Question

As an incentive to install solar equipment, a tax credit of \(42 \%\) of the first \(\$ 1100\) and \(25 \%\) of the next \(\$ 6400\) spent on solar equipment is proposed. How much credit, to the nearest dollar, would a homeowner get when installing \(\$ 5500\) worth of solar equipment?

Short answer

The homeowner would get a tax credit of \)1582 when installing $5500 worth of solar equipment.

Step-by-step solution

Calculate the Tax Credit for the First \(1100

Find the tax credit for the first \)1100 spent on solar equipment by multiplying \(1100 by the tax credit rate of 42%. Use the formula: Tax Credit = Amount * Credit Rate.

Calculate the Tax Credit for the Remaining Amount

Subtract \)1100 from the total spent (\(5500) to find the amount spent beyond the first \)1100, which is \(4400. Multiply this by the tax credit rate of 25% to find the tax credit for this portion.

Calculate the Total Tax Credit

Add the tax credit from the first \)1100 to the tax credit from the remaining \(4400 to find the total tax credit. This is the sum of the credits from Step 1 and Step 2.

Round to the Nearest Dollar

Round the total tax credit calculated in Step 3 to the nearest dollar, as tax credits are typically reported in whole dollars.

Key concepts

Tax Credit Percentage

Understanding tax credits is crucial when managing personal finances, especially when they are offered as an incentive to encourage specific purchases, like solar equipment. A tax credit percentage represents the portion of the expense that can be reduced directly from the taxes you owe.

For example, in the exercise, two different tax credit percentages are used: 42% for the first \(1100 and 25% for the next \)6400. To calculate the savings, you apply each percentage to the corresponding portion of your expenses. This makes a significant difference, as these are not deductions from taxable income, but credits subtracted from the total tax bill. The higher the percentage, the greater the savings.

Financial Mathematics

Financial mathematics encompasses the tools and techniques used to solve problems relating to money, investments, financing, and other financial instruments. It combines the principles of time, value, and money to make informed decisions about finances.

In our solar equipment tax credit calculation, we use fundamental financial mathematics to determine how much a homeowner saves. The process involves basic operations—multiplication for finding the credit of each portion and addition to sum them up. These calculations are essential for prudent financial planning and help individuals and businesses make better economic choices.

Mathematical Problem Solving

Mathematical problem solving is a critical thinking skill that allows us to tackle quantitative challenges in a structured manner using various strategies, such as breaking problems into smaller, more manageable parts, which we see in the step-by-step solution provided.

The homeowner's problem requires sequential steps: first calculating the credit for the initial expense up to $1100, then for the remaining spent beyond that, and finally combining these to reach the whole tax credit. This systematic approach simplifies complex problems into easier tasks, ensuring a thorough and accurate solution. In real-world applications, this skill is invaluable for making informed and financially sound decisions.