Question 935622
THE DIFFERENCE BETWEEN COMPOUND INTEREST AND SIMPLE INTEREST AT THE RATE FOR $5000 FOR 2 YEARS IS $72.  FIND THE RATE OF INTEREST p.a.?


equation for simple interest is i = n * r * p


equation for compound interest is i = p * (1+ r)^n - p


you are being told that the difference between them after 2 years is equal to 72.


replace what you know in both equations.


i = n * r * p becomes i = 2 * r * 5000


i = p * (1 + r)^n - p becomes i = 5000 * (1 + r)^2 - 5000


the difference between them is 72.
compound interest will be greater than simple interest.
the equation you now want to solve is:


5000 * (1 + r)^2 - 5000 - (2 * r * 5000) = 72


you need to use this equation to solve for r.


simplify the equation to get:


5000 * (1 + r)^2 - 5000 - 10000 * r = 72


since (1 + r)^2 = r^2 + 2r + 1, then your equation becomes:


5000 * (r^2 + 2r + 1) - 5000 - 10000 * r = 72


distribute the multiplication to get:


5000 * r^2 + 10000 * r + 5000 - 5000 - 10000 * r = 72


combine like terms to get:


5000 * r^2 = 72


divide both sides of this equation by 5000 to get:


r^2 = 72 / 5000


take the square root of both sides of this equation to get:


r = plus or minus sqrt(72/5000) = .12


your interest rate is 12%.


with simple interest, i = n * r * p becomes i = 2 * .12 * 5000 which becomes i =  1200


with compound interest, i = p * (1 + r)^n - p becomes i = 5000 * (1.12)^2 - 5000 which becomes i = 6272 - 5000 which becomes i = 1272.


the difference between them is 1272 - 1200 = 72.


the solution is that r = .12 and it is confirmed to be good.