 Hello and welcome to this session. In this session we are going to discuss the following question and the question says that a sum of $6,400 earns a compound interest of $1,008.80 in 18 months. When the interest is reckoned half early, find the rate of interest. We know that compound interest is equal to amount minus principal. Also, amount a is equal to p into 1 plus r upon 100 raised to the power n as 1 year is equal to 2.5 years r percent per annum means r by 2 percent half yearly. n years means 2 n half years. With this key idea, let us proceed with the solution. According to the question we need to find the rate of interest when a sum of $6,400 earns a compound interest of $1,008.80 in 18 months when the interest is reckoned half yearly. Here p is equal to $6,400 and compound interest is equal to $1,008.80. The time given to us is 18 months which is equal to 3 by 2 years. Therefore, 2 into 3 by 2 is equal to 3 half years. As we know compound interest is equal to amount minus principal. Therefore, amount is equal to principal plus compound interest which is equal to $6,400 plus $1,008.80. That is equal to $7,408.80. As we need to find the rate, let half yearly rate be equal to r percent. To find the compound interest formula, a is equal to p into 1 plus r upon 100 raised to power n. We have 7,408.80 is equal to $6,400 into 1 plus r upon 100 raised to power 3. Which implies 7,408.80 upon $6,400 is equal to 1 plus r upon 100 raised to power 3. Now, 2 times $3,200 is $6,400 and 2 times $3,704.40 is $7,408.80. So this implies 1 plus r upon 100 raised to power 3 is equal to $3,704.40 upon $3,200. Again, 2 times $1,600 is equal to $3,200 and 2 times $1,852.2 is equal to $3,704.40. This implies 1 plus r upon 100 whole cube is equal to $1,852.2 upon $1,600. Again, 2 times $800 is $600 and 2 times $926.1 is $1,852.2. So this implies 1 plus r upon 100 raised to power 3 is equal to $926.1 upon $800. Now here to remove the decimal point we multiply the numerator and denominator by 10 and we get this is equal to $9261 upon $8,000. As $9261 is the cube of $21 and $8,000 is the cube of $20 we have 1 plus r upon 100 whole cube is equal to $21 upon 20 cube. Taking the cube roots on both the sides we have 1 plus r upon 100 is equal to $21 upon 20 which implies r upon 100 is equal to $21 upon 20 minus 1 which is equal to $21 minus 20 upon 20 by taking the LCM which is equal to 1 upon 20. This implies r is equal to 100 upon 20 which is equal to r percent was the half yearly weight that is r is equal to 5 percent half yearly. This implies r is equal to 5 into 2 that is 10 percent per annum which is our answer. This completes our session. Hope you enjoyed this session.