What is the formula for solving the following problem?
What principal will grow to $1367.10 after 1 year at 10% compounded semi-annually?
If you invest a certain principal at 10% for a year then at the end of the year you receive
(the principle) + 10% of (the principal)
If you call the principal $P then, since 10% is 10/100 = 1/10, this is
$P + 1/10 &P = $P(1 + 1/10) = $P(1.10)
If the interest is compounded semi-annually then at the end of six months you get $P plus half the interest
$P(1 + 1/20) = $P(1.05)
which is then the principal which gets invested for the second six months. At the end of this six month period the expression is the same. You get the principal times 1.05. That is
$P(1.05)(1.05) = $P(1.05)2
Thus You have
$P(1.05)2 = $1367.10
Solve for P.
This argument leads to a general expression. If you invest $P at an interest rate r which is compounded m times per year then the amount after t years is
A = P(1 + r/m)mt
Claude and Penny