Problem 37

Question

PERSONAL FINANCE. Toshiro has $\$ 850$ to invest. He can invest in a savings account that has an annual interest rate of 1.7$\%$ , and he can invest in a money market account that pays about 3.5$\%$ per year. Write a polynomial to represent the amount of interest he will earn in 1 year if he invests $x$ dollars in the savings account and the rest in the money market account.

Step-by-Step Solution

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Answer

The polynomial is $-0.018x + 29.75$.

1Step 1: Define Variables

Let's define variables for this problem. Let $ x $ represent the dollars invested in the savings account. Since Toshiro has $ 850 $ dollars in total, the remaining amount $ 850 - x $ will be invested in the money market account.

2Step 2: Calculate Interest from Savings Account

The interest earned from the savings account can be calculated using the formula for simple interest: $ \, I = P \times r \, $ where $ P $ is the principal amount and $ r $ is the rate of interest. Here, $ P = x $ and $ r = 0.017 $. Thus, the interest from the savings account is $ 0.017x $.

3Step 3: Calculate Interest from Money Market Account

Similarly, the interest from the money market account is calculated as $ I = P \times r $. Here, $ P = 850 - x $ and $ r = 0.035 $. Thus, the interest from the money market account is $ 0.035 (850 - x) $.

4Step 4: Construct the Polynomial

To find the total interest earned, sum the interest from both accounts: \[ 0.017x + 0.035(850 - x) \]This is the polynomial that represents the total interest earned.

5Step 5: Simplify the Polynomial

Simplify the polynomial by distributing and combining like terms:\[ 0.017x + 0.035 \times 850 - 0.035x = 0.017x + 29.75 - 0.035x \]Combine like terms:\[ (0.017 - 0.035)x + 29.75 = -0.018x + 29.75 \]This simplified polynomial $-0.018x + 29.75$ represents the total interest Toshiro will earn in one year.

Key Concepts

Understanding Simple InterestExploring Investment StrategiesBuilding Financial Literacy

Understanding Simple Interest

Simple interest is a fundamental concept in personal finance that helps us understand how money can grow over time when invested. Unlike compound interest, which is calculated on the initial principal and the accumulated interest over periods, simple interest is calculated only on the initial amount of money invested. This makes it straightforward to compute and easy to track.

In Toshiro's situation, the formula for simple interest is applied to determine how much he earns from his investments in one year. Here's a quick refresher on the simple interest formula:

$ I = P \times r $
$ I $ is the interest earned
$ P $ is the principal amount (initial investment)
$ r $ is the rate of interest per year

For example, if Toshiro invests $ x $ dollars in a savings account with a 1.7% annual interest rate, the interest he will earn in one year is given by $ 0.017x $. This simple calculation highlights the straightforward nature of simple interest, making it an essential tool in financial literacy.

Exploring Investment Strategies

When planning how to allocate money between different investments, it's crucial to consider the potential returns and risks involved. Different investment vehicles offer varying interest rates and benefits. For Toshiro, the decision involves choosing between a low-risk savings account and a higher-yield money market account.

In this scenario:

A savings account offers stability with a lower interest rate (1.7%).
A money market account comes with a slightly higher risk but offers a better return at 3.5%.

These options present a classic opportunity for diversification - spreading investments to balance potential returns as well as risks. By investing a portion in the savings account and the remaining funds in the money market account, Toshiro can aim for a mix of safety and better returns. Polynomials like the one here provide a mathematical model showing how combining different investment strategies can yield optimal interest over time. Making smart investment choices after evaluating potential outcomes is key in financial planning.

Building Financial Literacy

Gaining financial literacy means understanding how various financial instruments work, including accounts with simple interest rates. It empowers individuals to make informed decisions regarding their personal finances.

Key elements of financial literacy include:

Being aware of different types of interest and how they affect savings and investments.
Understanding how to calculate returns using mathematical expressions, like the polynomial from the exercise.
Grasping the importance of diversification in investment strategies.

In our example, comprehending how the investments in Toshiro's scenario result in a polynomial equation helps individuals visualize potential earnings. It also enhances financial skills, enabling better decision-making. By studying scenarios like this one, learners can develop a strong foundation of financial literacy, equipping them to manage their money wisely in the real world. This foundation is beneficial not just for personal growth but also for making impactful financial decisions.

Problem 37

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