Example: We are investing $100,000 in an account that earns interest at a rate of 7.5%
for 54 months. Find out how much money will be in the account if,
(a) Interest is compounded quarterly.
(b) Interest is compounded monthly.
(c) Interest is compounded continuously.
Solution
Before getting into each part let's recognize the quantities which we will require in all the parts and won't change.
P = 100, 000 r = 7.5 /100= 0.075 t = 54/12 =4.5
Remember that interest rates have to be decimals for these computations and t has to be in years!
Now, Solve out the problems.
(a) Interest is compounded quarterly.
In this the interest is compounded quarterly and it means it is compounded 4 times a year. After 54 months then we have,
A = 100000 (1 + (0.075/ 4)^{( 4)( 4.5)}
= 100000 (1.01875)^{18}
=100000 (1.39706686207)
= 139706.686207 = $139, 706.69
Notice the amount of decimal places utilized here. We didn't do any rounding till the very last step. It is significant to not do too much rounding in intermediate steps along with these problems.
(b) Interest is compounded monthly.
In this compounding monthly and so it means we are compounding 12 times a year. Following is how much we'll have after 54 months.
A = 100000 (1 + 0.075 /12) (12)( 4.5)
= 100000 (1.00625)^{54}
= 100000 (1.39996843023)
= 139996.843023 = $139, 996.84
Thus, compounding more times per year will yield more money.
(c) Interest is compounded continuously.
At last, if we continuously compound then after 54 months we will have,
A =100000e^{(0.075)( 4.5)}
= 100000 (1.40143960839)
= 140143.960839 = $140,143.96