How Long Will It Take for an Investment to Triple Compounded Continuously at 6
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How long will it take for money to triple if it is invested at the following rates? (a) 5% compounded monthly. (b) 3% compounded annually. (a) It will take years (your answer should be a decimal number). (b) It will take years (your answer should be a decimal number)
(a) How long does it take for an investment to triple in value if it is invested at 6% compounded monthly? (b) How long does it take if the interest is compounded continuously?
Discussion
Video Transcript
Let's say that you want to invest somebody, and you're given two different plans. You're giving a plan that has a 6% annual interest rate, which we compounded monthly, or a plan that has a 6% annual interest rate that will be calm, pounded continuously. And you want to figure out how long it will take each plan to triple your investment. So let's do the monthly plan first. Since this is going to be calm pounded, we should use the compound interest formula or a equals. P has one plus R over and to the anti. We're a should equal three p because we want to triple our current value. So a the future value should be three p r equals ill 0.6 which is bit pushes just 6% in decimal form and is amount times it is calm, pounded in a year or 12 times because there are 12 months in a year and tease amount of time it will take. So let's plug everything into the formula. But we have three p. It was be because one plus still 10.6 over 12 into the 12 t, so let's divide both sides by piece, so you simply get three equals one plus 0.6 to the 12 to the 12 t Pau. Let's simplify all of this and the Prentice's by putting it into your calculator to get 1.5 So we have three equals 1.5 to the 12 t power. That means that we have to take the log of both sides, and the base should be what is here so that it will cancel out. So that means we have to take log based 1.5 of three again. Use your calculator. Use the one where you can change the base. So log based 1.5 three. So we get to So I changed the color. So let's just review log base of 1.5 of three equals log base 1.5 of one point. Oh, by to the 12th t, this will simply cancel out. And we would just bring down the exponents. So we have 220 weren't to seven again close on TV Calculator equals 12 t divide both sides by 12 and we get t equals 18 0.356 So take 18 0.356 years in order to triple your investment using that plan. Now, let's check the other plan. This one is a continuous compound plan, so we should use the continuous compound formula or a because PRT again is going to be three p and our A 0.6 So let's plug in the three p equals p times E to the 0.6 t divide both sides by P. We simply is three equals D dozo 0.6 t take Elena both sides because he will be a base and log based. Eat another way of saying that is Ln so we take l enough to be equals L, uh a E to 0.6 t We know that this cancels out to one and the exponents brought down. So let's plug in Ellen of three Interact calculator and we get 1.986 equals ill 0.0 60 Salvati. By dividing both sides by sale 0.6 we get t equals 18 0.3 years so we can see that the continuous compound, um plan will triple your investment a little faster than the one that is being compounded monthly. But it is relatively close, um, so there's not too much of a difference.
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