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Out of 300 vehicle-riders surveyed on a highway, 50% had a valid
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05 Sep 2026, 00:07
05 Sep 2026, 00:07
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Out of \(300\) vehicle-riders surveyed on a highway, 50 % had a valid registration, 40 % had a valid insurance card, and 30 % had a valid emissions certificate. Exactly 4 % had all three documents, and 10 % had none of the three. The number of riders with both a registration and an insurance card (including those with all three) was twice the number of riders with both an insurance card and an emissions certificate (including those with all three). If 18 riders had both a registration and an emissions certificate, how many riders had only a valid registration?
(A) 64
(B) 72
(C) 88
(D) 94
(E) 102
(A) 64
(B) 72
(C) 88
(D) 94
(E) 102
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Re: Out of 300 vehicle-riders surveyed on a highway, 50% had a valid
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08 Sep 2026, 10:32
08 Sep 2026, 10:32
1
Expert Reply
Solution
There are 300 riders.
Find the number in each category
Registration: (50%) of 300 = 150
Insurance: (40%) of 300 = 120
Emissions: (30%) of 300 = 90
All three: (4%) of 300 = 12
None: (10%) of 300 = 30
Therefore, the number with at least one document is:
\(300-30=270\)
Find the total of the pairwise overlaps
\(R+I+E-(RI+RE+IE)+RIE=270\)
Substitute:
\(150+120+90-(RI+18+IE)+12=270\)
\(372-(RI+IE+18)=270\)
\(RI+IE=84\)
We are told:
\(RI=2IE\)
So:
\(2IE+IE=84
IE=28\)
and therefore:
\(RI=56\)
Find riders with only registration
Start with all 150 registration holders.
Subtract those who also have insurance (56), and subtract those who also have emissions (18). But the 12 riders with all three were subtracted twice, so add them back once:
\(150-56-18+12 ={88}\)
Answer: C
There are 300 riders.
Find the number in each category
Registration: (50%) of 300 = 150
Insurance: (40%) of 300 = 120
Emissions: (30%) of 300 = 90
All three: (4%) of 300 = 12
None: (10%) of 300 = 30
Therefore, the number with at least one document is:
\(300-30=270\)
Find the total of the pairwise overlaps
\(R+I+E-(RI+RE+IE)+RIE=270\)
Substitute:
\(150+120+90-(RI+18+IE)+12=270\)
\(372-(RI+IE+18)=270\)
\(RI+IE=84\)
We are told:
\(RI=2IE\)
So:
\(2IE+IE=84
IE=28\)
and therefore:
\(RI=56\)
Find riders with only registration
Start with all 150 registration holders.
Subtract those who also have insurance (56), and subtract those who also have emissions (18). But the 12 riders with all three were subtracted twice, so add them back once:
\(150-56-18+12 ={88}\)
Answer: C
gmatclubot
Re: Out of 300 vehicle-riders surveyed on a highway, 50% had a valid [#permalink]
08 Sep 2026, 10:32
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