Solve each problem. See Example 4. Cody sells some property for \$240,000. The money will be paid off in two ways: a short-term note at 2% interest and a long-term note at 2.5%. Find the amount of each note if the total annual interest paid is \$5500.

Verified step by step guidance

1. Define variables for the amounts of the two notes. Let \(x\) be the amount of the short-term note at 2% interest, and \(y\) be the amount of the long-term note at 2.5% interest.
2. Write an equation representing the total amount of the property sold: \(x + y = 240000\).
3. Write an equation representing the total annual interest paid. The interest from the short-term note is \$0.02x\( and from the long-term note is \)0.025y\(. The total interest is \)5500\(, so \)0.02x + 0.025y = 5500$.
4. Use the first equation to express one variable in terms of the other, for example, \(y = 240000 - x\).
5. Substitute this expression for \(y\) into the interest equation and solve for \(x\). Once \(x\) is found, substitute back to find \(y\).

Key Concepts

Simple Interest Calculation

Simple interest is calculated using the formula I = P × r × t, where I is interest, P is principal, r is the interest rate, and t is time in years. Understanding this helps determine how much interest each note generates based on its principal and rate.

System of Linear Equations

A system of linear equations involves two or more equations with multiple variables. Solving such systems allows us to find the unknown amounts of the short-term and long-term notes that satisfy both the total principal and total interest conditions.

Problem Translation into Mathematical Expressions

Translating word problems into mathematical equations is essential. Here, the total amount ($240,000) and total interest ($5,500) must be expressed as equations involving the unknown note amounts, enabling the use of algebraic methods to find the solution.