Problem 30 - Beginning and Intermediate Algebra

Question

Natsumi receives \(\$ 126\) per year in simple interest from three investments. Part is invested at \(2 \%\), part at \(3 \%\), and part at \(4 \% .\) There is \(\$ 500\) more invested at \(3 \%\) than at \(2 \% .\) The amount invested at \(4 \%\) is three times the amount invested at \(3 \% .\) Find the amount invested at each rate.

Step-by-Step Solution

Answer

Natsumi invested approximately \(\$471.43\) at \(2\%\) interest rate, \(\$971.43\) at \(3\%\) interest rate, and \(\$2914.29\) at \(4\%\) interest rate.

Step 1: Translate the information into equations

Based on the given information, we can create the following equations: 1. Total interest: \(0.02x + 0.03y + 0.04z = 126\) 2. More invested at \(3\%\): \(y = x + 500\) 3. Invested amount at \(4\%\): \(z = 3y\).

Step 2: Solve the second equation for x

Since we now have three equations with three variables, we can solve the system of equations. Let's first solve equation 2 for \(x\): $$ x = y - 500 $$

Step 3: Substitute the expressions in equation 1 and solve for y

Now, we will substitute this expression for \(x\) and expression for \(z\) from equation 3 into equation 1: $$ 0.02(y-500) + 0.03y + 0.04(3y) = 126 $$ Combine and simplify the terms to get: $$ 0.14y - 10 = 126 $$ Now, we can solve for \(y\): $$ 0.14y = 136 $$ $$ y = \frac{136}{0.14} $$ $$ y = 971.43 \approx 971.43 $$

Step 4: Find x and z

Now, using the value for \(y\), we can find the values of \(x\) and \(z\): For x: $$ x = y - 500 $$ $$ x = 971.43 - 500 $$ $$ x = 471.43 \approx 471.43 $$ For z: $$ z = 3y $$ $$ z = 3(971.43) $$ $$ z = 2914.29 \approx 2914.29 $$

Step 5: Present the final answer

Natsumi invested the following amounts at each interest rate: 1. \(\$471.43\) at \(2\%\) interest rate. 2. \(\$971.43\) at \(3\%\) interest rate. 3. \(\$2914.29\) at \(4\%\) interest rate.

Key Concepts

Understanding Systems of Equations

A system of equations is essentially a set of two or more equations that share the same variables. To solve such systems, you need to find values for the variables that satisfy all equations simultaneously.
Imagine you have several strings tying the same knots at different ends. When you "pull" on the strings by solving, you're looking for that one point (or set of points) that every string leads to.
In the exercise, Natsumi's three investments create a system of equations involving three variables: the amounts invested at different rates. The goal is to find the exact amount of money in each investment, which adds another layer of puzzle-solving to our mathematical endeavor.

• The first equation represents the total interest received as a function of three unknown investments.
• The second and third equations express relationships between these investments, offering clues to their individual values.

By cracking this system, you not only answer a specific problem but use a method that applies to countless real-life mathematical analyses.

The Role of Investment Rates

Investment rates determine how much interest you'll earn over time. They are usually expressed as a percentage per year, reflecting the return on your investment over that period.
In simple interest, as used in this problem, the amount of money grows linearly, unlike compound interest, which grows exponentially.
For Natsumi:

• The rate of 2% yielded interest for one portion of her total investment.
• Similarly, the other portions attracted 3% and 4%, each contributing differently to her total annual interest.

While the rates appear small, over time, they significantly impact the total returns, especially when different portions of a principal amount are allocated across different rates as in this problem.
Understanding how these rates work helps in planning financial growth strategies effectively and forecasting potential earnings over a specific period.

Solving Equations to Unlock Answers

Once you have your equations set up, the core challenge is solving them to find the unknowns. This involves substituting values back and forth until all variables reveal their secrets.
In solving this system:

• Step two uses one equation to solve for a variable, simplifying the system.
• By substituting back, you decrease complexity, eventually working down to simpler expressions.
• This methodically untangles the variables, leading you to the individual amounts invested at each rate.

Essentially, each step in the solution reduces the unknowns, bringing you closer to the final answer, much like peeling an onion layer by layer.
These techniques are vital across numerous applications, from calculating finances as seen here, to analyzing scientific data, proving how versatile and powerful mastering equation-solving can be.