Kevin S. answered • 04/14/24
Mechanical Engineer with 10+ Years as Algebra 1 Tutor
Hi there! To tackle this question, you'll need the compound interest formula:
A = P(1 + r/n)nt
A = the final amount
P = the principal, the starting amount
r = the interest rate, as a decimal
n = the number of times interest is compounded over the time period
t = the time period
So, in this question, we are looking for the interest rate, so r is going to be our variable we're looking for. Let's look at what the other variables will be.
A and P are interesting, because we don't have specific numbers given, just that the final amount has to be double the starting amount. So, we can really pick any numbers we want that fits that! Let's use some easy numbers:
A = 20
P = 10
n depends on how many times we're going to compound over the period t. Since we're looking for t, the info we have to look at is the weekly bit. We can measure t in whatever units we want: years, months, weeks, etc., so it makes sense to use t in weeks. And since the compounding is once a week, we get the following:
n = 1
t = 60
So now we can plug in all our variables.
20 = 10(1 + r/1)(1*60)
or simplifying
20 = 10(1+r)60
To solve for r, the first step is getting rid of that number outside the parentheses through divison.
2 = (1+r)60
Now we have to deal with the exponent. The quickest way is to deal with it is using logarithms and log rules. So, let's take the log of both sides.
log(2) = log( (1+r)60 )
Now with the log rules about exponents inside logs, we can move the 60:
log(2) = 60*log(1+r)
Now we divide by 60 to get rid of it from the right side
log(2) / 60 = log(1+r)
That left side is something we can just put in the calculator to get 0.00502 (rounded)
0.00502 = log(1+r)
We rewrite that using the the definition of logs (remember that it's a base 10 log by default)
1+r = 10^(0.00502) = 1.0116 (rounded)
And lastly we just subtract that 1 off.
r = 0.0116, which translates to 1.16% as a percentage.
Hope that helps!
