Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Total pet spending (2006): Food: Vet care: Supplies/OTC medicine: Live animal purchases: Grooming and boarding:\(\$ 38.4\) billion \(\$ 15.2\) billion \(\$ 9.4\) billion \(\$ 9.3\) billion \(\$ 1.8\) billion \(\$ 2.7\) billion

Step-by-Step Solution

Answer

The total pet spending in 2006 was \(76.8\) billion dollars.

Step 1: Identify the amounts

Identify all the given values on each category, which are in billions: \nFood: \(38.4\) billion\nVet care: \(15.2\) billion\nSupplies/OTC medicine: \(9.4\) billion\nLive animal purchases: \(9.3\) billion\nGrooming and boarding: \(1.8\) billion

Step 2: Addition

Add together all these values using the addition operation: \(38.4 + 15.2 + 9.4 + 9.3 + 1.8 + 2.7 = 76.8\) billion

Step 3: Present the Result

The sum obtained, shows the total pet spending in 2006

Key Concepts

Addition

Addition is the mathematical process of combining two or more numbers into a single sum. In essence, it's about finding the total or whole when parts are put together. For the exercise mentioned, we are using addition to find the total spending on pets in several categories during 2006.

When adding numbers, the goal is to align the values and ensure each digit is added correctly based on its place value. For instance, in billions, millions, thousands, etc., we maintain order to avoid calculation errors. Here’s how addition can be effectively performed:

• Line up the numbers vertically by their decimal points.
• Add each column starting from the rightmost digit, moving towards the left.
• If a column sums to a number greater than nine, carry over the extra value to the next column on the left.

Although the numbers here are straightforward, always double-check to ensure every number is accounted for, especially in long lists or larger figures.

Problem Solving

Problem solving in mathematics involves identifying the problem, determining a strategy to tackle it, and executing the plan systematically. The given exercise is a classic problem-solving scenario where understanding the question is key before invoking the numerical operations. In the pet spending example:

• First, carefully read to identify all the components listed.
• Determine the goal - in this case, to find the total spending by adding specific costs.
• Systematically add each number to ensure no category is missed out.
• Finally, verify your work by reviewing each step and re-checking the final result.

Such methodical approaches can help avoid mistakes and are useful skills in various real-world situations beyond mathematics, ensuring a detailed understanding and solution to any given problem.

Numerical Operations

Numerical operations are processes that allow us to manipulate numbers to achieve desired outcomes. Among these operations are addition, subtraction, multiplication, and division. In this exercise, addition is employed to gauge the total spending across varied categories related to pets in 2006. These operations are foundational mathematical tools:

• Addition for combining values and getting totals.
• Subtraction is useful when needing to find the difference between values.
• Multiplication often applies when repeatedly adding the same number.
• Division is useful to break numbers into specified parts or to find averages.

Understanding when and how to use these operations is critical in problem solving. By mastering them, one can tackle a range of mathematical problems efficiently and accurately. Here, with all numbers summed, we understand the broader financial context of pet spending for that year. Familiarity with these operations equips anyone with the skills required for both everyday math challenges and more complex calculations that arise in various fields.