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23 Apr 2019, 01:57
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
28% (02:51) correct
72%
(03:01)
wrong
based on 82
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Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 365-day period will exactly two friends visit her?
(A) 48
(B) 54
(C) 60
(D) 66
(E) 72
(A) 48
(B) 54
(C) 60
(D) 66
(E) 72
23 Apr 2019, 05:40
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let : a = Alice's visits, b = Beatrix's visits, c = Claire's visits.
ab = Alice & Beatrix visits on same day ----> every 12 days (3*4) ---> in 365, ab = \(\frac{365}{12}\) = \(30.something\) = 30
bc = Beatrix & Claire visits on same day ---> every 20 days (4*5) ---> in 365, bc = \(\frac{365}{20}\) = \(18.something\) = 18
ca = Claire & Alice visist on same day ------> every 15 days (5*3) ---> in 365, ca = \(\frac{365}{15}\) = \(24.something\) = 24
abc = Alice, Beatrix & Claire visits on same day --> every 60 days (3*4*5) ---> in 365, abc = \(\frac{365}{60}\) = \(6.something\) = 6
Exactly 2 = (ab + bc + ca) - 3 (abc) = (30+18+24) - (3*6) = 54 :: B
ab = Alice & Beatrix visits on same day ----> every 12 days (3*4) ---> in 365, ab = \(\frac{365}{12}\) = \(30.something\) = 30
bc = Beatrix & Claire visits on same day ---> every 20 days (4*5) ---> in 365, bc = \(\frac{365}{20}\) = \(18.something\) = 18
ca = Claire & Alice visist on same day ------> every 15 days (5*3) ---> in 365, ca = \(\frac{365}{15}\) = \(24.something\) = 24
abc = Alice, Beatrix & Claire visits on same day --> every 60 days (3*4*5) ---> in 365, abc = \(\frac{365}{60}\) = \(6.something\) = 6
Exactly 2 = (ab + bc + ca) - 3 (abc) = (30+18+24) - (3*6) = 54 :: B
Updated on: 23 Apr 2019, 11:44
Originally posted by GMATinsight on 23 Apr 2019, 05:46.
Last edited by GMATinsight on 23 Apr 2019, 11:44, edited 1 time in total.
Last edited by GMATinsight on 23 Apr 2019, 11:44, edited 1 time in total.
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Bunuel
The cyclicity of Alice's visit = 3 days
The cyclicity of Beatrix visit = 4 days
The cyclicity of Clair visit = 5 days
A and B will meet every day that is multiple of 3 and 4 = 12th Day
i.e. Total cycles of 12 days in 365 days = 365 / 12 = 30 days
B and C will meet every day that is multiple of 4 and 5 = 20th Day
i.e. Total cycles of 20 days in 365 days = 365 / 20 = 18 days
A and C will meet every day that is multiple of 3 and 5 = 15th Day
i.e. Total cycles of 15 days in 365 days = 365 / 15 = 24 days
and They all will meet on a day which is multiple of 3, 4 and 5 all
i.e. All of them will meet on LCM(3,4,5) = 60 days
Total cycles of 60 days in 365 days = 365/60 = 6
i.e. Total distinct days = 30 +18 +24 - 3*6 = 54 days
Answer: Option B
