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

Amount

Amount

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%)

The value 1.045

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 y

Amount = Principal*(1+rate)

If this amount is the principal at the beginning of a new period, the amount at the end will be

Amount

_{2}= Amount_{1}*(1+rate) = Principal*(1+rate)*(1+rate)Amount

_{2}= 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*1.045^{4}The value 1.045

^{4}can be calculated or found in a lookup table. Its value is1.192518600625

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 y

^{x}key on some of the calculators I've seen. (Type 1.045 y^{x}4 =)