Problem 263 - Elementary Algebra

Question

Hattie had \(\$ 3,000\) to invest and wants to earn \(10.6 \%\) interest per year. She will put some of the money into an account that earns \(12 \%\) per year and the rest into an account that earns \(10 \%\) per year. How much money should she put into each account?

Step-by-Step Solution

Answer

Invest \(900 in the 12% account and \)2100 in the 10% account.

Step 1: Define Variables

Let the amount of money Hattie invests in the 12% account be represented by the variable x. The remaining amount she invests in the 10% account will then be represented by 3000 - x.

Step 2: Set Up the Interest Equations

The interest earned from the 12% account is represented by 0.12x. The interest earned from the 10% account is represented by 0.10(3000 - x).

Step 3: Formulate the Total Interest Equation

Formulate the equation based on the total interest Hattie wants to earn: 0.12x + 0.10(3000 - x) = 0.106 * 3000

Step 4: Simplify the Equation

Distribute and simplify the equation: 0.12x + 300 - 0.10x = 318 Combine like terms: 0.02x + 300 = 318

Step 5: Solve for x

Isolate the variable x: 0.02x = 318 - 300 0.02x = 18 Divide both sides by 0.02: x = 18 / 0.02 = 900

Step 6: Determine Amounts for Each Account

Hattie should invest \(900 in the 12% account and \)3000 - \(900 = \)2100 in the 10% account.

Key Concepts

Solving Linear Equations

Understanding linear equations is essential when solving investment word problems. A linear equation is an equation that makes a straight line when graphed. It is often written in the form ax + b = c. In Hattie's case, we set up a linear equation based on her total desired interest. This equation helps us determine how much money should go into each investment account. By isolating the variable, we can solve for the unknown amount. Remember to perform operations such as addition, subtraction, multiplication, and division to both sides of the equation to maintain equality.

Interest Calculations

Interest calculation is fundamental in finance and investments. It allows us to understand how much profit an investment will generate. The formula for simple interest is I = P * r * t, where I is the interest, P is the principal amount, r is the rate of interest, and t is the time period. For Hattie's problem, the interest earned from the 12% account is calculated as 0.12x, and from the 10% account, it's 0.10(3000 - x). Combining these helps us see how close we are to her desired interest of 10.6% on her total investment.

Defining Variables in Algebra

Defining variables is critical to setting up equations that model real-world scenarios. In Hattie's investment problem, we need to represent unknown quantities with variables. Here, we let x be the amount invested in the 12% account. Therefore, the amount in the 10% account is the remainder of her total investment, which is 3000 - x. Clear definition of variables reduces complexity and helps in solving the equations effectively. Always make sure to clearly specify what each variable represents to avoid confusion.

Distributive Property

The distributive property is a vital algebraic property used to simplify expressions and solve equations. It states that a(b + c) is equivalent to ab + ac. In Hattie's problem, we apply the distributive property when dealing with the interest from the 10% account: 0.10(3000 - x) becomes 0.10 * 3000 - 0.10 * x, simplifying to 300 - 0.10x. By distributing, we make the equation easier to solve by breaking it into more manageable parts.

Combining Like Terms

Combining like terms is an essential step in solving algebraic equations. It means simplifying expressions by adding or subtracting terms that have the same variables raised to the same power. In our example, after distributing, we have the equation 0.12x + 300 - 0.10x = 318. By combining like terms (0.12x - 0.10x), we simplify it further to 0.02x + 300 = 318. This simplification step is crucial for solving the equation efficiently. Make sure to group and combine terms with the same variable to streamline your solution process.