Problem 133 - College Algebra

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

The total interest earned at the end of the year if 6000 dollars are invested in the safe, government-insured certificate of deposit is $600.

Step 1: Simplify the Algebraic Expression

Simplify the given algebraic expression \(0.05x + 0.12(10,000-x)\) to its simplest form. Begin by distributing \(0.12\) across \(10,000 - x\), this gives us \(0.05x + 1200 - 0.12x\). Further simplify by combining like terms, which results into \(-0.07x + 1200\).

Step 2: Substitute the value of x Into the Simplified Expression

Substitute the value of \(x = 6000\) into the simplified expression \(-0.07x + 1200\). This gives \(-0.07(6000) + 1200\), which simplifies to \(600\).

Step 3: Interpret the Result

The result from step 2 is the total amount of interest earned in a year when 6000 dollars is invested in the safe, government-insured certificate of deposit.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. It is a key component in algebra and is used to represent real-world situations mathematically. In our exercise example, \(0.05x + 0.12(10,000-x)\) represents the total interest earned from two different types of investments.

To simplify an algebraic expression, you combine like terms and perform arithmetic operations. Simplification helps to make the expression easier to understand and work with, especially in solving equations or evaluating the expression for different values of the variables involved. It’s like tidying up your room so you can move around easily without tripping over anything!

Investment Interest Calculation

The concept of investment interest calculation is used to determine the return on investment for various financial instruments. Interest can be calculated using different methods, and it's essential to understand the terms such as 'principal,' which is the initial amount of money invested, and 'interest rate,' which is the percentage of the principal earned or paid. In the given exercise, there are two investments with different interest rates: a certificate of deposit (CD) with a lower, safer rate and corporate bonds with a higher, riskier rate.

By using algebraic expressions to represent the total interest, we can easily plug in different investment amounts to see how they will affect the return, optimizing investment strategies according to individual preferences for risk and reward.

Substitution in Algebra

In mathematics, substitution is the process of replacing a variable in an expression with its actual value. This is fundamental in algebra when you want to find out the result of the expression for particular values of the variables.

For instance, the exercise asks to substitute \(x\) with \(6000\) into the simplified interest expression. By doing this, we’re essentially saying, 'What if we invested \($6000\) into the government-insured CD?' By substituting the value, computing, and interpreting the result, we can make informed decisions about where to place our investments.

Combining Like Terms

The technique of combining like terms is a cornerstone of simplifying algebraic expressions. Like terms have the same variable(s) raised to the same power(s). Essentially, they’re terms that look alike and can logically be added or subtracted.

Different coefficients are the only distinction between like terms. In our problem, combining \(0.05x\) and \(-0.12x\) allows us to consolidate the expression into a single term with the variable \(x\), streamlining the calculation and setting the stage for easily substituting values for \(x\). The aim of combining like terms is to reduce complexity, make calculations more straightforward, and clarify the overall structure of the algebraic expression.