Problem 47 - Intermediate Algebra

Question

Write a mathematical model for each situation. Answers may vary depending on the variables chosen. Taxes. \(\quad\) A married couple has decided to split the money equally when they receive their federal income tax refund. Furthermore, the husband is going to donate \(\$ 75\) of his share to charity. Describe the relationship between the amount of money that the husband will keep and the amount of the couple's refund.

Step-by-Step Solution

Answer

The husband's amount after donation is \( H = \frac{R}{2} - 75 \).

Step 1: Define Variables

Let's define variables to represent the quantities in question. Let \( R \) represent the couple's total federal income tax refund, and let \( H \) represent the amount of money the husband will keep after his donation to charity.

Step 2: Determine Initial Share

Since the couple has decided to split the money equally, the husband's initial share of the refund will be half of the total refund. Therefore, we can express the husband's initial share as \( \frac{R}{2} \).

Step 3: Account for Donation

The husband plans to donate \( \$75 \) to charity from his share of the refund. To find out how much he will keep, we subtract the donation amount from his initial share: \( H = \frac{R}{2} - 75 \).

Step 4: Formulate the Mathematical Model

The relationship between the money the husband will keep \( H \) and the total tax refund \( R \) is given by the equation: \[ H = \frac{R}{2} - 75 \]

Key Concepts

Income Tax Refunds

An income tax refund is the amount of money returned to a taxpayer when they have paid more tax than was due to the government.
At the end of the tax year, individuals and couples report their income and taxes paid. If it turns out they overpaid compared to their actual tax liability, they receive a refund.
In this exercise, a married couple receives a total income tax refund, which they plan to split equally. Understanding income tax refunds can help you calculate how much money you might have available after taxes, which is essential for financial planning.
When dividing the refund, each person receives half of the total amount. However, external factors, like one person donating to charity, can affect how much they get to keep.

Variable Definition

In mathematical modeling, variables are symbols used to represent unknown quantities.
They help us form equations and express relationships between different quantities without needing exact numbers.
In our example, we defined two crucial variables:

• \( R \): Represents the total amount of the income tax refund received by the couple.


• \( H \): Represents the amount the husband keeps after making a donation from his share of the refund.

Choosing the right variables simplifies solving the problem. It guides us towards forming an accurate equation that describes the situation. Using variables makes it easier to adjust the problem's parameters, like changing the donation amount, without redoing the entire exercise.

Equation Formation

Forming an equation is about creating a mathematical statement that represents the real-world situation you're examining.
It's often the most crucial step in mathematical modeling because it enables you to solve for unknown variables.
Here's how we crafted our equation:

• We first determined how much each person receives from the refund if it's split equally: \( \frac{R}{2} \).


• The husband plans to donate \( \$75 \) from his portion, reducing the amount he keeps by this figure.

To express this relationship mathematically, we subtract the donation from the husband's share:\[ H = \frac{R}{2} - 75 \]This equation links his final amount (\( H \)) to the total refund (\( R \)). Such equations are vital for breaking down complex situations into manageable, solvable components.