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In a survey of 248 people, 156 are married, 70 are self-employed and 25% of those who are married are self-employed. If a person is to be randomly selected from those surveyed, what is the probability that the person selected will be self-employed but not married?
Let's break down what we're looking for in everyday language. We have 248 people surveyed, and we want to find the probability of randomly picking someone who is self-employed BUT NOT married.
Think of this like having two overlapping circles - one for married people and one for self-employed people. Some people belong to both circles (married AND self-employed), some belong to just one circle, and some belong to neither.
We're specifically looking for people who are ONLY in the self-employed circle - not in the overlap area.
Given information:
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding
Now we need to figure out how many people are in that overlap area - people who are both married AND self-employed.
The problem tells us that 25% of the married people are self-employed. Since there are 156 married people:
People who are both married AND self-employed = \(25\% \text{ of } 156\)
= \(0.25 \times 156 = 39\) people
So 39 people belong to both groups - they're married and self-employed.
Now we can find our target group: people who are self-employed but NOT married.
We know that 70 people total are self-employed. Of these 70 people, 39 are also married. So the remaining self-employed people must be the ones who are NOT married:
Self-employed but NOT married = Total self-employed - (Self-employed AND married)
= \(70 - 39 = 31\) people
So there are 31 people who are self-employed but not married.
Process Skill: INFER - Drawing the non-obvious conclusion about the non-overlapping portion
Finally, we calculate the probability by dividing our target group by the total number of people surveyed:
Probability = (Number of people who are self-employed but not married) ÷ (Total people surveyed)
= \(31 \div 248\)
Let's simplify this fraction. We need to find the greatest common divisor of 31 and 248:
\(248 \div 31 = 8\)
So: \(\frac{31}{248} = \frac{1}{8}\)
The probability that a randomly selected person will be self-employed but not married is \(\frac{1}{8}\).
This matches answer choice A: \(\frac{1}{8}\)
Verification: Let's double-check our work:
Faltering Point 1: Misinterpreting "self-employed but not married"
Students often confuse what "but not married" means. They might think they need to find people who are NOT self-employed and NOT married, or they might forget the "not married" constraint entirely and just calculate the probability of being self-employed. The key is recognizing this requires finding the non-overlapping portion of the self-employed group.
Faltering Point 2: Misunderstanding the percentage constraint
When the problem states "25% of those who are married are self-employed," students might incorrectly interpret this as "25% of self-employed people are married." This reversal leads to calculating 25% of 70 instead of 25% of 156, giving a completely wrong overlap value.
Faltering Point 3: Not recognizing the need for set subtraction
Students might try to directly calculate "self-employed but not married" without first finding the overlap. They fail to see that they need to subtract the "both married and self-employed" group from the total self-employed group to get their target population.
Faltering Point 1: Arithmetic errors in percentage calculations
When calculating 25% of 156, students might make basic arithmetic mistakes like computing \(0.25 \times 156 = 36\) instead of 39, or forgetting to convert the percentage to a decimal (using \(25 \times 156\) instead of \(0.25 \times 156\)).
Faltering Point 2: Incorrect subtraction for finding the target group
After finding that 39 people are both married and self-employed, students might subtract incorrectly: either \(156 - 39 = 117\) (subtracting from married instead of self-employed) or making a basic arithmetic error in \(70 - 39\).
Faltering Point 3: Fraction simplification errors
When simplifying \(\frac{31}{248}\), students might not recognize that \(248 = 31 \times 8\), or they might make division errors. Some students might leave the answer as \(\frac{31}{248}\) without simplifying, missing that it reduces to \(\frac{1}{8}\).
Faltering Point 1: Selecting an intermediate calculation result
Students might select answer choice C (\(\frac{117}{248}\)) if they incorrectly calculated \(156 - 39 = 117\) and used that as their numerator, thinking 117 represents people who are self-employed but not married. This comes from confusing "married but not self-employed" with the target group.
Faltering Point 2: Choosing the unsimplified fraction equivalent
Students might arrive at \(\frac{31}{248}\) correctly but fail to recognize this equals \(\frac{1}{8}\). If there were an answer choice showing \(\frac{31}{248}\), they might select it instead of the simplified form, not realizing that GMAT answers are typically in simplest form.