You inherit one hundred thousand dollars. You invest it all...

You inherit one hundred thousand dollars. You invest it all in three accounts for one year. The first account pays 4% compounded annually, the second account pays 3% compounded annually, and the third account pays 2% compounded annually. After one year, you earn $\$ 3,650$ in interest. If you invest five times the money in the account that pays 4$\%$ compared to 3$\%$ , how much did you invest in each account?

Key Concepts

Algebraic Modeling

Algebraic modeling is the process of translating a real-world situation into mathematical expressions or equations. This enables a systematic approach to identify unknowns, establish relationships between variables, and ultimately solve for these unknowns. In the problem at hand, algebraic modeling is key to representing the conditions for total investment, interest earnings, and the stipulated proportional investment into the different accounts.

Investment Allocation

Investment allocation involves distributing a total sum of money among various accounts or assets according to specific criteria or constraints, such as differing interest rates or risk profiles. In the problem, this concept is applied by dividing a fixed amount across three different interest-bearing accounts while satisfying additional conditions regarding the relationships between the investments.

Compound Interest

Compound interest refers to the process of earning interest on both the initial principal and the accumulated interest from previous periods. In many financial scenarios, including investments and savings, it explains how money grows over time, taking into account the effect of reinvesting earned interest. Even though the given problem involves a one-year investment period where the effect is straightforward, understanding this concept is crucial in many financial calculations.

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. It is used to model and solve problems where multiple relationships exist between variables. In the context of the investment problem, setting up equations based on the total amount invested, the interest earned, and the proportional allocation between accounts is essential to find the precise amounts allocated to each account.

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