You inherit one million dollars. You invest it all in three...

You inherit one million dollars. You invest it all in three accounts for one year. The first account pays 3$\%$ compounded annually, the second account pays 4$\%$ compounded annually, and the third account pays 2$\%$ compounded annually. After one year, you earn $\$ 34,000$ in interest. If you invest four times the money into the account that pays 3$\%$ compared to 2$\%$ , how much did you invest in each account?

You inherit one million dollars. You invest it all in three accounts for one year. The first account pays
3$\%$ compounded annually, the second account pays 4$\%$ compounded annually, and the third account pays 2$\%$ compounded annually. After one year, you earn $\$ 34,000$ in interest. If you invest four times the money into the account that pays 3$\%$ compared to 2$\%$ , how much did you invest in each account?

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Key Concepts

Mixture Problems

Mixture problems involve combining different components—in this case, investments at different interest rates—to achieve a specific overall outcome, such as a target amount of interest earned. The approach requires balancing the contributions of each component according to their respective rates and amounts.

Ratio and Proportion

Ratio and proportion play a critical role when a problem specifies a relationship between investments in different accounts. This concept allows setting up a proportional relationship between amounts invested in various accounts, which simplifies the process of forming equations to solve for the unknown values.

Compound Interest

Compound interest refers to the process where interest is calculated not only on the initial principal but also on the accumulated interest from previous periods. This concept is central to problems involving investments with rates that vary by account and compounding frequency, as it determines how the principal grows over time.

Systems of Equations

In many investment allocation problems, unknown amounts are distributed among different accounts under several constraints, such as total investment and total interest earned. These constraints naturally lead to forming a system of equations that can be solved simultaneously to find the individual amounts invested.

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