A68 APPENDIX Review Interest charged according to formula (1) is called simple interest. When using formula (1), be sure to express r as a decimal. For example, if the rate of interest is 4%, then = r 0.04. Finance: Computing Interest on a Loan Suppose that Juanita borrows $500 for 6 months at the simple interest rate of 9% per annum. What is the interest that Juanita will be charged on the loan? How much does Juanita owe after 6 months? Solution EXAMPLE 2 The rate of interest is given per annum, so the actual time that the money is borrowed must be expressed in years. The interest charged would be the principal, $500, times the rate of interest ( ) = 9% 0.09 , times the time in years, 1 2 : I Prt Interest charged 500 0.09 1 2 $22.50 = = = ⋅ ⋅ = After 6 months, Juanita will owe what she borrowed plus the interest: + = $500 $22.50 $522.50 Financial Planning Lin has $2500 to invest and wants an annual return of $100, which requires an overall rate of return of 4%. She can invest in a safe, government-insured certificate of deposit, but it pays only 2%.To obtain 4%, she agrees to invest some of her money in noninsured corporate bonds paying 7%. How much should be placed in each investment to achieve her goal? Solution EXAMPLE 3 Step 1 The question is asking for two dollar amounts: the principal to invest in the corporate bonds and the principal to invest in the certificate of deposit. Step 2 Let b represent the amount (in dollars) to be invested in the bonds. Then −b 2500 is the amount that will be invested in the certificate. (Do you see why?) Step 3 We set up a table: Principal ( )$ Rate Time (yr) Interest ( )$ Bonds b = 7% 0.07 1 b 0.07 Certificate −b 2500 = 2% 0.02 1 ( ) −b 0.02 2500 Total 2500 = 4% 0.04 1 0.04 2500 100 ⋅ = Since the combined interest from the investments is equal to the total interest, we have ( ) + = + − = b b Bond interest Certificate interest Total interest 0.07 0.02 2500 100 (Note that the units are consistent: the unit is dollars on both sides.) Step 4 + − = = = b b b b 0.07 50 0.02 100 0.05 50 1000 Lin should place $1,000 in the bonds and − = $2500 1000 $1500 in the certificate. Step 5 The interest on the bonds after 1 year is ⋅ = 0.07 $1000 $70; the interest on the certificate after 1 year is ⋅ = 0.02 $1500 $30. The total annual interest is $100, the required amount. Simplify. Divide both sides by 0.05. Now Work problem 19 NOTE We could have also let c represent the amount invested in the certificate and −c 2500 the amount invested in bonds. j
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