Financial Mathematics → Interest → Simple Interest
After the accumulation of Interest, the final amount (A) is the sum of original amount plus the interest earned.
A = Cash + Interest Earned
That is
So, we have the Simple Interest Formula
Example 1. Suppose $1,000 is deposited in a bank that earns 5% simple interest. How much will be in the account after:
(a) 2 years
(b) 2.5 years
(c) 6 months
(d) 10 weeks
(e) 24 days
Solution:
(a) After 2 years
![A=C(1+ni) \\ =(1000)[1+(2)(.05)] \\ =(1000)[1+0.1] \\ =(1000)[1.1] \\ =\$1100](http://s0.wp.com/latex.php?latex=A%3DC%281%2Bni%29+%5C%5C++%3D%281000%29%5B1%2B%282%29%28.05%29%5D+%5C%5C++%3D%281000%29%5B1%2B0.1%5D+%5C%5C++%3D%281000%29%5B1.1%5D+%5C%5C++%3D%5C%241100+&bg=ffffff&fg=000&s=0 "A=C(1+ni) \\ =(1000)[1+(2)(.05)] \\ =(1000)[1+0.1] \\ =(1000)[1.1] \\ =\$1100")
(b) After 2.5 years
![A=C(1+ni) \\ =(1000)[1+(2.5)(.05)] \\ =(1000)[1+0.125] \\ =(1000)[1.125] \\ =\$1125](http://s0.wp.com/latex.php?latex=A%3DC%281%2Bni%29+%5C%5C++%3D%281000%29%5B1%2B%282.5%29%28.05%29%5D+%5C%5C++%3D%281000%29%5B1%2B0.125%5D+%5C%5C++%3D%281000%29%5B1.125%5D+%5C%5C++%3D%5C%241125+&bg=ffffff&fg=000&s=0 "A=C(1+ni) \\ =(1000)[1+(2.5)(.05)] \\ =(1000)[1+0.125] \\ =(1000)[1.125] \\ =\$1125")
(c) After 6 months
![A=C(1+ni) \\ =(1000)[1+(6/12)(.05)] \\ =(1000)[1+0.025] \\ =(1000)[1.025] \\ =\$1025](http://s0.wp.com/latex.php?latex=A%3DC%281%2Bni%29+%5C%5C++%3D%281000%29%5B1%2B%286%2F12%29%28.05%29%5D+%5C%5C++%3D%281000%29%5B1%2B0.025%5D+%5C%5C++%3D%281000%29%5B1.025%5D+%5C%5C++%3D%5C%241025+&bg=ffffff&fg=000&s=0 "A=C(1+ni) \\ =(1000)[1+(6/12)(.05)] \\ =(1000)[1+0.025] \\ =(1000)[1.025] \\ =\$1025")
(d) After 10 weeks
![A=C(1+ni) \\ =(1000)[1+(10/52)(.05)] \\ =(1000)[1+0.0096] \\ =(1000)[1.0096] \\ =\$1009.6](http://s0.wp.com/latex.php?latex=A%3DC%281%2Bni%29+%5C%5C++%3D%281000%29%5B1%2B%2810%2F52%29%28.05%29%5D+%5C%5C++%3D%281000%29%5B1%2B0.0096%5D+%5C%5C++%3D%281000%29%5B1.0096%5D+%5C%5C++%3D%5C%241009.6+&bg=ffffff&fg=000&s=0 "A=C(1+ni) \\ =(1000)[1+(10/52)(.05)] \\ =(1000)[1+0.0096] \\ =(1000)[1.0096] \\ =\$1009.6")
(e) After 24 days
![A=C(1+ni) \\ =(1000)[1+(24/365)(.05)] \\ =(1000)[1+0.00329] \\ =(1000)[1.00329] \\ =\$1003.29](http://s0.wp.com/latex.php?latex=A%3DC%281%2Bni%29+%5C%5C++%3D%281000%29%5B1%2B%2824%2F365%29%28.05%29%5D+%5C%5C++%3D%281000%29%5B1%2B0.00329%5D+%5C%5C++%3D%281000%29%5B1.00329%5D+%5C%5C++%3D%5C%241003.29+&bg=ffffff&fg=000&s=0 "A=C(1+ni) \\ =(1000)[1+(24/365)(.05)] \\ =(1000)[1+0.00329] \\ =(1000)[1.00329] \\ =\$1003.29")
NOTE: In Simple Interest; Accumulated Interest from prior periods is not used in calculations for the following periods.
Example 2. An investor puts $1,000 in a savings account that pays 10% simple interest at the end of each year. Compare how much the investor would have after 2 years if the money was:
(a) invested continuously for 2 years
(b) invested for 1 year, then immediately reinvested for a further year.
Solution:
(a) invested continuously for 2 years
![A=C(1+ni) \\ =(1000)[1+(2)(.1)] \\ =(1000)[1+0.2] \\ =(1000)[1.2] \\ =\$1200](http://s0.wp.com/latex.php?latex=A%3DC%281%2Bni%29+%5C%5C++%3D%281000%29%5B1%2B%282%29%28.1%29%5D+%5C%5C++%3D%281000%29%5B1%2B0.2%5D+%5C%5C++%3D%281000%29%5B1.2%5D+%5C%5C++%3D%5C%241200+&bg=ffffff&fg=000&s=0 "A=C(1+ni) \\ =(1000)[1+(2)(.1)] \\ =(1000)[1+0.2] \\ =(1000)[1.2] \\ =\$1200")
(b) first invested for 1 year
![A=C(1+ni) \\ =(1000)[1+(1)(.1)] \\ =(1000)[1+0.1] \\ =(1000)[1.1] \\ =\$1100](http://s0.wp.com/latex.php?latex=A%3DC%281%2Bni%29+%5C%5C++%3D%281000%29%5B1%2B%281%29%28.1%29%5D+%5C%5C++%3D%281000%29%5B1%2B0.1%5D+%5C%5C++%3D%281000%29%5B1.1%5D+%5C%5C++%3D%5C%241100+&bg=ffffff&fg=000&s=0 "A=C(1+ni) \\ =(1000)[1+(1)(.1)] \\ =(1000)[1+0.1] \\ =(1000)[1.1] \\ =\$1100")
Now, the investor has cash $1100. And, he immediately reinvested the $1100 for a further year
![A=C(1+ni) \\ =(1100)[1+(1)(.1)] \\ =(1100)[1+0.1] \\ =(1100)[1.1] \\ =\$1210](http://s0.wp.com/latex.php?latex=A%3DC%281%2Bni%29+%5C%5C++%3D%281100%29%5B1%2B%281%29%28.1%29%5D+%5C%5C++%3D%281100%29%5B1%2B0.1%5D+%5C%5C++%3D%281100%29%5B1.1%5D+%5C%5C++%3D%5C%241210+&bg=ffffff&fg=000&s=0 "A=C(1+ni) \\ =(1100)[1+(1)(.1)] \\ =(1100)[1+0.1] \\ =(1100)[1.1] \\ =\$1210")
When the investor withdrew his cash after one year and immediately reinvested it again for a further a year, he obtained more money than the continuous investment for two years. So, there is inconsistency in the simple interest. And, hence, simple interest in not practicable. That is why, we always use Compound Interest in calculations.
Next: Compound Interest