Question 1090036: You deposit $3,000 in a savings account that earn 9% simple interest per year. How many years will it take to double your balance? If instead you deposit the $3,000 in another savings account that earns 8% interest compounded yearly, how many years will it take to double your balance?
Answer by EMStelley(208)   (Show Source):
You can put this solution on YOUR website! The formula for the total amount, using simple interest is
Where A is the total amount, P is the principal (initial) amount, r is the interest rate in decimal form, and t is the number of years.
In this case, P = 3000 and r = 0.09. The first question is how many years until you double your balance. Double the balance would be 6000. So we want to find what t is when A is 6000.
Divide both sides by 3000
Divide both sides by 0.09
The problem itself does not say how to round. If the answer should be in whole years, the answer would be 23 years. Why? If you round down to 22, note that your balance wouldn't have fully doubled because it takes 22.222 (repeating) years to double. So you would have to round up. If the answer dictates how many decimal points to round to, round accordingly.
The second part of the problem asks the same question, but with r = 0.08 compounded yearly. The formula for compound interest is
Where A is the total amount, P is the principal amount, r is the rate in decimal form, n is the number of times per year you compound, and t is the number of years. So we have P = 3000, A = 6000, r = 0.08, n = 1, and we are solving for t.
To solve this, we need to use a natural logarithm since our variable is in an exponent. Taking the ln of both sides, we have
Due to the laws of logarithms, we can pull the t down.
So
So again, if your answer should be in whole years, it would technically be 10, because the balance wouldn't be quite doubled in just 9 years. Otherwise, round accordingly.
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