Problem 133 - College Algebra Essentials

Question

You had \(\$ 10,000\) to invest. You put \(x\) dollars in a safe, government- insured certificate of deposit paying \(5 \%\) per year. You invested the remainder of the money in noninsured corporate bonds paying \(12 \%\) per year. Your total interest earned at the end of the year is given by the algebraic expression $$0.05 x+0.12(10,000-x)$$ a. Simplify the algebraic expression. b. Use each form of the algebraic expression to determine your total interest earned at the end of the year if you invested \(\$ 6000\) in the safe, government-insured certificate of deposit.

Step-by-Step Solution

Answer

a. The simplified algebraic expression is \(-0.07x + 1200\). b. The total interest earned at the end of the year when $6000 is invested in the government-insured certificate is $780.

Step 1: Simplify the Expression.

To simplify, start by applying the distributive property to the term \(0.12(10000 - x)\). This gives: \(0.05x + 1200 - 0.12x\). Now combine like terms by subtracting \(0.12x\) from \(0.05x\). This gives you a simplified algebraic expression: \(-0.07x + 1200\).

Step 2: Substitute the given value of x into the Original Expression.

Replace \(x\) with \(6000\) in the original expression, so: \(0.05*6000 + 0.12(10000 - 6000) = 300 + 0.12*4000 = 300 + 480 = 780\).

Step 3: Substitute the given value of x into the Simplified Expression.

Replace \(x\) with \(6000\) in the simplified expression, so: \(-0.07*6000 + 1200 = -420 + 1200 = 780\).