Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. Michael Perez has a total of $$\$ 2000$$ on deposit with two savings institutions. One pays interest at the rate of $$6 \% /$$ year, whereas the other pays interest at the rate of $$8 \% /$$ year. If Michael earned a total of $$\$ 144$$ in interest during a single year, how much does he have on deposit in each institution?

Step-by-Step Solution

Answer

Michael has deposited \(800\) in the institution paying 6% interest and \(1200\) in the institution paying 8% interest.

Step 1: Define the variables

Let x be the amount Michael has deposited in the institution paying 6% interest and y be the amount he has deposited in the institution paying 8% interest.

Step 2: Set up the equations

We can use the total deposited amount and the interest earned to create two equations: Total deposited amount: \(x + y = 2000\) Total interest earned: \(0.06x + 0.08y = 144\)

Step 3: Solve the system of equations

We can use any method to solve this system of equations, such as substitution, elimination, or matrices. Here we will use substitution. From the first equation, we can express x in terms of y: \(x = 2000 - y\) Now, we can substitute this expression for x into the second equation: \(0.06(2000 - y) + 0.08y = 144\)

Step 4: Solve for y

Expand and simplify the equation to solve for y: \(120 - 0.06y + 0.08y = 144\) Combine like terms: \(0.02y = 24\) Now divide by 0.02 to find the value of y: \(y = 1200\)

Step 5: Solve for x

Now that we have the value of y, we can find the value of x using the expression we found in step 3: \(x = 2000 - y = 2000 - 1200 = 800\)

Step 6: State the Answer

Michael has deposited \(800 in the institution paying 6% interest and \)1200 in the institution paying 8% interest.

Key Concepts

Interest Rates

Interest rates play a crucial role in how money grows over time in savings accounts. When you deposit money in a savings institution, the bank pays you a percentage of your deposit as interest. This percentage is known as the interest rate, and it is usually expressed on an annual basis, such as 6% or 8% per year. These rates determine how much extra money you earn based on your initial deposit amount.

In Michael's case, the two institutions pay different interest rates: one offers a 6% rate and the other an 8% rate. The interest you earn can be calculated by multiplying the deposit amount by the interest rate. For example, if you deposit $100 at a 6% interest rate, you would earn $6 in a year. Understanding how interest rates impact your savings helps in making informed financial decisions.

Algebra

Algebra provides the tools needed to solve problems with unknown variables, like knowing how much Michael deposited in each institution. In this problem, two main algebraic equations are used to find the answer. The first equation is the total amount deposited across both savings accounts:

• \(x + y = 2000\)

The second equation is the total interest earned from both deposits:

• \(0.06x + 0.08y = 144\)

Solving these equations simultaneously allows us to determine the values of \(x\) and \(y\), which represent the amounts deposited in the 6% and 8% interest accounts, respectively. By substituting \(x\) from the first equation into the second, we make it possible to solve for \(y\), and subsequently \(x\), illustrating how algebra can be used to work through real-world financial scenarios.

Financial Mathematics

Financial mathematics combines mathematical methods with financial concepts to solve problems like Michael's deposit situation. It involves using equations to model and solve financial questions, like how much interest is gained or how money should be allocated to maximize returns.

In Michael's case, financial mathematics comes into play in determining the division of money between two accounts to achieve a certain interest, using different interest rates. By applying mathematical models to financial assets, it's possible to predict outcomes and make better financial decisions. The system of equations created with Michael's accounts provides a straightforward approach to solve for unknowns, aiding in successful financial planning.

Using financial mathematics, one can not only calculate interest but also make informed choices about future investments and savings, showcasing the practical application of math in everyday financial management.