₹15,000 is lent for one year at the rate of 20% per annum, the interest being compounded annually. If the compounding of the interest is done half - yearly, then how much more interest will be obtained at the end of the one-year period on the same initial sum?

₹15,000 is lent for one year at the rate of 20% per annum, the interest being compounded annually.

compound interest for one year when it is compounded annually = $$15000\left[\left(1+\frac{20}{100}\right)^1-1\right]$$

= $$15000\left[\left(\frac{120}{100}\right)^1-1\right]$$

If the compounding of the interest is done half - yearly.

As we know that when interest is compounded half-yearly, then the rate will be half and time will be double.

compound interest for one-year when it is compounded half yearly = $$15000\left[\left(1+\frac{\frac{20}{2}}{100}\right)^2-1\right]$$

= $$15000\left[\left(1+\frac{10}{100}\right)^2-1\right]$$

= $$15000\left[\left(\frac{110}{100}\right)^2-1\right]$$

= $$15000\left[\left(\frac{11}{10}\right)^2-1\right]$$

= $$15000\left[\frac{121}{100}-1\right]$$

= $$15000\times\frac{21}{100}$$

= 3150    Eq.(ii)

Difference between both interests = Eq.(ii)-Eq.(i)

= 3150-3000

= 150

So ₹150 more interest will be obtained at the end of the one-year period on the same initial sum when interest is compounded half yearly.