Problem 63 - Elementary Algebra

Question

Set up an algebraic equation and then solve. Alice puts money into two accounts, one with \(2 \%\) annual interest and another with \(3 \%\) annual interest. She invests 3 times as much in the higher yielding account as she does in the lower yielding account. If her total interest for the year is \(\$ 27.50\), how much did she invest in each account?

Step-by-Step Solution

Answer

Alice invested $250 at 2% and $750 at 3%.

Step 1: Identify Variables

Let the amount Alice invested in the account with a 2% annual interest be \(x\). Since she invested 3 times as much in the account with a 3% interest, the amount invested in that account is \(3x\).

Step 2: Formulate Interest Equations

The interest from the account with 2% interest is \(0.02x\). The interest from the account with 3% interest is \(0.03 \times 3x = 0.09x\).

Step 3: Set Up Total Interest Equation

Set up the equation for the total interest: \[0.02x + 0.09x = 27.50\]

Step 4: Combine Terms

Combine the terms on the left side of the equation:\[0.11x = 27.50\]

Step 5: Solve for \(x\)

Divide both sides by 0.11 to find the value of \(x\):\[x = \frac{27.50}{0.11} = 250\]

Step 6: Calculate Investments

Since \(x = 250\), Alice invested \(250\) dollars in the account with 2% interest, and \(3x = 750\) dollars in the account with 3% interest.

Key Concepts

Interest Calculation

The calculation of interest plays a crucial role in understanding how money grows over time when invested. Interest is essentially the cost of borrowing money or the reward for saving or investing. In this exercise, Alice earns interest from two different accounts: one with an interest rate of 2% and another with 3%.
Calculating the interest from each account involves understanding simple interest, which is determined using the formula:

• Interest = Principal × Rate

Here, the "principal" refers to the initial amount of money deposited in the account, and the "rate" is the annual interest rate expressed as a decimal. For instance, a 2% interest is represented as 0.02.
By applying this formula to each account, Alice's interest from the 2% account becomes \(0.02x\), and from the 3% account, it becomes \(0.03 \times 3x = 0.09x\). Thus, the sum of these interests equals her total annual interest income.

Variable Identification

Identifying the correct variables is essential in setting up an algebraic equation for this problem. We start by defining a variable to represent the unknown quantity.
In the given exercise, let’s denote the amount Alice invested in the account with a 2% annual interest as \(x\). Since Alice invested three times this amount in the account with a 3% interest rate, the amount invested in that account becomes \(3x\).
This process of identifying and assigning variables allows us to translate the word problem into a mathematical equation, making it easier to solve. By clearly defining these variables, we can systematically follow through with solving the problem without confusion.

Problem Solving

Problem-solving is the core ability needed to tackle algebraic equations effectively. In this exercise, we apply a logical approach to find the solution.
First, with identified variables, we formulate the interest equations for each account. The problem prompts us to add these amounts to reach a total interest of $27.50. This setup is given by:

• \(0.02x + 0.09x = 27.50\)

Next, we follow basic algebraic steps: combining like terms to simplify the equation to \(0.11x = 27.50\).
Then, solving for \(x\), we get the equation \(x = \frac{27.50}{0.11}\). Solving this gives \(x = 250\), indicating the amount invested in the 2% account. Problem-solving involves not only mathematical calculations but also logic and reasoning to reach the correct solution.

Investment Distribution

The correct investment distribution is vital for achieving desired financial returns. In our exercise, Alice strategically invests her money in two accounts with different interest rates.
Since Alice invests three times as much in the account with a higher interest rate (3%) than in the one with a lower rate (2%), the distribution directly affects her total interest earned.
After solving the equation, we find that Alice invested \(250\) dollars at 2% and \(750\) dollars at 3%, confirming the three-fold relationship between the amounts in the two accounts.

• Investment in 2% account = \(x = 250\) dollars
• Investment in 3% account = \(3x = 750\) dollars

This demonstrates how understanding and planning the distribution of funds can maximize interest gains efficiently.