In a set of 5 measurements with an average (arithmetic mean) of 60, one of the measurements is found to...

1. Translate the problem requirements: We have 5 measurements averaging 60, meaning their total sum is 300. One incorrect measurement gets replaced with 80, making the new average 70 (total sum 350). We need to find the value of the incorrect measurement that was removed.
2. Connect average changes to sum changes: When the average increases from 60 to 70, the total sum increases from 300 to 350, so the sum increased by 50.
3. Relate sum change to value replacement: The sum increased by 50 because we removed the incorrect value and added 80. This means 80 minus the incorrect value equals 50.
4. Solve for the incorrect measurement: Set up the equation: 80 - incorrect value = 50, therefore the incorrect value = 30.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know in everyday terms. We have 5 test scores (or measurements) and their average is 60. Think of it like this: if you add up all 5 scores and divide by 5, you get 60.

Now, here's a key insight about averages: if 5 numbers average to 60, then their total sum must be \(5 \times 60 = 300\). This is like saying if 5 people each contribute an average of $60 to a group fund, the total fund has $300.

Next, we're told that one score was wrong and gets replaced with the correct value of 80. After this replacement, the new average becomes 70. Using the same logic, if 5 numbers now average to 70, their new total sum must be \(5 \times 70 = 350\).

Process Skill: TRANSLATE - Converting the problem's language about averages into concrete sum values

2. Connect average changes to sum changes

Now let's think about what happened to the total sum when the average changed. We started with a sum of 300, and we ended with a sum of 350.

The increase in the sum is: \(350 - 300 = 50\)

This makes intuitive sense - when the average goes up for the same number of items, the total sum must have increased. The sum increased by exactly 50.

3. Relate sum change to value replacement

Here's where we connect the dots. The sum increased by 50, and this happened because we took out one incorrect measurement and put in the correct value of 80.

Let's call the incorrect measurement 'x'. When we remove x and add 80, the net change to the sum is: \(80 - \mathrm{x}\)

But we know from step 2 that the sum increased by exactly 50. So we can say: \(80 - \mathrm{x} = 50\)

Process Skill: INFER - Connecting the sum change to the specific value replacement

4. Solve for the incorrect measurement

Now we solve the simple equation: \(80 - \mathrm{x} = 50\)

To find x, we rearrange: \(\mathrm{x} = 80 - 50 = 30\)

Let's verify this makes sense: if we remove 30 and add 80, the sum increases by \(80 - 30 = 50\). ✓
If the sum increases by 50, the average increases by \(50 \div 5 = 10\), from 60 to 70. ✓

Final Answer

The incorrect measurement was 30.

Looking at our answer choices:

1. 10
2. 20
3. 25
4. 30
5. 50

The answer is D. 30

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the relationship between average and sum
Students often struggle to connect that if 5 measurements have an average of 60, then their total sum must be \(5 \times 60 = 300\). They might try to work directly with the average values instead of converting to sums, making the problem much more complex than necessary. This foundational step is crucial for setting up the entire solution approach.

Faltering Point 2: Not recognizing that sum change equals net value change
When one value is replaced with another, students may not immediately see that the change in total sum equals (new value - old value). They might attempt more complicated approaches like setting up multiple equations or trying to find all individual measurements, rather than focusing on the simple replacement relationship.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in basic calculations
Students may make simple computational mistakes when calculating \(5 \times 60 = 300\), \(5 \times 70 = 350\), or \(350 - 300 = 50\). These seemingly trivial errors can derail the entire solution, especially under time pressure during the GMAT.

Faltering Point 2: Setting up the equation incorrectly
When establishing that \(80 - \mathrm{x} = 50\), students might confuse the order and write \(\mathrm{x} - 80 = 50\) instead. This happens because they may think about "the change from the incorrect value" rather than "the net addition to the sum," leading to \(\mathrm{x} = 130\), which doesn't match any answer choice.

Errors while selecting the answer

Faltering Point 1: Selecting the correct value instead of the incorrect value
After solving \(80 - \mathrm{x} = 50\) and finding \(\mathrm{x} = 30\), some students might mistakenly select 80 (the correct measurement value) from the answer choices if it were present, rather than 30 (the incorrect measurement they were asked to find). Always re-read what the question is specifically asking for in the final step.