(a) How long does it take for an investment to triple in value if it is invested at 6 % compound

(a) How long does it take for an investment to triple in...

Key Concepts

Logarithms for Solving Exponential Equations

Logarithms are a crucial mathematical tool for solving equations where the unknown variable appears as an exponent. When determining the time required for an investment to reach a target value in compound interest or continuous compounding problems, logarithms allow us to isolate the time variable by converting exponential equations into linear ones. This concept is fundamental in various financial calculations and other applications in mathematics and science.

Continuous Compounding

Continuous compounding involves interest that is calculated and added to the account balance at every possible instant. The formula used is A = Pe^(rt), where e is the base of the natural logarithm. This method simplifies the calculation of the growth of an investment when time is treated as a continuous variable, which is particularly useful for theoretical financial models and for solving problems where the compounding frequency tends to infinity.

Compound Interest Formula

This concept involves calculating the future value of an investment based on periodic compounding. It uses the formula A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n compounding periods per year over t years, r is the annual interest rate, and P is the principal amount. This formula is essential when determining the time required for an investment to grow by a certain multiple under discrete compounding conditions.

Transcript

00:01 In this question, we're asked how long it would take for investment to triple in value.

00:06 So we know that we have to use the compound interest formula, where a is the amount after two years, p is the principal, or the amount deposited, therefore it can be also thought of as the investment, because it's how much you're putting in.

00:30 N is the number of compounds per year, and t is the number of years, while r is just the rate of interest.

00:36 So we're told that the amount of money is invested at 6 % compounded monthly.

00:47 So we know that r is equal to 6%, which is equal to 0 .06.

00:54 And we know that n is equal to 12, since it is compounded monthly.

01:03 So now all that we're going to do is plug this information into our compound interest formula.

01:13 However, since we're trying to find out how long it will take for the initial investment to triple, we know that the amount that we should end up with should be three times the principle.

01:27 So we can just set three times the principle.

01:31 To the principle and then just plug all the information we have so 1 plus 0 .06 over n to the power of n times t which is the unknown that's what we're trying to figure out in this problem so now we're just going to have to simplify this down so 3p is equal to p times 1 .005 to the power of 12 t so now we can just divide by p so the p cancels out so you get 3 is equal to 1 .005 to the power of 12t so now if you just take log on each side we get log you get log log of 3 is equal to the log of 0, 05 to the power of 12 t.

03:16 Now you can just bring this to the front.

03:21 So you get a log of 3 is equal to 12 t times the log of 1 .0 .0.

03:38 5.

03:43 So now we can just start dividing things over to rearrange for t.

03:50 To divide log 1 .005 and 12 over...

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