Problem 59 Set up an algebraic equation and... [FREE SOLUTION]

Chapter 2: Problem 59

Set up an algebraic equation and then solve. Joe invested last year's $\$ 2,500$ tax return in two different accounts. He put most of the money in a money market account earning $5 \%$ simple interest. He invested the rest in a CD earning $8 \%$ simple interest. How much did he put in each account if the total interest for the year was $\$ 138.50 ?$

Short Answer

Expert verified

Joe invested $2,050 in the money market account and $450 in the CD.

Step by step solution

Define Variables

Let $ x $ be the amount Joe invested in the money market account at 5% interest, and let $ 2500 - x $ be the amount he invested in the CD at 8% interest.

Set Up Interest Equations

The interest from the money market account is $ 0.05x $ and the interest from the CD is $ 0.08(2500 - x) $. The sum of these interests is equal to \$138.50.

Write the Equation

Write the equation based on the information: \[ 0.05x + 0.08(2500 - x) = 138.50 \].

Simplify the Equation

Distribute the 0.08 in the equation: \[ 0.05x + 200 - 0.08x = 138.50 \].

Combine Like Terms

Combine $ x $-terms and constants: \[-0.03x + 200 = 138.50 \].

Solve for x

Subtract 200 from both sides: \[-0.03x = -61.5 \]. Then divide by $-0.03$ to solve for $ x $: \[ x = 2050 \].

Solve for Remaining Investment

Subtract the value of $ x $ from the total investment to find the amount in the CD: \[ 2500 - 2050 = 450 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Interest

Simple interest is a way to calculate the interest earned or paid on a principal amount over a certain period of time. It is called "simple" because the interest is calculated only on the initial amount, or principal, which does not change over time.

Here's how you can understand simple interest more easily:

The formula to calculate simple interest is: \[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \] where the Rate is usually expressed as a decimal and Time is the period for which the money is invested or borrowed.
It is straightforward to compute as the amount of interest does not change over time.
It is commonly used in short-term loan or investment scenarios.

In Joe's case, each account calculates interest simply, using the principal times the rate.

Investment Accounts

Investment accounts are types of accounts where money is deposited or invested, and they earn interest over time. Different accounts can offer different returns based on the amount invested and the interest rates.

Here are key points to understand about investment accounts:

Money Market Account: This account typically offers a higher interest rate than traditional savings accounts. It may come with restrictions on withdrawals but is known for security and decent interest returns.
Certificate of Deposit (CD): A CD offers fixed interest rates for a specified period. It generally provides higher rates than a regular savings account but requires the money to stay in the account for a set term.
Both types of accounts in Joe's situation use simple interest to calculate returns on the amount invested.

Investment accounts like these offer different levels of liquidity and interest, making them a diverse choice for varied financial goals.

Solving Equations

Solving equations is a key process in algebra, which involves finding the value of unknown variables that satisfy a given equation. In this type of problem, you are often provided with an equation derived from the problem's context.

When solving equations like Joe's investment problem:

First, write down the equation that represents the problem.
Use arithmetic operations to simplify and isolate terms, combining like terms when necessary.
Rearrange the equation to get the variable on one side, allowing you to solve for it.
Ensure every step logically follows, keeping track of both operations and initial conditions.

In Joe鈥檚 problem, solving the equation allows us to determine the exact amounts he invested in each account.

Variable Definition

Variable definition is an essential step in forming and solving equations, particularly in word problems where numbers are represented by symbols. A variable typically represents an unknown value that you need to find.

To effectively define variables:

Identify the quantities in the problem that are unknown but required to be solved.
Select a simple letter, often $ x $, to represent one of these unknown amounts.
Understand that other unknown quantities can be expressed in relation to the variable set, like $ 2500 - x $ for the other amount Joe invested.
Clearly defining variables helps in setting up equations correctly and avoids confusion.

In Joe's example, defining $ x $ as the money in the money market account and $ 2500 - x $ as the investment in the CD made the equation straightforward to solve.

91影视

Short Answer

Step by step solution

Define Variables

Set Up Interest Equations

Write the Equation

Simplify the Equation

Combine Like Terms

Solve for x

Solve for Remaining Investment

Key Concepts

Simple Interest

Investment Accounts

Solving Equations

Variable Definition

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