Question

Solve. By insulating their water heater, adding a storm door, and installing new double-pane windows, the Goto family decreased their electric and gas bills from a total of \(\$ 1536.98\) in one year to \(\$ 1020.57\) in the next year. What was the percent of decrease in the Goto family's electric and gas charges?

Short answer

Answer: The percent of decrease is approximately 33.58%.

Step-by-step solution

Find the difference in charges between the two years

To find the difference in charges between the two years, we will subtract the charges in the second year from the charges in the first year. Difference = \(\$1536.98 - \$1020.57\)

Calculate the difference

Now, let's calculate the difference in charges: Difference = \(\$1536.98 - \$1020.57 = \$516.41\)

Find the percent decrease

To find the percent decrease, we will divide the difference by the original amount (the charges in the first year) and multiply by 100: Percent decrease = \(\frac{Difference}{Original\:Amount} \times 100\)

Calculate the percent decrease

Now, let's calculate the percent decrease: Percent decrease = \(\frac{\$516.41}{\$1536.98} \times 100\)

Simplify the expression

Let's simplify the expression: Percent decrease ≈ 33.58% So, the percent of decrease in the Goto family's electric and gas charges after their home improvements is approximately 33.58%.

Key concepts

Prealgebra

Prealgebra serves as the foundation for all higher-level math, including the calculations required to determine percent decreases. This field encompasses basic arithmetic operations like addition, subtraction, multiplication, and division, which are used to solve real-world problems.

For instance, in the case of the Goto family, prealgebra allowed them to understand how energy-saving measures led to financial savings on their electric and gas bills. An important prealgebra skill displayed in this scenario was the ability to calculate differences between numbers. This skill is vital not only in mathematics but in many financial circumstances where budgeting or analyzing expenses is required.

Percent Decrease

Percent decrease is a mathematical concept used to express a reduction in quantity relative to the original amount. It is calculated by finding the difference between the original and new value, dividing that by the original value, and then multiplying by 100 to get a percentage.

To make sense of this: imagine you had a jar of 100 marbles and you lost 20; the jar now has 80 marbles. The percent decrease in marbles would be \( (100 - 80) / 100 \times 100 = 20% \). In terms of the Goto family, the percent decrease in their bills highlights the effectiveness of their home improvements in saving money, which is a common application of this concept in financial planning and budget management.

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics and are used extensively in percent decrease calculations. The four basic operations are:

• Addition (summing numbers)
• Subtraction (finding the difference between numbers)
• Multiplication (repeated addition of a number)
• Division (determining how many times a number is contained within another number)

These operations allow us to quantify and compare changes, such as the Goto family's utilities savings. It's essential to master these operations to handle more complex financial tasks.

Financial Mathematics

Financial mathematics combines mathematical methods with financial theory to solve problems related to financial markets and investments. The Goto family's situation, where they calculate the percent decrease in their utilities, is a practical example. This concept helps individuals and businesses make informed decisions by analyzing the costs and benefits of their financial actions.

Understanding how math, particularly prealgebra and percentages, intersects with financial decision-making is crucial for effective budgeting and financial planning. Saving on utility bills is just one way this comes into play – similar principles are used in calculating interest rates, loan payments, and investment returns.