac{5}{100} \right)}^{2}} \\

& \text{ }=8000{{\left( \dfrac{100+5}{100} \right)}^{2}} \\

& \text{ }=8000{{\left( \dfrac{105}{100} \right)}^{2}} \\

& \text{ }=8000\times \dfrac{105}{100}\times \dfrac{105}{100} \\

& \therefore {{A}_{2}}=\text{Rs }8820. \\

\end{align}\]

Hence, the credited amount after two years is \[\text{Rs }8820\].

(ii) Now, to calculate the interest for the third year we need to subtract the amount ${{A}_{2}}$ obtained after two years form the amount ${{A}_{3}}$ obtained after three years.

\[\begin{align}

& \therefore {{A}_{3}}=8000{{\left( 1+\dfrac{0.05}{1} \right)}^{1\times 3}} \\

& \text{ }=8000{{\left( 1+\dfrac{5}{100} \right)}^{3}} \\

& \text{ }=8000{{\left( \dfrac{100+5}{100} \right)}^{3}} \\

& \text{ }=8000{{\left( \dfrac{105}{100} \right)}^{3}} \\

& \text{ }=8000\times \dfrac{105}{100}\times \dfrac{105}{100}\times \dfrac{105}{100} \\

& \therefore {{A}_{3}}=\text{Rs 9261}. \\

\end{align}\]

The interest $I$ for the third year is given by:

$\begin{align}

& I={{A}_{3}}-{{A}_{2}} \\

& \text{ }=9261-8820 \\

& \text{ }=441. \\

\end{align}$

Hence, the interest for the third year is \[\text{Rs }441\].

Note: Here, the value of $n$ must be substituted carefully. We have to read the question carefully as it is given that the rate is compounded annually, therefore, $n=1$ is substituted. We must divide the given rate by 100 and then substitute in the equation.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

What is the past participle of wear Is it worn or class 10 english CBSE

What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

The coefficient of xn 2 in the polynomial left x 1 class 9 maths CBSE

How do you know if an equation is linear or nonlin class 10 maths CBSE