The range of a set of measurements is 112.4. If each measurement in the set is increased by 10 and...

1. Translate the problem requirements: We need to understand what "range" means (difference between largest and smallest values) and how transformations affect this range when applied to all measurements in a set.
2. Analyze the effect of adding a constant: Determine how adding \(10\) to each measurement impacts the range of the dataset.
3. Analyze the effect of dividing by a constant: Determine how dividing each measurement by \(4\) impacts the range after the addition step.
4. Apply transformations to calculate final range: Combine the effects of both transformations to find the new range value.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what "range" means in everyday terms. Think of range as the "spread" of your data - it's simply how far apart your highest and lowest values are.

For example, if your test scores were \(70, 75, 80, 85, 90\), then:

• Highest score = \(90\)
• Lowest score = \(70\)
• Range = \(90 - 70 = 20\)

So when the problem tells us "the range of a set of measurements is \(112.4\)," it means that if we took the biggest measurement and subtracted the smallest measurement, we'd get \(112.4\).

We don't need to know what the actual measurements are - we just need to understand how transformations affect this "spread" between the highest and lowest values.

Process Skill: TRANSLATE - Converting the problem language into mathematical understanding

2. Analyze the effect of adding a constant

Now let's think about what happens when we add the same number (\(10\)) to every measurement.

Imagine you have those test scores again: \(70, 75, 80, 85, 90\).
If we add \(10\) to each score: \(80, 85, 90, 95, 100\).

Notice what happened:

• Original range: \(90 - 70 = 20\)
• New range: \(100 - 80 = 20\)

The range stayed exactly the same! This makes perfect sense when you think about it - we're shifting all the numbers by the same amount, so the "gap" between the highest and lowest stays the same.

This is like moving a group of people - if everyone takes \(10\) steps forward together, the distance between the person at the front and the person at the back doesn't change.

So after adding \(10\) to each measurement in our problem, the range is still \(112.4\).

3. Analyze the effect of dividing by a constant

Next, let's see what happens when we divide every measurement by the same number (\(4\)).

Let's continue with our example. We had: \(80, 85, 90, 95, 100\) after adding \(10\).
Now divide each by \(4\): \(20, 21.25, 22.5, 23.75, 25\).

Let's check the range:

• Range before dividing: \(100 - 80 = 20\)
• Range after dividing: \(25 - 20 = 5\)

Notice that \(5 = 20 \div 4\). The range got divided by the same number we used to divide all the measurements!

This makes intuitive sense too - if we shrink everything by the same factor, then the distance between any two points also gets shrunk by that same factor.

So in our problem, when we divide each measurement by \(4\), the range of \(112.4\) becomes: \(112.4 \div 4 = 28.1\)

4. Apply transformations to calculate final range

Let's put it all together step by step:

1. Original range = \(112.4\)
2. After adding \(10\) to each measurement: range = \(112.4\) (unchanged)
3. After dividing each measurement by \(4\): range = \(112.4 \div 4 = 28.1\)

We can verify this with our general understanding:

• Adding the same constant to all values: range stays the same
• Multiplying (or dividing) all values by the same constant: range gets multiplied (or divided) by that same constant

Therefore: New range = \(112.4 \div 4 = 28.1\)

Final Answer

The range of the resulting set of measurements will be \(28.1\).

Looking at our answer choices, this matches choice E.

Answer: E. \(28.1\)

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding what "range" means
Students might confuse range with other statistical measures like mean, median, or standard deviation. They may try to calculate how these other measures change with the transformations, rather than focusing on the simple difference between maximum and minimum values.

Faltering Point 2: Thinking they need the actual data values
Students often get stuck thinking they need to know the specific measurements in the set to solve this problem. They might try to create sample data or feel frustrated that the problem doesn't give them the actual numbers, not realizing that range transformations follow predictable patterns regardless of the specific values.

Faltering Point 3: Misunderstanding the sequence of operations
Students might misread the problem and think they need to divide by \(4\) first, then add \(10\), rather than the correct sequence of adding \(10\) first, then dividing by \(4\). This confusion about order of operations will lead to incorrect analysis of how the range changes.

Errors while executing the approach

Faltering Point 1: Incorrectly applying the addition transformation
Students might think that adding \(10\) to each measurement also adds \(10\) to the range. They would calculate: new range = \(112.4 + 10 = 122.4\), failing to understand that adding the same constant to all values shifts the entire dataset but doesn't change the spread.

Faltering Point 2: Arithmetic errors in division
When dividing \(112.4\) by \(4\), students might make simple calculation errors, potentially getting \(28.6\) instead of \(28.1\), or other incorrect decimal results. This is especially common when working under time pressure.

Faltering Point 3: Applying transformations in wrong order to the range
Even if students understand the individual effects, they might apply them incorrectly to the range calculation. For example, they might calculate \((112.4 + 10) \div 4 = 30.6\), incorrectly thinking that both operations affect the range.

Errors while selecting the answer

Faltering Point 1: Choosing the intermediate calculation result
Students who incorrectly calculated \((112.4 + 10) \div 4 = 30.6\) might select answer choice D (\(30.6\)), not realizing they made an error in their approach to how transformations affect range.

Faltering Point 2: Selecting the original range value
Students who get confused about whether any transformation actually changes the range might default to choosing answer C (\(112.4\)), thinking that somehow the range remains completely unchanged by all the operations.