Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Investment Mix You invest \(\$ 30,000\) in two funds paying \(3 \%\) and \(4 \frac{1}{2} \%\) simple interest. The total annual interest is \(\$ 1230\). How much do you invest in each fund?

Step-by-Step Solution

Answer

You should invest $8,000 in the 3% fund and $22,000 in the 4.5% fund.

Step 1: Define the equations

Let's denote x as the amount invested in the 3% fund and y as the amount invested in the 4.5% fund. From the problem, it's clear that there are two conditions: \n1) The total amount invested is $30,000 i.e., x + y = 30000 \n2) The total yearly interest is $1230. This can be expressed using the interest rates as 0.03*x + 0.045*y = 1230.

Step 2: Substitute and solve for one variable

Let's isolate x in the first equation: x = 30000 - y. Now, we can substitute (30000 - y) instead of x in the second equation: 0.03*(30000 - y) + 0.045*y = 1230. By multiplying out and combining like terms, we get: 900 - 0.03*y + 0.045*y = 1230, which simplifies to 0.015*y = 330.

Step 3: Solve for the other variable

Now, solve for y in the equation 0.015*y = 330, we get y = 330/0.015, which gives y = $22,000. Substitute y = 22,000 into the first equation, we get x = 30000 - 22000 = $8,000.

Key Concepts

Simple Interest

Simple interest is a straightforward method to calculate the interest on an investment. It is determined by multiplying the principal amount (the initial sum of money), the interest rate, and the time for which the money is invested. The formula for simple interest is given by: \[ \text{Simple Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]For example, if you invest 100 dollars for one year at an interest rate of 5%, the interest will be 5 dollars. One key aspect of simple interest is that it does not take into account any interest on previously earned interest, unlike compound interest. Therefore, it grows linearly over time. This makes it easier to calculate, but also means the returns may be less compared to compound interest over long periods.
In the context of the problem, we used this principle to calculate the expected interest for two different funds.

Equation Solving

Equation solving is a fundamental aspect of mathematics that involves finding the value of variables that make an equation true. In this context, the problem involves solving two linear equations simultaneously. The first equation represents the total amount invested, while the second equation represents the total interest earned.
To solve these equations, the substitution method is used. This involves expressing one variable in terms of the other and then substituting this expression into the second equation. This technique allows you to simplify the problem by reducing it to a single variable equation.
By solving these types of equations step-by-step, you can determine the exact values needed for each variable, which in this case are the amounts invested in each fund.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They can include numbers, variables, and arithmetic operations. In solving the problem, we dealt mainly with linear algebraic equations.
Linear equations have the general form \( ax + by = c \), where \(a\), \(b\), and \(c\) are constants. The solution to these equations aims to determine the values of the variables that make the equation valid.
In our problem, we had two algebraic equations:

• \( x + y = 30000 \) for the total investment
• \( 0.03x + 0.045y = 1230 \) for the interest earned

By simultaneously solving these equations using algebraic techniques, such as substitution and simplification, we found the exact investment amounts needed for each fund.