In a survey of 200 people, 80 people read Magazine M and 96 people read magazine R. If the number...

1. Translate the problem requirements: We need to clarify what 'people who read both magazines' means (the overlap between the two groups) and understand that 'people who do not read either magazine' means those completely outside both groups. The key relationship is that non-readers = \(3 \times \mathrm{both-readers}\).
2. Set up the overlap relationship using given constraints: Use the constraint that non-readers equal 3 times the both-readers to create our primary equation with one unknown variable.
3. Apply the inclusion-exclusion principle: Account for the fact that people reading both magazines are counted twice when we add individual magazine readers, so we need to subtract the overlap once to get the total number of readers.
4. Solve for the overlap using total survey size: Since total readers + non-readers must equal 200 people, substitute our expressions to find the number reading both magazines.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what each piece of information means in everyday terms:

• 200 people were surveyed in total
• 80 people read Magazine M
• 96 people read Magazine R
• Some people read both magazines (this is our unknown that we need to find)
• Some people don't read either magazine
• The key relationship: People who read neither = \(3 \times \mathrm{People\ who\ read\ both}\)

Think of it like this: imagine you're looking at all 200 people. Some read only M, some read only R, some read both M and R, and some read neither. The people who read both are counted in both the "80 who read M" and the "96 who read R" groups.

Let's call the number of people who read both magazines "x".
Then the number of people who read neither magazine is "\(3x\)".

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

2. Set up the overlap relationship using given constraints

Now we establish our key constraint equation. We know that:

People who read neither = \(3 \times \mathrm{People\ who\ read\ both}\)
So: Number who read neither = \(3x\)

This gives us a direct relationship between two unknown quantities, reducing our problem to finding just one variable (x).

3. Apply the inclusion-exclusion principle

Here's the crucial insight: when we add up the 80 people who read M and the 96 people who read R, we're double-counting the people who read both magazines.

In plain English: If someone reads both magazines, they get counted once in the "M readers" group and once again in the "R readers" group. To find the actual total number of people who read at least one magazine, we need to subtract the overlap once.

So the total number of people who read at least one magazine is:
\(80 + 96 - x = 176 - x\)

This means:

• People reading at least one magazine = \(176 - x\)
• People reading neither magazine = \(3x\)

Process Skill: INFER - Recognizing that double-counting occurs and must be corrected

4. Solve for the overlap using total survey size

Since everyone in the survey either reads at least one magazine OR reads neither magazine, these two groups must add up to the total of 200 people.

People reading at least one + People reading neither = Total surveyed
\((176 - x) + 3x = 200\)

Simplifying:
\(176 - x + 3x = 200\)
\(176 + 2x = 200\)
\(2x = 200 - 176\)
\(2x = 24\)
\(x = 12\)

Let's verify: If 12 people read both magazines, then \(3 \times 12 = 36\) people read neither magazine.
People reading at least one magazine = \(176 - 12 = 164\)
Total check: \(164 + 36 = 200\) ✓

4. Final Answer

The number of people who read both magazines is 12.

This matches answer choice B.

Verification:

• Both magazines: 12 people
• Neither magazine: \(3 \times 12 = 36\) people
• At least one magazine: \(80 + 96 - 12 = 164\) people
• Total: \(164 + 36 = 200\) people ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the constraint relationship
Students often misread "the number of people who do not read either magazine is 3 times the number who read both magazines" as "the number who read both is 3 times the number who read neither." This reversal leads to setting up the equation as \(x = 3(\mathrm{neither})\) instead of \(\mathrm{neither} = 3x\), fundamentally changing the problem setup.

2. Forgetting about the overlap in Venn diagram logic
Many students fail to recognize that people who read both magazines are counted twice when we add the 80 M-readers and 96 R-readers. They incorrectly assume that 80 + 96 = 176 represents the actual number of people reading at least one magazine, missing the crucial inclusion-exclusion principle.

3. Not establishing the total constraint equation
Students sometimes focus only on the given numbers (80, 96) and the 3:1 relationship, but fail to recognize that all survey participants must be accounted for in the 200 total. They miss setting up the fundamental equation: (at least one magazine) + (neither magazine) = 200.

Errors while executing the approach

1. Algebraic manipulation errors
When solving \((176 - x) + 3x = 200\), students commonly make sign errors, writing \(176 - x + 3x\) as \(176 - 4x\) instead of \(176 + 2x\). This leads to incorrect equations like \(176 - 4x = 200\), yielding negative values.

2. Arithmetic calculation mistakes
Students may correctly set up \(176 + 2x = 200\) but then make basic arithmetic errors: calculating \(200 - 176\) as 34 instead of 24, or dividing incorrectly to get \(x = 17\) or \(x = 6\) instead of \(x = 12\).

Errors while selecting the answer

1. Selecting the wrong calculated value
After correctly solving and finding \(x = 12\) (people reading both), students sometimes confuse this with other values they calculated, such as selecting 36 (people reading neither) or 24 (the intermediate calculation \(2x\)) as their final answer instead of 12.