The formula for compound interest is the following:
A = P(1 + r/n)nt
A is the final amount, P is the initial principal, r is the annual interest rate, n is the number of compounding periods in a given time t.
In this case here are the values:
A = 20000 (the question says we want to end up at double the initial principal of 10000)
P = 10000
r = .14 or 14%
n = 2 (the reason n is 2 is because the question tells us that the compounding happens semi-annually. This means that it happens 2 times in one year which is the time period that the interest is applied. The words in the question that tell us this are: "at an annual interest of 14 percent compounded semi- annually"
Ok, let us substitute this information into the compound interest formula. We will get the following equation:
A = P(1 + r/n)nt
20000 = 10000(1+0.14/2)^(2t)
Now, we need to solve for t in order to get the answer to our question:
Step 1:
20000 = 10000 (2.14/2)^(2t)
Since 2.14/2 is 1.07, we can rewrite this as:
20000 = 10000 (1.07)^(2t)
Step 2:
Divide both sides by 10000:
20000/10000 = (10000/10000)*(1.07^2t)
This becomes:
2 = (1.07)^(2t)
Step 3:
To solve for a variable that is an exponent, we need to use the log function and take log of both sides:
log (2) = log (1.07 ^(2t))
At this point we have to know a log formula which says the following:
log (x^d) = d * log(x)
As you can see, the log of x raised to a power d can be written as d * log (x). Let us do this to our equation above:
log (2) = log (1.07 ^(2t))
log (2) = 2t * log (1.07) ---> Note what is going on here. 2t was the power or exponent. It is the same as d from the above formula. Thus, it becomes the multiple.
Step 3:
Now we have this equation to solve:
log (2) = 2t * log(1.07)
Divide both sides by log(1.07)
log(2)/log(1.07) = 2t *log(1.07)/log(1.07)
That becomes:
log(2)/log(1.07) = 2t
Use your calculator to divide the logs and you get this:
10.244768 = 2t
Step 4:
Now we have a simple equation that we can solve for t:
2t = 10.244768
Divide both sides by 2
2t/2 = 10.244768/2
t = 5.122384
This is your answer.
It will take 5.122384 years to double the principal of 10,000 if there was an interest rate of 14% compounded semi-anually.
