You invested \$30,000 in two accounts paying 2.19% and 2.45% annual interest. If the total interest earned for the year was \$705.88, how much was invested at each rate?

검증된 단계별 안내

Define variables for the amounts invested: let $x$ be the amount invested at 2.19%, and $y$ be the amount invested at 2.45%.

Write an equation representing the total amount invested: $x + y = 30000$.

Write an equation representing the total interest earned: $0.0219x + 0.0245y = 705.88$.

Use the first equation to express one variable in terms of the other, for example, $y = 30000 - x$.

Substitute $y = 30000 - x$ into the interest equation and solve for $x$, then use that value to find $y$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Systems of Linear Equations

This concept involves solving two or more equations with multiple variables simultaneously. In this problem, the total investment and total interest create two equations that must be solved together to find the amounts invested at each interest rate.

Simple Interest Calculation

Simple interest is calculated using the formula I = P × r × t, where I is interest, P is principal, r is the annual interest rate, and t is time in years. Understanding this helps translate the interest earned from each account into algebraic expressions.

Variable Assignment and Equation Setup

Assigning variables to unknown quantities (e.g., amounts invested) and setting up equations based on the problem's conditions is crucial. This step transforms the word problem into a solvable mathematical model.