3. Section 9.3
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The formula that gives the balance y of an account earning compound interest, where P is the principal, r is the annual interest rate, t is the time in years, and n is the number of times the interest is compounded in one year.
278 months, or about 23 years
Compound interest is the interest earned on the principal and on previously earned interest. Let's recall the formula that gives the balance A of an account earning compound interest.
A= P( 1+r/n )^(n t)
In this formula, P is the principal, or initial amount, r is the annual interest rate written in decimal form, t is the time in years, and n is the number of times the interest is compounded in one year. Let's carefully consider the given exercise.
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$ 1000 is deposited into a savings account that pays 6 % annual interest compounded monthly. |
We can immediately identify P as 1000. The annual interest rate, written as a decimal number, is 0.06. Finally, since the interest is compounded monthly and there are 12 months in one year, we have that n= 12. We want to calculate how long we have to wait until the account is worth A = 4000. Let's substitute these values into the formula and simplify.
Substitute values
Calculate quotient
Add terms
.LHS /1000.=.RHS /1000.
Now we want to solve the simplified equation. To do so, we will use logarithm, since the logarithm is the inverse function of exponentiation used in the equation. Let's do it!
log_(1.005)(LHS)=log_(1.005)(RHS)
log_(1.005)(1.005^m)=m
Calculate logarithm
Round to nearest integer
.LHS /12.=.RHS /12.
Calculate quotient
Round to nearest integer
Rearrange equation
We found that t is approximately 23. We have to wait about 23 years until the account is worth $ 4000.