Logical Reasoning

PT155 · S4 · Q21 All of the students at Harrison

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All of the students at Harrison University live in one of two residence complexes, either Pulham or Westerville.

Facts

All students live in Pulham or Westerville. 38% of students take a night class. 29% of students in W take a night class.

Evaluate

We primarily read Inference questions looking for ways to combine ideas using Conditional, Causal, Comparative, or Math-y thinking. This one definitely seems math-y.

We can infer that more than 38% of Pulham's students a night class, since the average of Pulham and Westerville comes out to be 38%. If W is below that number, than Pulham has to be above that number.

Suppose all students at a school or girls or boys. If 70% of the boys play soccer and 80% of the girls play soccer, what % of the students at the school play soccer?

If there were an equal number of boys and girls, it would be 75% of students. But, not knowing whether there's an equal amount, we would just know that the % of students who play soccer is somewhere between 70% and 80%.

If there are more girls at the school, it would be higher than 75% (the simple average of 70 and 80), because the 80% of girls statistic would have more influence than the 70% of boys statistic. If there were more boys, it would be between 70-75%.

This is the concept of a weighted average. If you mix a beer that's 5% alcohol with whiskey that's 45% alcohol, the resulting beverage will be somewhere between 5-45% alcohol.

Goal

The mathematical inference they're setting up is that since all students come from Pulham or Westerville, and the students at Westerville are lower than the school average for night school (38%), the students at Pulham have to be higher than 38% for night school, in order for the overall average to be 38%.

If both Pulham and Westerville were lower than 38%, then the school's overall average couldn't be 38%.

21.

If the statements above are true, which one of the following must also be true?

Correct
More than 38 percent of
Why this is right
If you know that a mix of two things comes out to be 38%, then either both things are 38%, or one of the things is lower than 38% and one is higher than 38%. If I know a mixed drink with two ingredients is 20% alcohol, then either both ingredients are themselves 20% alcohol, or it means that one of the ingredients is lower than 20% alcohol and the other one is greater than 20% alcohol. If I know the average age at a company is 35 years old, then either the male and female employees both have an average age of 35, or one group has an average age below 35 and the other group has an average age above 35. We know that when you mix Pulham and Westerville together you get a 38% rate of students taking night classes. If W is below 38%, then P must be above 38%. NERDY MATH EXAMPLE (if you want it) Pretend there are 100 students who live in W and 100 who live in Pulham. 200 total students go to this school. 29% of W takes a night class = 29 out of 100 38% of students overall do = 76 out of 200 So that would mean the other 47 students who take a night class (76 - 29 = 47) come from Pulham. 47% of Pulham take a night class. That's how the math looks with equal sized Pulham and Westerville dorms. Since W is 9% below the overall average, Pulham would be 9% above the overall average in order to make the average be what it is. If Pulham and Westerville aren't equal sized, then Pulham's gap above 38% doesn't have to equal Westerville's gap below 38%, but it does still have to be the case that Pulham is above 38%. The school's overall rate of students taking night classes will always be somewhere in between the rate of Westerville students taking a night class and that of Pulham students taking a night class. since those two dorms comprise the whole school.
Skill tested: Must be True · how this choice captures the argument's function is the move to repeat next time.
46% picked this
More than 50 percent of
Too Strong: more than 50%
This is possible but not mandatory. If there were a lot more students in Westerville than in Pulham, then Pulham could easily have a higher than 50% rate of night students, but without that information there's no reason to think Pulham is above 50%. Pulham has to be higher than 38% for the overall average to settle at 38%, but we can't derive how much farther above 38% Pulham would be without knowing the ratio of students living in Pulham vs. Westerville.
26% picked this
More students at Harrison live
Out of Scope Comparison: # vs. %
This discussion only involves relative ideas. We can't derive any real number comparison. However, if we knew the night school rates were like this ... Westerville = 29% overall = 38% Pulham = 55% ... that would prove that there are more students in W than in Pulham (because the overall average is closer to Westerville 's average than it is to Pulham's average). If you combine 5% alcohol beer and 45% alcohol whiskey and get a drink that is 7% alcohol, then you know that it's mostly beer, with a few drops of whiskey in there. If the combo was 42% alcohol, you would know that it's mostly whiskey, with a few drops of beer in there.
10% picked this
Harrison students living in Pulham
Out of Scope
Out of Scope: more than one night class We couldn't possibly derive whether they take only one or more than one night class, since the data is only expressed in terms of whether or not you take at least one night class.
14% picked this
Night classes at Harrison have
Out of Scope Comparison
Out of Scope Comparison: Night classes vs. Day classes Enrollment We don't have any information about students per class for night or for day, so we can't possibly make this comparison about how dense, on average, night vs. day classes are.
3% picked this

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