Interest rate........?

Norton borrowed $1800 form the bank. If he repaid the loan with $3966.76 at the end of 16 years, what interest rate did the bank charge if it was compounded semiannually?





Please explain this to me...I am confused. Thanks|||Let me try to explain.





Imagine, the interest rate is 'a'





Every six months the interest is compounded.





So after the first six months, the principle 'P' becomes





P+ (P*a)/(2*100)





( /2 is because it is interest for half year. )





= P * (200+a) / 200





On completion of the next six months, the principle becomes





P * (200+a) / 200 * (200+a) / 200





So, for every six months, it get multiplied by the factor (200+a) / 200





Let us assume (200+a) / 200 = x





For 16 years or after 32 half years, the principle becomes





P * [(200+a) / 200]^32





Final amount = P * x^32





Putting the known figures, we get





3966.76 = 1800 * x^32





So, x^32 = 3966.76 / 1800 = 2.20376





Hence x = 1.025





ie. (200+a) / 200 = 1.025





There for, The interest rate a = ( 1.025 * 200 ) - 200 = 5





So, Interest rate, a = 5% annually, compounded half yearly.





or





Interest rate is 2.5% half yearly, compounded





You can verify this





1800* ( 1.025 )^32 = 1800 * 2.20376 = 3966.76





Hence proved|||4.9% anually so his interest rate was 4.9/2 %