A certain state's milk production was 980 million pounds in 2007 and 2.7 billion pounds in 2014. Approximately how many...

1. Translate the problem requirements: We need to find the difference in milk production between 2014 and 2007, but convert from pounds to gallons and express the answer in million gallons. Key conversions: \(1 \mathrm{billion} = 10^9\), \(1 \mathrm{gallon} = 8.6 \mathrm{pounds}\).
2. Calculate the difference in pounds: Find how many more pounds of milk were produced in 2014 compared to 2007.
3. Convert pounds to gallons: Use the given conversion rate to transform the pound difference into gallons.
4. Express in million gallons: Convert the final result to match the required units in the answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. We want to know how many MORE million gallons of milk the state produced in 2014 compared to 2007.

Here's what we know:

• 2007 production: 980 million pounds
• 2014 production: 2.7 billion pounds
• Conversion factor: \(1 \mathrm{gallon} = 8.6 \mathrm{pounds}\)
• We need our final answer in million gallons

The key insight is that we're asked for the DIFFERENCE between the two years, not the total for either year. Also, notice that one number is given in millions and the other in billions - we'll need to make sure we're working with the same units.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical requirements

2. Calculate the difference in pounds

First, let's get both numbers in the same units. Since \(1 \mathrm{billion} = 10^9\), we can convert:

• \(2.7 \mathrm{billion\ pounds} = 2.7 \times 1,000 \mathrm{million\ pounds} = 2,700 \mathrm{million\ pounds}\)

Now we can find the difference:

• Difference = 2014 production - 2007 production
• Difference = \(2,700 \mathrm{million\ pounds} - 980 \mathrm{million\ pounds} = 1,720 \mathrm{million\ pounds}\)

So the state produced 1,720 million pounds more milk in 2014 than in 2007.

3. Convert pounds to gallons

Now we need to convert this difference from pounds to gallons. We know that \(1 \mathrm{gallon} = 8.6 \mathrm{pounds}\), which means:

• Number of gallons = Number of pounds ÷ 8.6

So our difference in gallons is:

• \(1,720 \mathrm{million\ pounds} \div 8.6 \mathrm{pounds\ per\ gallon}\)
• = \(1,720 \div 8.6 \mathrm{million\ gallons}\)
• = \(200 \mathrm{million\ gallons}\)

To make this calculation easier: \(1,720 \div 8.6 = 1,720 \div 8.6 \approx 1,720 \div 8.5 = 1,720 \times \frac{2}{17} = 3,440 \div 17 \approx 200\)

4. Express in million gallons

Our answer is already in the correct units: 200 million gallons.

Let's verify this makes sense by checking against the answer choices. We calculated approximately 200 million gallons, which matches choice B exactly.

Final Answer

The state produced approximately 200 million gallons more milk in 2014 than in 2007.

Answer: B. 200

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what the question is asking for: Students may think they need to calculate the total production for each year in gallons, rather than finding the DIFFERENCE between the two years. This leads to calculating individual conversions instead of focusing on the increase.

2. Unit confusion from the start: Students may get overwhelmed by the mixed units (millions vs billions, pounds vs gallons) and fail to plan a systematic approach to handle the conversions in the right order.

3. Misinterpreting the final answer format: Students may not notice that the answer choices are in simple numbers (like 200) while the question asks for "million gallons," leading them to set up incorrect calculations.

Errors while executing the approach

1. Billion to million conversion errors: Students may incorrectly convert 2.7 billion to millions, either forgetting that 1 billion = 1,000 million or making arithmetic mistakes like writing 2.7 billion as 270 million instead of 2,700 million.

2. Division approximation mistakes: When calculating \(1,720 \div 8.6\), students may struggle with the decimal division and either make computational errors or use poor approximations that lead them to wrong answer choices.

3. Forgetting to maintain consistent units: Students may correctly find the difference in pounds but then forget to convert to gallons, or mix up whether they're working in millions throughout the calculation.

Errors while selecting the answer

1. Magnitude confusion: Students who get an answer around 200 might second-guess themselves because it seems "too small" compared to the large numbers in the problem (980 million, 2.7 billion), leading them to select a larger answer choice like 1,700 or 8,200.

2. Selecting intermediate calculations: Students might select 1,700 (choice C) if they calculated the difference in pounds correctly (1,720 million pounds) but forgot to convert to gallons, seeing 1,700 as close to their intermediate result.