Use the compound interest formulas to solve Exercises 10–11. Supp...

Question

Use the compound interest formulas to solve Exercises 10–11. Suppose that you have $5000 to invest. Which investment yields the greater return over 5 years: 1.5% compounded semiannually or 1.45% compounded monthly?

Accepted Answer

Hello. Today we're going to be calculating the returns of a bank using the compound interest formula. So before we jump into this problem let's just quickly recall what the compound interest formula is and that's represented as A. Is equal to p times one plus R. Over N. Race the power of N. Times T. What do those variables mean though? Well A. Is going to be our final return. P. Is the starting point or our initial investment. R. Is going to be our interest rate. N. Is the number of times that interest is applied and t. Is going to represent time. So how long were giving interest to compound? So let's go ahead and jump into this problem. We are told that Diane wants to invest her $3000 on a bank. Bank A offers 1.2% compounded monthly interest. And bank B. Offers 1.65 compounded semiannually interest. Which bank will give her the greater amount after seven years. So Two things that we can go ahead and label right off the bat. We are given a starting point or a starting price. She is starting off with $3,000. So we can let p equal to 3000. And we want to calculate how much our return is gonna be after seven years. So we're gonna go ahead and let t equal to seven. Now what we need to do is go ahead and calculate how much each bank will return in seven years. So let's go ahead and start with bank A. So we are told that bank a offers a 1.2% interest rate. So for bank a. R. is going to equal to 1.2%. Now we don't want to work with the percentage we want to work with the decimal in order to convert this percentage to a decimal we're gonna divide this value by 100. So 1.2% divided by 100 is gonna give us a rate of 0.012. Now we are told that this is compounded monthly. So there's 12 months in one year. So that means that we're gonna let n equal to 12 because this that means that this interest is gonna be applied 12 times per year. So we have P. T. R. And n. Let's go ahead and apply this to our compound interest formula. So plugging in everything together is going to give us A. Is equal to P. Which is 3000 times one plus R. Which is 0.0 12 divided by N. Which is 12 race of the power of N times T. Which is seven. So simplifying this 0.012 divided by 12 is going to give us a value of 0.001 and one plus 0.1 is gonna give us 1.1. So A. Is going to equal to 3000 times 1.1 raised to the power of 12 times seven which is 84. Now 1.001 race to the power of 84 is gonna give us a value of 1.09. So a is going to equal to 3000 times 1.09 And 2 3000 times 1.09 is going to give us a final value of 3270. So for bank a we are getting a total return of $3,270 by the end of seven years Now, let's go ahead and calculate the total return for Bank B. So for bank B we are told that it offers an interest rate of 1.65%. So R is going to equal to 1.65% and just like bank a. We want this to be a decimal. So we're going to divide this value by 101. divided by 100 is going to give us 0.0165. And we are told that this interest is going to be compounded semiannually now, semiannually means every six months. That means that this interest is going to be applied twice a year. So N is going to equal to two. Now we have all the information we need for plank B. So let's go ahead and apply this to our compound interest formula. So A is going to equal to P which again is times one plus R which is 0.165 divided by N. Which is to raise to the power of N. Two times T. Which is seven because seven years. Well 0.0165 divided by two is gonna give me a value of 0. plus 0.825 is going to give me the value Of 1.008-5. And that's going to be raised the power of two times seven which is 14. Now 1.825. Race of the power of 14 is going to give me a value of 1.12. So A. Is equal to 3000 times 1. and 3000 times 1.12 is going to give me a total value of $3360. And that's gonna be the total return that will be getting from bank be after seven years now. Let's just go ahead and quickly take a look at both of the final returns. Again. We want to go ahead and choose the bank that's going to give us the greater amount. And in this case here it's gonna be bank B. Because they're giving us a return of $3360 compared to 3270. So the answer to this problem is going to be be we want to invest in bank B. So I hope this video helps you in understanding how to use the compound interest formula, and I'll go ahead and see you all in the next video.

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