On a recent trip, Cindy drove her car 290 miles, rounded to the nearest 10 miles, and used 12 gallons...

1. Translate the problem requirements: We need to find the range of actual miles per gallon. Since both distance (290 miles) and gas used (12 gallons) are rounded values, we must determine what the actual unrounded values could have been, then find the minimum and maximum possible miles per gallon.
2. Determine the actual ranges for rounded values: Find what the true distance and gas consumption could have been before rounding to establish the boundaries for our calculation.
3. Identify extreme cases for miles per gallon: Recognize that maximum mpg occurs when actual miles is highest and actual gallons is lowest, while minimum mpg occurs when actual miles is lowest and actual gallons is highest.
4. Calculate the boundary values: Compute the minimum and maximum possible miles per gallon using the extreme combinations identified in the previous step.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this problem is really asking. We have two pieces of information that have been rounded:

• Distance driven: 290 miles (rounded to nearest 10 miles)
• Gas used: 12 gallons (rounded to nearest gallon)

We want to find the range of actual miles per gallon. Think of it this way: if someone tells you they drove "about 290 miles" and used "about 12 gallons," what's the best and worst gas mileage they could have actually gotten?

The key insight is that since both numbers are rounded, we don't know the exact values. We need to figure out what the actual unrounded numbers could have been, then calculate the range of possible miles per gallon.

Process Skill: TRANSLATE - Converting the rounded values into ranges of actual possible values

2. Determine the actual ranges for rounded values

When a number is rounded, we need to think about what values could have produced that rounded result.

For the distance (290 miles, rounded to nearest 10):

• The actual distance could have been anywhere from 285 miles up to (but not including) 295 miles
• Why? Because anything from 285.0 to 294.9 would round to 290
• So actual distance range: \(285 ≤ \mathrm{actual\ miles} < 295\)

For the gas consumption (12 gallons, rounded to nearest gallon):

• The actual gas used could have been anywhere from 11.5 gallons up to (but not including) 12.5 gallons
• Why? Because anything from 11.5 to 12.49 would round to 12
• So actual gas range: \(11.5 ≤ \mathrm{actual\ gallons} < 12.5\)

3. Identify extreme cases for miles per gallon

Now here's the crucial reasoning: Miles per gallon = Distance ÷ Gas Used

To get the MAXIMUM miles per gallon:

• We want the highest possible distance and the lowest possible gas consumption
• Highest distance: just under 295 miles
• Lowest gas: just over 11.5 gallons
• Maximum mpg approaches: \(295 ÷ 11.5\)

To get the MINIMUM miles per gallon:

• We want the lowest possible distance and the highest possible gas consumption
• Lowest distance: just over 285 miles
• Highest gas: just under 12.5 gallons
• Minimum mpg approaches: \(285 ÷ 12.5\)

Process Skill: INFER - Recognizing that extreme ratios come from extreme combinations of numerator and denominator

4. Calculate the boundary values

Since we're looking at ranges that approach but don't quite reach the boundary values, the actual miles per gallon must have been between:

Minimum: \(\frac{285}{12.5}\) (approaches from above)
Maximum: \(\frac{295}{11.5}\) (approaches from below)

Let's verify this makes sense:

• \(285 ÷ 12.5 = 22.8\) mpg
• \(295 ÷ 11.5 ≈ 25.65\) mpg

So the actual mpg was between \(\frac{285}{12.5}\) and \(\frac{295}{11.5}\).

Final Answer

The actual number of miles per gallon must have been between \(\frac{285}{12.5}\) and \(\frac{295}{11.5}\).

This matches answer choice D: \(\frac{285}{12.5}\) and \(\frac{295}{11.5}\).

Answer: D

Common Faltering Points

Errors while devising the approach

• Misunderstanding what "rounded to the nearest" means: Students often think that if 290 is rounded to the nearest 10, the actual value could be anywhere from 280 to 300, instead of correctly identifying the range as 285 to just under 295. This leads to using wrong boundary values throughout the solution.
• Not recognizing the need to find extreme cases: Students may try to simply calculate \(290÷12\) without realizing they need to find the minimum and maximum possible values of miles per gallon by considering all possible combinations of actual distance and gas consumption.
• Confusing which combinations give maximum vs minimum ratios: Students might incorrectly think that maximum distance with maximum gas gives the highest mpg, rather than understanding that maximum mpg requires maximum distance with minimum gas consumption.

Errors while executing the approach

• Using inclusive vs exclusive boundaries incorrectly: Students may use 295 and 12.5 as exact achievable values rather than understanding these are limits that can be approached but not reached, leading to incorrect boundary calculations.
• Mixing up numerator and denominator in ratio calculations: When calculating the extreme cases, students might accidentally put the wrong values in numerator vs denominator positions (e.g., calculating \(\frac{12.5}{285}\) instead of \(\frac{285}{12.5}\)).

Errors while selecting the answer

• Selecting the range in wrong order: Students might correctly calculate \(\frac{285}{12.5}\) and \(\frac{295}{11.5}\) but then select an answer choice that lists them in the wrong order (smaller value second instead of first), not recognizing that \(\frac{285}{12.5} < \frac{295}{11.5}\).
• Choosing decimal approximations instead of exact fractions: Students might calculate the decimal values (22.8 and 25.65) and look for answer choices with these decimals, missing that the correct answer uses the exact fractional forms.