To solve each problem, refer to the formulas for compound inte...

Question

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)^tn and A = Pe^rt At what interest rate, to the nearest hundredth of a percent, will $16,000 grow to $20,000 if invested for 7.25 yr and interest is compounded quarterly?

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're going to be looking at this math question that states what interest rate rounded to the nearest 100th of a percent is needed for an investment of 28,000 to grow to 35,000 over 12.25 years with quarterly compounding of interest. Now, the S provided are a 1.1% B 3.98% C 2.57% and D 1.83%. Now with a question like this where we're talking about rates and compounding and, and money, how it's gonna grow over years. We have a formula that states that A is equal to P multiplying a parentheses with one plus R divided by N inside of that parentheses. And that parentheses raced to the N T exponent. No, the A is the final amount, the P is the initial amount or the proper word would be principle. The R is the annual interest, right? The end is the number of times interest is compounded per year, per year. And finally T is the time in years. So let's rewrite our formula with the numbers that they gave us. So A is the final amount. Our final amount is $35,000 because that's what we want to grow to. So we'll have $35,000. It's equal to P or our initial amount, which in this case is the 28,000, which we invest. And that'll multiply one plus R we're solving for the rate. So we just leave it as R divided by N which is the number of times compounded per year. And they tell us this is quarterly. So that'll be four times a year. So four is our end. And that parentheses is raised to the four to NT power. So four multiplied by T which is the time in years. And they tell us it is 12.25 years. Now it's a matter of simplifying and solving for R. So the first simplification we can do is divide both sides by 28,000. And also multiply the exponent that we have the four by the 12.25. So once we do that, we'll have one plus R divided by four, raced to the exponent 49 which is four multiplied by 12.25. And that'll be equal to the 35,000 divided by the 28,000, which is 1.25. Now, we can take the log of both sides to get rid of the exponent that we have. So we'll have log of one plus R divided by four and the one plus R divided by four raised to the 49 exponent. And that'll be equal to the dog of 1.25. And now we can do a simple recall of a rule that we have for logarithms on how we can manipulate exponents, how move the exponent down. So I'm gonna write a general formula, let's say you have log of base B of A to the exponent C that is equal to C multiplying the log of base B of A. So all you do is bring the exponent down in front of the log and multiply it by the log. So we'll do that. We have the 49 is our exponent here. So that's the C in our general formula, we'll move it forward in front of the exponent or in front of the log. So we'll have 49 multiplied by log of one plus R divided by four. And that'll still be equal to the log of 1.25. Now, we can continue to do some simplification here. So I'm going to move the 49 over from the left hand side to the right hand side by dividing. So I'll have log of one plus R divided by four and that'll be equal to here. I, I will multiply it by the fraction of one divided by 49. It's the same as dividing by 49. So I have that fraction multiplying the log of 1.25. And now we have to plug that into our calculator to get an answer. So I'll have log of one plus R divided by four is equal to. And now in my calculator, I'll write log of 1 25 or, or log of 1.25. And that answer, I'll divide it by 49 and we should get something like 0.00 19778. And that's where it will stop. And then we can recall another, another formula for logs. So let's say you have log of base B of A equals C. That can be rewritten as B to the exponent C equals A. So this is a way that we can manipulate a logarithm to turn it into an exponential equation. So in our case, we have log of one plus R divided by four equals or answer. So the one plus R divided by four will be the A and our answer will be the C and our base here is 10 because they didn't specify or we're just working with a normal lab which are have a base of 10. So in that case, we'll have one plus R divided by four equals 10 to the exponent 0.0019778. And once again, we'll have to plug that into our calculator. So we'll have one plus R divided by four equals 1.0. 0456433502. And now we can bring the one over from the left hand side to the right hand side. So we have R divided by four is equal to R and minus one. Which one I'll be 0.00456433502. And now if we multiply both sides by four, we get rid of the four under the denominator and we'll have the R is equal to 0.01 825734. And now to go from a decimal to a percentage, we just move the decimal point over to the right twice. Which means that the rate and we'll have to estimate now to the 100th place like they asked. So it will be that R is around 1.8 three percent because we had a 10.01825. So we cut it off at the two and because we have a five after the two means we have to round up, which means we have 1.83%. And that'll be our answer for the rate or in this case, answer choice. D so I hope that was helpful. And until next time.

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)^tn and A = Pe^rt At what interest rate, to the nearest hundredth of a percent, will \$16,000 grow to \$20,000 if invested for 7.25 yr and interest is compounded quarterly?

Key Concepts

Compound Interest Formula

Solving for the Interest Rate

Compounding Frequency and Time

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