 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that find the rate percent per annum if $4000 amount to $6,912 in one and a half years. Interest being compounded half-yearly. We know that the compound interest formula is amount A is equal to P into 1 plus R upon 100 raise to power n. With this key idea we shall proceed with the solution. According to the question we have to find the rate percent if $4000 amount to $6,912 in one and a half years. So let the rate of interest be equal to R percent half-yearly. Then the rate will be equal to 2R percent per annum. We are given the principal P equal to $4000. The amount after one and a half years is equal to A which is equal to $6,912. The time n is equal to one and a half years which is equal to 3 by 2 years that is 2 into 3 by 2 which is equal to 3 half years n is equal to 3. Now as amount A is equal to P into 1 plus R upon 100 raise to power n therefore here $6,912 is equal to $4000 into 1 plus R upon 100 raise to power 3 which implies 1 plus R upon 100 raise to power 3 is equal to $6,912 upon $4000. Both $6,912 and $4000 are divisible by 4 and we will obtain $1,728 upon $1,000. This implies 1 plus R upon 100 raise to power 3 is equal to 12 upon 10 raise to power 3 as $1,728 is the cube of 12 and $1,000 is the cube of 10. So on taking cube roots of both the size 1 plus R upon 100 is equal to 12 upon 10 which implies R upon 100 is equal to 12 upon 10 minus 1 which is equal to 12 minus 10 upon 10 that is 2 by 10. This implies R is equal to 2 upon 10 into 100 which is equal to 20. Therefore R is equal to 20% half yearly. Rate of interest is equal to 2 into 20 which is equal to 40% per annum which is our answer. This completes our session. Hope you enjoyed the session.