Deconstructing the Question10% of all motorists receive speeding tickets.
Among those who exceed the speed limit, 20% do not receive tickets, so 80% do.
Let the percentage of motorists who exceed the speed limit be
\(x\).
Step-by-stepThe percentage who both exceed the limit and receive tickets is
\(0.8x\)This equals the total percentage receiving tickets:
\(0.8x = 10\%\)Solve:
\(x = \frac{10\%}{0.8} = 12.5\%\)Answer: BANOTHER APPROACHDeconstructing the QuestionLet
\(S\) be the set of motorists who exceed the speed limit, and let
\(T\) be the set of motorists who receive speeding tickets.
The problem says that
\(10\%\) of all motorists are in
\(T\).
It also says that
\(20\%\) of motorists in
\(S\) do not receive tickets, so
\(80\%\) of motorists in
\(S\) do receive tickets.
Step-by-stepLet the total set of motorists be
\(U\).
Then
\(|T| = 0.10|U|\)Since
\(20\%\) of speeders do not get tickets,
\(80\%\) of speeders do get tickets, so
\(|T| = 0.80|S|\)Now equate the two expressions for
\(|T|\):
\(0.80|S| = 0.10|U|\)So
\(|S| = \frac{0.10}{0.80}|U| = 0.125|U|\)Thus the percentage of motorists who exceed the speed limit is
\(12.5\%\)Answer: B