Of the 66 people in a certain auditorium, at most 6 people have their birthdays in any one given month. Does at least one person in the auditorium have a birthday in January?
(1) More of the people in the auditorium have their birthday in February than in March.
(2) Five of the people in the auditorium have their birthday in March.
OA D
Source: Official Guide
Of the 66 people in a certain auditorium, at most 6 people h
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$$0\,\, \leqslant \,\,J,F,M, \ldots ,N,D\,\, \leqslant \,\,6\,\,\,\left( * \right)\,\,\,\,\,\,\left( {{\text{ints}}} \right)$$BTGmoderatorDC wrote:Of the 66 people in a certain auditorium, at most 6 people have their birthdays in any one given month. Does at least one person in the auditorium have a birthday in January?
(1) More of the people in the auditorium have their birthday in February than in March.
(2) Five of the people in the auditorium have their birthday in March.
Source: Official Guide
$$J + F + M + \ldots + N + D = 66\,\,\,\,\,\,\, \Rightarrow \,\,\,{\mu _{{\text{all}}}} = 5.5\,\,\frac{{{\text{birthdays}}}}{{{\text{month}}}}$$
$$?\,\,\,:\,\,\,J\,\,\mathop \geqslant \limits^? \,\,1$$
$$\left( 1 \right)\,\,F > M\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\left\langle {{\text{YES}}} \right\rangle $$
$$\left( {**} \right)\,\,\,J = 0\,\,\,\, \Rightarrow \,\,\,{\mu _{{\text{all}}\, - \,\left\{ J \right\}}} = 6\,\,\frac{{{\text{birthdays}}}}{{{\text{month}}}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\underline {F = M} = \ldots = N = D = 6\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\underline {{\text{impossible}}} $$
$$\left( 2 \right)\,\,\,M = 5\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\left\langle {{\text{YES}}} \right\rangle $$
$$\left( {***} \right)\,\,\,J = 0\,\,\,\, \Rightarrow \,\,\,{\mu _{{\text{all}}\, - \,\left\{ J \right\}}} = 6\,\,\frac{{{\text{birthdays}}}}{{{\text{month}}}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,F = \underline M = \ldots = N = D = \underline 6 \,\,\,\,\, \Rightarrow \,\,\,\,\,\,\underline {{\text{impossible}}} $$
We follow the notations and rationale taught in the GMATH method.
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Fabio.
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Let's take each statement one by one.BTGmoderatorDC wrote:Of the 66 people in a certain auditorium, at most 6 people have their birthdays in any one given month. Does at least one person in the auditorium have a birthday in January?
(1) More of the people in the auditorium have their birthday in February than in March.
(2) Five of the people in the auditorium have their birthday in March.
OA D
Source: Official Guide
(1) More of the people in the auditorium have their birthday in February than in March.
Since to exclude January, from February to December, in each month, 6 persons b'day must fall in each of these 11 months. However, we know that in March, the number of persons whose b'day falls in it must be 5 or less; thus, at least one person's b'day must be in January. Sufficient.
(2) Five of the people in the auditorium have their birthday in March.
Since to exclude January, from February to December, in each month, 6 persons b'day must fall in each of these 11 months. However, we know that in March, there are only 5 persons; thus, at least one person's b'day must be in January. Sufficient.
The correct answer: D
Hope this helps!
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Solution:BTGmoderatorDC wrote: ↑Sun Jan 27, 2019 3:52 pmOf the 66 people in a certain auditorium, at most 6 people have their birthdays in any one given month. Does at least one person in the auditorium have a birthday in January?
(1) More of the people in the auditorium have their birthday in February than in March.
(2) Five of the people in the auditorium have their birthday in March.
OA D
Source: Official Guide
We need to determine whether at least one person in the auditorium has a birthday in January. The only way this can’t happen (i.e., no one has a birthday in January) isif exactly 6 people have a birthday in each of the other 11 months. In other words, if we know there is a month (besides January) that has fewer than 6 people whose birthdays are in that month, then at least one person in the auditorium will have a birthday in January.
Statement One Alone:
More of the people in the auditorium have their birthday in February than in March.
Since more people in the auditorium have their birthday in February than in March, there must be fewer than 6 people whose birthdays are in March (since the most February could have is 6). Therefore, from our stem analysis, there is at least one person whose birthday is in January. Statement one alone is sufficient.
Statement Two Alone:
Five of the people in the auditorium have their birthday in March.
From our stem analysis, we see that there must be at least one person whose birthday is in January. Thus, statement two is also sufficient.
Answer: D
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