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Question 1054884: Of 110 clock radios sold recently in a department store, 60 had FM circuitry, and 80 had AM circuitry. How many could receive FM transmission only?
10
60
20
50
40
Please explain. Thank you.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 60 could receive FM because they had FM circuitry.
80 could receive AM because they have AM circuitry.
if you add 60 and 80, you get 140.
since there are only 110 radios, then some of the radios had to have both.
in order to get from 140 to 110, you have to subtract 30.
that is the number of radios that had both.
how does this work?
you are given that 60 had FM circuitry.
if 30 of those also had AM circuitry, then you can break the total down into:
30 had FM circuitry only.
30 had both.
you are given that 80 had AM circuitry.
if 30 of those also had FM circuitry, then you can break the total down into:
50 had AM circuitry only.
30 had both.
the ones that had both are being double counted.
they were included in the ones that had AM circuitry and they were included in the ones that had FM circuitry.
to remove the double counting, you have to remove the count of one of them.
your total count becomes:
30 had FM only.
50 had AM only.
30 had both.
add that up and you get a total of 110 radios.
the official formula for this phenomenon is:
(A or B) = A + B - (A and B).
A would be the set of FM radios.
B would be the set of AM radios.
the formula of (A or B) = A + B - (A and B) becomes:
(FM or AM) becomes FM + AM - (AM and FM).
replace with the numbers given and you get:
110 = 60 + 80 - (AM and FM).
simplify to get 110 = 140 - (AM and FM).
subtract 110 from both sides of the equation and add (AM and FM) to both sides of the equation to get:
(AM and FM) = 140 - 110.
simplify to get (AM and FM) = 30.
the implied assumption in a problem such as this is that the number of FM radios contains those radios that have both FM and AM circuitry in them.
likewise, the implied assumption in a problem such as this is that the number of AM radios contains those radios that have both FM and Am circuitry in them.
as such, the radios that have both FM and AM circuitry in them are being double counted.
this is why one set of them has to be subtracted.
the general formula is, once again.
(A or B) = A + B - (A and B)
this means that the number of elements in set A or set B is equal to the number of elements in set A plus the number of elements in set B minus the number of elements that are in both set A and B.
the question was:
How many could receive FM transmission only?
the answer to that would be 30, because.
FM DISABLED_event_only= 30
AM DISABLED_event_only= 50
AM and FM combined would be 30.
you may have misstated the problem because 30 isn't one of the selections.
if you meant 60 had AM and 80 had FM, then the answer would be 50.
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