Simple interest results in a final Amount for a given rate at the end of one period as follows
Amount = Principal*(1+rate)
If this amount is the principal at the beginning of a new period, the amount at the end will be
Amount2 = Amount1*(1+rate) = Principal*(1+rate)*(1+rate)
Amount2 = Principal*(1+rate)2
You can see that the exponent on (1+rate) will match the number of periods over which the principal is compounded.
As you have noted in your problem statement, when compounding occurs at an interval less than 1 year, the rate is divided according to the number of times compounding occurs in the year. If we compound 2 times per year, the rate used in the formula is the annual rate divided by 2, and there are 2 periods in the year.
Compounding 9% semiannually for 2 years results in rate=4.5% for one period and the number of periods = 4.
Amount = Principal*(1+4.5%)4 = Principal*1.0454
The value 1.0454 can be calculated or found in a lookup table. Its value is
1.192518600625
My mortgage company claims to use 6 decimal digits for such calculations, so they would use 1.192519.
These days, calculators or computers make it so easy to compute such numbers that tables have all but disappeared from use. You would use the yx key on some of the calculators I've seen. (Type 1.045 yx 4 =)
Amount = Principal*(1+rate)
If this amount is the principal at the beginning of a new period, the amount at the end will be
Amount2 = Amount1*(1+rate) = Principal*(1+rate)*(1+rate)
Amount2 = Principal*(1+rate)2
You can see that the exponent on (1+rate) will match the number of periods over which the principal is compounded.
As you have noted in your problem statement, when compounding occurs at an interval less than 1 year, the rate is divided according to the number of times compounding occurs in the year. If we compound 2 times per year, the rate used in the formula is the annual rate divided by 2, and there are 2 periods in the year.
Compounding 9% semiannually for 2 years results in rate=4.5% for one period and the number of periods = 4.
Amount = Principal*(1+4.5%)4 = Principal*1.0454
The value 1.0454 can be calculated or found in a lookup table. Its value is
1.192518600625
My mortgage company claims to use 6 decimal digits for such calculations, so they would use 1.192519.
These days, calculators or computers make it so easy to compute such numbers that tables have all but disappeared from use. You would use the yx key on some of the calculators I've seen. (Type 1.045 yx 4 =)