$A = P ( 1 + r ) ^ { n }$
gives amount A in an account after n years for an initial investment P that earns interest at an annual rate r. How long will it take for $250 to increase to$500 at 4% annual interest?

Solution

Verified

500 = 250(1 + 0.4)^{n}

2 = (1 + 0.4)^{n}

\log_{1.04}(2) = n

\dfrac{\log_{10}(2)}{\log_{10}(1.04)} = n

n = 17.673

Answer: 18 years

Since you are trying to find when the compounded interest will turn 250 dollars into 500 dollars, you initial investment P = 250 and A = 500.

Since the annual interest rate is 4 percent, r = 0.04.

Since you are trying to find how long it will take for 250 dollars to increase to 500 dollars, you are looking for "n" in years.

Plug these values into the equation $A=P(1+r)^n$ so that you get $500 = 250(1 + 0.4)^{n}$ .

First, divide the 250 over so that you get $2 = 1.04^n$

Next, take the log base 1.04 of both sides so that "n" will no longer be an exponent and you can solve for it.

Next, use the change of base formula:

\log_{b}(x) = \dfrac{\log_{a}(x)}{\log_{a}(b)}

where b = 1.04, a = 10, and x = 2

Put this in your calculator and you should get n = 17.673. However, since the interest is only compounded every year, it will take a full year to get at least 500 dollars. So, the answer is 18 years.

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