Problem 50 - Algebra A Combined Function

Question

The number of farms in the United States is decreasing. In \(1940,\) there were approximately 6.3 million farms, while in 2007 there were only 2.1 million farms. Find the percent decrease in the number of farms. Round to the nearest tenth of a percent.

Step-by-Step Solution

Answer

The percent decrease in the number of farms is 66.7%.

Step 1: Determine the Initial and Final Values

Identify the initial value as the number of farms in 1940, which is 6.3 million, and the final value as the number of farms in 2007, which is 2.1 million.

Step 2: Calculate the Decrease in the Number of Farms

Subtract the number of farms in 2007 from the number of farms in 1940. This gives us the decrease: \(6.3\, \text{million} - 2.1\, \text{million} = 4.2\, \text{million}\).

Step 3: Calculate the Percent Decrease

Use the formula for percent decrease: \(\text{Percent Decrease} = \left(\frac{\text{Decrease}}{\text{Initial Value}}\right) \times 100\). Substitute the known values: \(\left(\frac{4.2}{6.3}\right) \times 100 \approx 66.67\%\).

Step 4: Round the Percent Decrease

Round the number calculated to the nearest tenth of a percent. Thus, \(66.67\%\) becomes \(66.7\%\).

Key Concepts

Initial Value and Final Value

Before you can calculate any changes, it's crucial to understand the initial and final values. This is your starting and ending point. In the context of percent decrease calculation, these values represent the quantity at the beginning and at the end of the period you're examining.

• The initial value is the number or amount you begin with. For the farms example, the initial value was the number of farms in 1940, which is 6.3 million.
• The final value is the number or amount you end with after the decrease. The farms' final value was 2.1 million in 2007.

Understanding these values helps set the stage for determining how much change has occurred over time. Always identify these values clearly to avoid any confusion when proceeding with calculations.

Decrease in Quantity

Once you have the initial and final values, the next step is to calculate the decrease in quantity. This is simply the difference between the two values, showing how much was lost or reduced. It's a straightforward subtraction problem.To find the decrease, subtract the final value from the initial value:- **Decrease**: Initial Value - Final ValueFor instance, in our farm example:- Initial Value: 6.3 million- Final Value: 2.1 millionSo, the decrease is calculated as: \[6.3 \, \text{million} - 2.1 \, \text{million} = 4.2 \, \text{million}\]Understanding the quantitative decrease is essential to measure how significant the change is. This value will then be used in the next steps to determine the percent change.

Rounding Percentages

After calculating the percent decrease, the final touch is rounding the result to the nearest tenth of a percent. Rounding can make the percentage easier to understand and communicate. Consider the formula: \[\text{Percent Decrease} = \left(\frac{\text{Decrease}}{\text{Initial Value}}\right) \times 100\]Once you plug in your numbers, you might get a result that has many decimal places. For example, the decrease in farms calculated to about 66.67%. To simplify, you round to one decimal place:- Look at the second decimal place (hundredths) to check if it's 5 or above.- If it is, round up. Otherwise, leave it.Here, 66.67% becomes:- 66.7%, rounded to the nearest tenth.Rounding accurately ensures clarity and precision in your results, which is crucial for any serious mathematical or statistical work.