Tara and Lamar decide to place 20,000 of their savings into...

Tara and Lamar decide to place $$ 20,000$ of their savings into investments. They put some in a money market account earning $3 \%$ interest, some in a mutual fund that has been averaging $7 \%$ a year, and some in a stock that rose $10 \%$ last year. If they put 600 dollars more in the money market than in the mutual fund and the mutual fund and stocks experience the same growth the next year as they did the previous year, they will earn $$ 1180$ in a year. How much money did Tara and Lamar put in each of the three investments?

Key Concepts

System of Linear Equations

In problems where several variables represent unknown amounts or quantities, a system of linear equations is used to model the relationships and constraints provided in the problem. These systems allow for the simultaneous consideration of all conditions, such as total investment amount and specific differences between investment amounts, enabling one to solve for the individual unknowns.

Substitution Method

The substitution method is an algebraic technique used to solve a system of equations by expressing one variable in terms of another. This approach simplifies the system by reducing the number of variables in one or more equations, making it easier to solve for the remaining unknowns. It is particularly useful when one of the equations in the system already relates two of the variables directly.

Investment Allocation

This concept involves dividing a total amount of money among various investment vehicles to achieve certain financial goals. In problems of this type, each investment option typically has its own rate of return or risk profile, and the allocation must adhere to the constraints provided, such as total investment amount and specific relationships between the individual amounts invested.

Interest Rate Calculations

Interest rate calculations concern computing the gains or losses on an investment over a period, typically expressed as a percentage of the principal. Understanding how different rates affect different portions of an investment is crucial for determining the overall expected return, especially in setups where multiple investment options yield different percentages.

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