Problem 26 - Prealgebra

Question

Which represents the percent of change in the number of German shepherd dogs from 2003 to \(2004 ?\) \begin{array}{|l|c|c|} \hline \multicolumn{1}{|c|}\text { Breed } & 2003 & 2004 \\ \hline \text { Labrador Retriever } & 144,896 & 146,692 \\ \hline \text { Golden Retriever } & 52,520 & 52,550 \\ \hline \text { Beagle } & 45,021 & 44,555 \\ \hline \text { German Shepherd Dog } & 43,938 & 46,046 \\ \hline \end{array} $$\begin{array}{l} \text { A } 2.1 \% \\ \text { B } 2.5 \% \end{array}$$ $$\begin{array}{l} \text { C } 4.6 \% \\ \text { D } 4.8 \% \end{array}$$

Step-by-Step Solution

Answer

The percent change in the number of German shepherd dogs from 2003 to 2004 is 4.8% (Option D).

Step 1: Identify Initial and Final Values

From the given table, note the number of German Shepherd dogs in 2003 and 2004. In 2003, there were 43,938 dogs, and in 2004, there were 46,046 dogs.

Step 2: Calculate the Change in Quantity

Subtract the number of German Shepherd dogs in 2003 from the number in 2004 to find the change: Change = 46,046 - 43,938 = 2,108.

Step 3: Determine the Original Quantity

The original quantity is the number of German Shepherd dogs in 2003, which is 43,938.

Step 4: Calculate the Percent Change

Use the formula for percent change: \[\text{Percent Change} = \left(\frac{\text{Change}}{\text{Original Quantity}}\right) \times 100\%\]Substitute the values:\[\text{Percent Change} = \left(\frac{2,108}{43,938}\right) \times 100\%\approx 4.796\%\]

Step 5: Round and Match to Options

Round the calculated percent change to match the given options. The value \(4.796\%\) rounds to \(4.8\%\), which corresponds to option D.

Key Concepts

Step-by-Step Solution

Solving math problems can be much simpler if broken down into smaller steps. In a step-by-step solution, like the one given in this problem, you follow a logical sequence to reach the answer. Each step builds upon the previous one, making it easier to understand and find the solution. For instance, in our problem about calculating percent change in the number of German Shepherd dogs, the steps were precisely laid out:

• First, you identify what you have, the initial and final numbers.
• Then, calculate the difference in values to determine the change in quantity.
• Third, use the initial value to calculate the percent change.
• Finally, round the answer to the nearest option given in the problem.

By breaking down the problem into these smaller parts, it ensures that no steps are skipped, and students can follow along easily to understand the underlying math concepts without feeling overwhelmed.

Calculating Percent Change

Calculating percent change is an important skill in mathematics, often used to determine how much a quantity increases or decreases over time. The percent change is essentially a measure of how significant the change is relative to the original value. The formula used in calculations, \[\text{Percent Change} = \left(\frac{\text{Change}}{\text{Original Quantity}}\right) \times 100\%\]shows the mathematical relationship between the change and the original quantity.
To find the percent change:

• Compute the change in quantity first, which is the difference between the new and the old values.
• Divide the change by the original value to analyze the magnitude of change relative to the original amount.
• Multiply the result by 100 to convert it into a percentage, making it easier to understand.

This method helps you understand how significant or minor a change is by turning it into an easily comparable format.

Change in Quantity

In mathematics, the term 'change in quantity' refers to the difference between two values over a given period or between two states. To find the change in quantity:

• Identify the initial and final values of the quantity in question.
• Subtract these values to determine the increase or decrease.

For the example of German Shepherd dogs, we identified 43,938 dogs in 2003 and 46,046 in 2004. Calculating the change involves simply subtracting the initial number from the final number:
\[\text{Change in Quantity} = 46,046 - 43,938 = 2,108\]
This value tells us how many more or fewer units there are compared to the original. Comprehending the change in quantity is often the first step in analyzing trends and making predictions based on numerical data. It's the backbone of calculating percent change and understanding shifts in data.