Question

In what time will Rs 1500 yield Rs 496.50 as compound interest at 20 percent per year if the interest is compounded half yearly?

Answer 1

Hint: Now first we will find the final amount. Final amount is given by Principal amount + Interest. Now once we have the final amount we will use the formula for compound interest.
$A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}$ Now since the Interest in compounded half yearly we will take n=2. Hence we have $A=P{{\left( 1+\dfrac{r}{200} \right)}^{2t}}$. Now we know P which is principal amount, A which is final amount and r which is rate of interest. Hence we will substitute the values and find t.

Complete step-by-step answer:
Now we are given that Rs 1500 yields Rs 496.50.
Hence we have Principal = 1500 and interest = 496.50
Now final amount = Principal + interest = 1500 + 496.5.
Hence final amount = 1996.5 Rs.
Now we know that the final amount of compound interest compounded annually is given by
$A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}$
Here we have the final amount = 1996.5, principal amount = 1500.
rate of interest = 20 percent = $\dfrac{20}{100}=0.2$ , and n = 2 since the compound interest will be compounded 2 times a year.
Now substituting these values in the formula we have.
$1996.5=1500{{\left( 1+\dfrac{0.2}{2} \right)}^{2t}}$
Dividing the equation by 1500 we get
$\dfrac{1996.5}{1500}={{\left( 1+0.1 \right)}^{2t}}$
$\Rightarrow \dfrac{1331}{1000}={{\left( 1.1 \right)}^{2t}}$
$\Rightarrow \dfrac{1331}{1000}={{\left( \dfrac{11}{10} \right)}^{2t}}$
Hence we get $2t=3$. Now dividing this equation by 2 we get
$t=\dfrac{3}{2}=1\dfrac{1}{2}$
Hence we have one and half years.
Hence we get the money that is supposed to be kept for one and a half year so that will be Rs 1500 yield Rs 496.50.

Note: Note that in the formula of compound interest n stands for the number of times the interest will be compounded in a year. Hence if the interest is compounded half yearly, in one year the interest will be compounded 2 times. Hence n = 2 and not $\dfrac{1}{2}$ .