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Question 86093: When a student is questioned about his school, he replies that there
are at least as many freshmen as there are juniors and at least as many
juniors as there are sophomores. If the student is correct, which of the
following statements must be true?
A. There are just as many sophomores as there are freshmen.
B. There are at least as many sophomores as there are freshmen.
C. There are at least as many freshmen as there are sophomores.
D. There are more freshmen than there are sophomores
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let f=# of freshmen, s=# of sophomores, and j=# of juniors
If the problem states:
"there are at least as many freshmen as there are juniors"
that means that the number of freshmen equal or exceed the number of juniors. Lets say we had 10 juniors. That means we must have 10 or more freshmen. So this can be represented by this inequality:
Also, if the problem states:
"there are at least as many juniors as there are sophomores"
that means that the number of juniors equal or exceed the number of sophomores. Lets say we had 20 sophomores. That means we must have 20 or more juniors. So this can be represented by this inequality:
So we have this system of inequalities
,
By intuition, we can clearly see that since f is greater than j, and j is greater than s, then f must be greater than s. Since there is a possibility that everything is equal, f can also equal s. So we can represent our ideas into this combined inequality:
However,if intuition is not good enough, what we just applied was the Transitive Property of Inequalities
So looking at the given answers, the only one that matches with our inequality is answer C). Remember, the statement
"There are at least as many freshmen as there are sophomores"
looks like this
and this matches with our answer
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