10.3 Compound Interest 607 Compound Interest Formula A p r n 1 n t ( ) = ⎛ + ⎝ ⎞ ⎠ ⋅ where A is the amount that accumulates in the account, p is the principal, r is the annual interest rate as a decimal number, n is the number of compounding periods per year, and t is the time in years. When using the compound interest formula, if the interest is compounded semiannually, use n 2; = quarterly, use n 4; = monthly, use n 12; = weekly, use n 52; = daily, use n 360. = Recall from Section 10.2, that the Banker’s Rule considers a year to have 360 days. We will now use this formula to show how an investment can grow using compound interest. Example 1 Using the Compound Interest Formula Kathy invested $3000 in a savings account with an interest rate of 1.8% compounded monthly. If Kathy makes no other deposits into this account, determine the amount in the savings account after 2 years. Solution We will use the formula for compound interest, A p r n 1 . n t ( ) = ⎛ + ⎝ ⎞ ⎠ ⋅ The principal, p, is the amount of money invested, so p $3000. = The interest rate, r, is 1.8%, so r 0.018. = Because the interest is compounded monthly, there are 12 compounding periods per year and n 12. = Because the money is invested for 2 years, t 2. = Now try Exercise 21 A 3000 1 0.018 12 3000(1 0.0015) 3000(1.0015) 3000(1.0366279) 3109.88 (12 2) 24 24 = ⎛ + ⎝ ⎞ ⎠ = + = ≈ ≈ ⋅ Thus, the amount in the account after 2 years would be about $3109.88. 7 For the fourth quarter, use a principal of $5113.34. i A $5113.34 0.03 0.25 $38.35 $5113.34 $38.35 $5151.69 = × × ≈ = + = Hence, the $5000 grows to a final value of $5151.69 over the 1-year period. This example shows the effect of earning interest on interest, or compounding interest. In 1 year, the amount of $5000 has grown to $5151.69, compared with $5150 that would have been obtained with a simple interest rate of 3%. Thus, in 1 year alone the gain was $1.69 more with compound interest than with simple interest. A simpler and less time-consuming way to calculate compound interest is to use the compound interest formula and a calculator.
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