Answer
452.7k+ views
Hint: Use compound interest formula which is given as $A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}-P$, where ‘r’ is interest in percentage, ‘n’ is time period in years, ‘P’ is principal amount and ‘A’ is amount after ‘n’ years.
Complete step-by-step answer:
We know that the amount ‘A’ at the end of ‘n’ years at the rate of r% per annum when the interest is compounded annually is given by,
$A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}...........\left( i \right)$
Where P = Initial amount or principal amount that increases by r % annually.
Here, we have 12000 as the principal amount which is increasing at a rate of 12% for the time period of 10 years.
Hence, from the given equation, we have
P = 12000 Rs
r = 12%
n = 10 years
Now, using the equation, we can get amount ‘A’ after 10 years as,
$\begin{align}
& A=12000{{\left( 1+\dfrac{12}{100} \right)}^{10}} \\
& A=12000{{\left( 1+\dfrac{3}{25} \right)}^{10}} \\
\end{align}$
Now, taking LCM inside the bracket, we get
$A=12000{{\left( \dfrac{28}{25} \right)}^{10}}...........\left( ii \right)$
Now, we can solve equation (ii) by multiplying $\dfrac{28}{25}$ to 10 times. But this process would be very lengthy for calculating the value of ‘A’.
So, we can use logarithm and antilogarithm to get value of A as follows
Taking log to both sides to equation (ii), we get.
$\log A=\log \left( \left( 12000 \right){{\left( \dfrac{28}{25} \right)}^{10}} \right)..........\left( iii \right)$
We can use identity of logarithm as
log ab = log a + log b
Applying above identity with equation (iii) we get,
$\log A=\log 12000+\log {{\left( \dfrac{28}{25} \right)}^{10}}$
Now, we can use another identity of ‘log’ as
$\log {{m}^{n}}=n\log m$
So, we get
$\log A=\log 12000+10\log \dfrac{28}{25}........\left( iv \right)$
We have another identity of logarithm
$\log \left( \dfrac{a}{b} \right)=\log a-\log b$
Hence, equation (iv) can be given as
$\log A=\log 12+\log 1000+10\left( \log 28-log25 \right)$
Now, we get values of logarithms of above by using logarithm table as
log 28 = 1.4472
log 25 = 1.3979
log 12 = 1.0792
log 1000 = $\log {{10}^{3}}$ = 3log 10 = 3
Hence, on putting values in above equation, we get
log A = 4.0792 + 0.493
log A = 4.5722
Taking antilog to both sides, we get
A = antilog (4.5722) = 37350
So, the amount after 10 years is Rs 37350.
As we know compound interest can be given by relation.
Compound interest = Total amount after interest – principal value/amount.
So, compound interest after 10 years is,
37350 – 12000 = Rs.25350
Note: One may apply the formula of simple interest as $\dfrac{P\times R\times T}{100}$ where ‘P’ is principal amount, ‘R’ is interest and ‘T’ is time period but that has become wrong. So, be clear with both terms i.e simple and compound interest. One can answer compound interest as ‘A’ i.e. total amount after 10 years. So, don’t confuse compound interest and total amount.
Complete step-by-step answer:
We know that the amount ‘A’ at the end of ‘n’ years at the rate of r% per annum when the interest is compounded annually is given by,
$A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}...........\left( i \right)$
Where P = Initial amount or principal amount that increases by r % annually.
Here, we have 12000 as the principal amount which is increasing at a rate of 12% for the time period of 10 years.
Hence, from the given equation, we have
P = 12000 Rs
r = 12%
n = 10 years
Now, using the equation, we can get amount ‘A’ after 10 years as,
$\begin{align}
& A=12000{{\left( 1+\dfrac{12}{100} \right)}^{10}} \\
& A=12000{{\left( 1+\dfrac{3}{25} \right)}^{10}} \\
\end{align}$
Now, taking LCM inside the bracket, we get
$A=12000{{\left( \dfrac{28}{25} \right)}^{10}}...........\left( ii \right)$
Now, we can solve equation (ii) by multiplying $\dfrac{28}{25}$ to 10 times. But this process would be very lengthy for calculating the value of ‘A’.
So, we can use logarithm and antilogarithm to get value of A as follows
Taking log to both sides to equation (ii), we get.
$\log A=\log \left( \left( 12000 \right){{\left( \dfrac{28}{25} \right)}^{10}} \right)..........\left( iii \right)$
We can use identity of logarithm as
log ab = log a + log b
Applying above identity with equation (iii) we get,
$\log A=\log 12000+\log {{\left( \dfrac{28}{25} \right)}^{10}}$
Now, we can use another identity of ‘log’ as
$\log {{m}^{n}}=n\log m$
So, we get
$\log A=\log 12000+10\log \dfrac{28}{25}........\left( iv \right)$
We have another identity of logarithm
$\log \left( \dfrac{a}{b} \right)=\log a-\log b$
Hence, equation (iv) can be given as
$\log A=\log 12+\log 1000+10\left( \log 28-log25 \right)$
Now, we get values of logarithms of above by using logarithm table as
log 28 = 1.4472
log 25 = 1.3979
log 12 = 1.0792
log 1000 = $\log {{10}^{3}}$ = 3log 10 = 3
Hence, on putting values in above equation, we get
log A = 4.0792 + 0.493
log A = 4.5722
Taking antilog to both sides, we get
A = antilog (4.5722) = 37350
So, the amount after 10 years is Rs 37350.
As we know compound interest can be given by relation.
Compound interest = Total amount after interest – principal value/amount.
So, compound interest after 10 years is,
37350 – 12000 = Rs.25350
Note: One may apply the formula of simple interest as $\dfrac{P\times R\times T}{100}$ where ‘P’ is principal amount, ‘R’ is interest and ‘T’ is time period but that has become wrong. So, be clear with both terms i.e simple and compound interest. One can answer compound interest as ‘A’ i.e. total amount after 10 years. So, don’t confuse compound interest and total amount.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)