/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 You deposit \(\$ 3000\) in an ac... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 41

You deposit \(\$ 3000\) in an account that pays \(7 \%\) interest compounded semiannually. After 10 years, the interest rate is increased to \(7.25 \%\) compounded quarterly. What will be the value of the account after 16 years?

Short Answer

Expert verified
To compute the final value of the account, calculate the compound interest for the first 10 years, then compute the compound interest for the next 6 years. The sum of these two will give you the final balance of the account after 16 years.

Step by step solution

01

Calculate the value of the account after the first 10 years

The formula for compound interest is \(A = P(1 + r/n)^{nt}\), where \n - \(P\) is the principal amount (the initial amount of money), \n - \(r\) is the annual interest rate (in decimal form), \n - \(n\) is the number of times that interest is compounded per year, and \n - \(t\) is the time in years. \n Here, \(P = $3000\), \(r = 7/100 = 0.07\), \(n = 2\) (as the interest is compounded semi-annually), and \(t = 10\). Inserting these values into the formula gives us the value \(A\) which is the value of the account after the first 10 years.
02

Calculate the value of the account for the next 6 years

After the first 10 years, the interest rate changes to 7.25% and it is compounded quarterly for the next 6 years. Here, \(P\) is the value of the account after the first 10 years, \(r = 7.25/100 = 0.0725\), \(n = 4\) (as the interest is compounded quarterly), and \(t = 6\). Again applying the compound interest formula will give us the final value of the account after the complete 16 years.
03

Calculate the total value of the account

Add the values obtained from step 1 and step 2. This gives the total value of the account after 16 years.

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