Question

Find the amount and compound interest on Rs. 8000 for $ 1\dfrac{1}{2}years $ at a 10% rate per annum compounded to half-yearly.

Answer 1

Hint: We have to find amount and interest compounded to half-yearly so, for this, we have a direct formula which is given as $ A=P{{\left( 1+\dfrac{R}{200} \right)}^{2n}} $ where A is the amount, P is the principal amount, R is the rate of interest and n is time period. On substituting the values and solving we will get an amount. Then to find interest, we will put values in the formula $ A=P+I $ . Here, on putting values of P and A, we will get an amount of I i.e. interest amount.

Complete step-by-step answer:
Here, we have to find the total amount and interest at the end of $1\dfrac{1}{2}years$ i.e. 1.5 years compounded to half-yearly. So, whenever we are given that compounded to half-yearly terms, then there is the direct formula which we can apply i.e. given as $A=P{{\left( 1+\dfrac{R}{100} \right)}^{n}}$. From this we will find the total amount, P is the principal amount, R is the rate of interest and n is the time period which is to be multiplied with 2 as it is told that compounded half-yearly. We get as $n=2\times 1.5=3$ .
On substituting all the values, we get as
$A=8000{{\left( 1+\dfrac{10}{100} \right)}^{3}}$
On further solving, we can write it as
$A=8000{{\left( \dfrac{100+10}{100} \right)}^{3}}$
$A=8000{{\left( \dfrac{110}{100} \right)}^{3}}$
So, on expanding the equation we get as
$A=8000\left( 1.1\times 1.1\times 1.1 \right)$
On simplification, we get as
$A=8000\left( 1.331 \right)$
$A=Rs.10648$
Thus, the amount is Rs. 10648. Now, we have to find an interest, which we will get by using the formula $A=P+I$. “I” is the interest, so on putting the values we get as $10648=8000+I$.
Thus, on solving we get interested as
$I=10648-8000=Rs.2648$
Thus, amount and compound interest on Rs. 8000 for $1\dfrac{1}{2}years$ at a 10% rate per annum compounded to half yearly is Rs.10648 and Rs.2648, respectively.

Note: Be careful while using the formula $A=P{{\left( 1+\dfrac{R}{100} \right)}^{n}}$ which is almost similar to formula $A=P{{\left( 1+\dfrac{R}{200} \right)}^{2n}}$ .If we put value in $A=P{{\left( 1+\dfrac{R}{200} \right)}^{2n}}$ and on finding the amount, we will get as Rs. 9621 which is incorrect. So, be careful when and where to use formulas in order to get the correct answer. Also, remember to multiply the time period i.e. 1.5 years with 2 as it is told that amount compounded half-yearly. If not multiplied, then the answer will be wrong.