Problem 40 - Algebra A Combined Function

Question

The lettuce consumption per capita in 1990 was 31.5 pounds, and in 2007 the consumption dropped to 29.5 pounds. Find the percent decrease.

Step-by-Step Solution

Answer

The percent decrease is approximately 6.35%.

Step 1: Identify the Original and New Values

The original value of lettuce consumption per capita in 1990 was 31.5 pounds, and the new value in 2007 is 29.5 pounds.

Step 2: Calculate the Decrease in Consumption

Subtract the new value from the original value to find the decrease in consumption: \[31.5 - 29.5 = 2 \, ext{pounds}\]

Step 3: Compute the Relative Decrease

To find the relative decrease, divide the decrease by the original value:\[\frac{2}{31.5}\]

Step 4: Convert to Percentage

Multiply the result from Step 3 by 100 to convert the relative decrease to a percentage:\[\left(\frac{2}{31.5} \right) \times 100 \approx 6.35\%\]

Step 5: Conclusion

The percent decrease in lettuce consumption per capita from 1990 to 2007 is approximately 6.35%.

Key Concepts

Lettuce Consumption

Lettuce consumption refers to the amount of lettuce that each person eats on average during a specific period. In this problem, we are looking at lettuce consumption per capita in the United States, where ‘per capita’ means the average per person.

In the context of the problem, lettuce consumption is recorded as pounds per person for the years 1990 and 2007. By calculating the changes over these years, we can better understand trends in dietary habits. Such data is often used by nutritionists and policymakers to make informed decisions about food supply and health guidelines.

• 1990: 31.5 pounds per person
• 2007: 29.5 pounds per person

These figures help quantify how the popularity of certain foods, like lettuce, changes over time.

Per Capita

The term 'per capita' originates from Latin, meaning 'by heads.' It is used to describe an average per person within a given area or population. When we calculate something like lettuce consumption per capita, we are averaging the total quantity of lettuce consumed by dividing by the number of people.

This measure helps in understanding consumption or production in a fair and comparable way regardless of the total population size. For example, even if the total amount of lettuce consumed in a country increases, if the population grows at the same rate, the 'per capita' consumption might stay the same. This method is particularly useful to standardize data, allowing for meaningful comparisons across different countries or time periods.

• Gives a personal average
• Normalizes data across populations

Calculate Decrease

To calculate the decrease in lettuce consumption per capita, we need to find the difference between the original and new values. This involves a simple subtraction of the later value from the initial value:

• Original value in 1990: 31.5 pounds
• New value in 2007: 29.5 pounds
• Decrease: 31.5 - 29.5 = 2 pounds

Calculating the decrease helps to clearly quantify the change over time. Once we know how much consumption has dropped, it allows us to move on to find the relative change, which sets the stage to calculate the percent decrease. Each step builds upon the previous one, paving the way for converting this decrease into a percent change.

Convert to Percentage

To convert the numerical decrease into a percent decrease, we begin by understanding how much the decrease is in relation to the original value. This is done by dividing the change by the original amount. Using the figures from our lettuce consumption scenario, this gives us:

• Decrease: 2 pounds
• Original consumption: 31.5 pounds
• Relative decrease: \( \frac{2}{31.5} \)

Once the relative decrease is determined, we convert it to a percentage. This is achieved by multiplying the relative decrease by 100. This calculation leads us to understand the change as a percent of the original consumption:

• Percent decrease: \( \left( \frac{2}{31.5} \right) \times 100 \approx 6.35\% \)

This final calculation provides a clear and intuitive representation of how much lettuce consumption has decreased by placing it in a percentage context, making it a helpful metric for understanding the significance of the change.