Problem 18 - Algebra Form and Function

Question

The interest paid by a savings account in one year is proportional to the starting balance, with constant of proportionality 0.06 . Write a formula for \(I,\) the amount of interest earned, in terms of \(B\), the starting balance. Find the interest earned if the starting balance is (a) \(\$ 500\) (b) \(\quad \$ 1000\) (c) \(\$ 5000\).

Step-by-Step Solution

Answer

Answer: The interest earned for starting balances of \(500 is \)30, for \)1000 is \)60, and for \(5000 is \)300.

Step 1: Understand the proportionality

The problem states that the interest earned (I) is proportional to the starting balance (B) with a constant of proportionality of 0.06. This can be represented as: I ∝ B

Step 2: Write the formula for I in terms of B

Since the interest earned (I) is proportional to the starting balance (B), we can write this proportionality as a formula by introducing the constant of proportionality, which is 0.06: I = 0.06 * B

Step 3: Find the interest earned for a starting balance of \(500

To find the interest earned when the starting balance is \)500, plug B = 500 into the formula: I = 0.06 * 500 I = 30 So, the interest earned when the starting balance is \(500 is \)30.

Step 4: Find the interest earned for a starting balance of \(1000

To find the interest earned when the starting balance is \)1000, plug B = 1000 into the formula: I = 0.06 * 1000 I = 60 So, the interest earned when the starting balance is \(1000 is \)60.

Step 5: Find the interest earned for a starting balance of \(5000

To find the interest earned when the starting balance is \)5000, plug B = 5000 into the formula: I = 0.06 * 5000 I = 300 So, the interest earned when the starting balance is \(5000 is \)300.

Key Concepts

Interest Calculation

Interest calculation is a fundamental concept in finance, especially when managing savings. When you deposit money into a savings account, interest is the reward you earn for keeping your money there. In this context, the problem described that the interest earned is proportional to the starting balance.

The term "proportional" indicates that the relationship between the amount of interest and the starting balance is linear. Here, the constant of proportionality is 0.06, which means for every dollar in the savings account, you earn $0.06 of interest annually. This makes it easier to predict how much interest you can earn on different balances.

To calculate the interest, you use the formula:

• Interest = Constant of Proportionality × Starting Balance

So, if you want to find out the interest on a starting balance of $500, $1000, or $5000, simply plug the appropriate values into the formula. This simple method allows you to manage expectations and plan effectively.

Linear Equations

Linear equations are a crucial part of algebra that describe relationships with a constant rate of change. When you express the interest calculation as an equation, it becomes a linear equation:

• \( I = 0.06 \times B \)

This equation is a straightforward representation of how the interest (I) depends on the starting balance (B) with a slope (constant rate of change) of 0.06.

The form of this equation is similar to the familiar equation for a straight line:

• \( y = mx + c \)

where \( y \) is the output variable, \( x \) is the input variable, \( m \) is the slope, and \( c \) is the y-intercept. In the case of interest, there is no \( c \) (y-intercept) term because when the starting balance is zero, the interest is also zero.

Understanding this linear equation helps one see that interest grows at a constant rate as the balance increases. It's a simple yet powerful way to predict and calculate financial outcomes.

Constant of Proportionality

The constant of proportionality is a fixed number that defines how two variables are related in a proportional relationship. In our problem, the constant is 0.06, indicating the interest earned per dollar. This constant is crucial because it standardizes the relationship between the starting balance and the interest.

Let's break it down: if the starting balance is multiplied by the constant of proportionality, the result is the interest. This relationship is expressed with:

• Interest = 0.06 × Starting Balance

Whether you start with $500, $1000, or $5000, multiplying by 0.06 yields the interest for each scenario.

The concept of a constant of proportionality is not just for finance, it's a vital idea in mathematics offering predictable outcomes in linear relationships. Understanding it allows for consistent calculations across various fields.