Before getting Answer of Week 3, Lets Practice some Analytical Portion
Six actors ---- Bob, Carol, Dave Ed, Frank, and Grace audition for a part in an
off-Broadway play. The auditions will take place over four consecutive days,
starting on a Thursday. Each actor will have one audition; the days on which
the different actors will audition must conform to the following conditions.
i. At least one audition will take place each day.
ii. No more than two auditions will take place on any day.
iii. No more than three auditions will take place on any two
consecutive days.
iv. Bob’s audition must take place on Saturday.
v. Carol’s audition must take place on the same day as another
audition.
vi. Frank’s auditions must take place on the day before Grace’s
audition.
vii. Dave’s audition must take place on a day after Ed’s audition.
1 If only one audition takes place on Thursday which actor could have that
audition?
(A) Bob (B) Carol (C) Dave (D) Frank (E) Grace
2 If Bob’s and Frank’s auditions are on the same day, which of the following
must be true
(A) Dave’s audition will take place on Thursday
(B) Dave’s audition will take place on Friday
(C) Grace’s audition will take place on Thursday
(D) Carol’s audition will take place on Sunday
(E) Ed’s audition will take place on Sunday
3 If the director decides to hold two auditions on Thursday and two on
Sunday, how many actors would be eligible to audition on Friday?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
4 If Ed and Grace have their auditions on the same day which of the following
must be true?
(A) Ed’s audition will take place on Thursday.
(B) Frank’s audition will take place on Friday.
(C) Carol’s audition will take place on Saturday.
(D) Grace’s audition will take place on Saturday.
(E) Carol’s audition will take place on Sunday.
5 If Ed’s audition is on Saturday, which of the following actors cannot
audition on the same day as any other actor?
(A) Bob
(B) Carol
(C) Ed
(D) Frank
(E) Gr
ace
Questions 6-10:
During the first half of the year, from January through June, the chairperson of
the mathematics department will be on sabbatical. The dean of the college has
asked each of the six professors in the department --- Arkes, Borofsky, Chang,
Denture, Hobbes, and Lee--- to serve as acting chairperson during one ofthose months. The mathematicians can decide the order in which they will
serve, subject only to the following criteria established by the dean.
i. Chang will serve as chairperson in February.
ii. Arkes will serve as chairperson before Hobbes does.
iii. Borofsky and Dexter will serve as chairpersons in consecutive
months.
6 Which of the following professors could serve as chairperson in January?
(A) Borodfsky (B) Chang (C) Dexter (D) Hobbes (E) Lee
7 In how many ways can the schedule be made up if Lee has to serve as
chairperson in May?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 6
8 If Lee serves in April, all of the following could be true EXCEPT
(A) Arkes serves in January
(B) Hobbes serves in march
(C) Borofsky serves in may
(D) Borofsky serves in June
(E) Hobbes serves in June
9 If Borofsky serves in May, what is the latest month in which Arkes could
serve?
(A) January
(B) February
(C) March
(D) April
(E) June
10 Which of the following CANNOT be true?
(A) Arkes and Lee serve in consecutive months.
(B) Lee and Hobbes serve in consecutive months.
(C) Hobbes and Dexter serve in consecutive months.
(D) Arkes and Chang serve in consecutive months.
(E) Borofsky and Chang serve in consecutive months.
Solutions 1-5:
First express each of the conditions symbolically:
B, C, D, E, F, and G: 1 audition each
Days: Thu, Fri, Sat, Sun
Each day: 1 or 2 auditions
2 consecutive days: 2 or 3 auditions
B=Sat Cx F<G E<D
1 A violates the condition that Bob’s audition will take place on Saturday
(B=Sat). B violates the condition that Carol’s audition cannot be the only
audition on a particular day (Cx). Choices C and E are impossible. Since
Dave’s audition must take place on a day after Ed’s audition (E<D) and Grace’s
audition must take place on a day after Frank’s audition (F<G) neither can
take place on Thursday. Only choice D does not violate any of the given
conditions, so this is the correct answer.
2 The condition that Bob’s and Frank’s auditions are on the same day completely
determines the schedule. They must take place on Saturday (B=Sat). To
avoid having more than three auditions on two consecutive days, there can be
only one audition on Friday and one on Sunday, which means there will be two
on Thursday. Since Frank must have to precede Grace (F<G), Grace’s audition
will take place on Sunday. Since Ed must precede Dave, Ed’s audition will take
place on Thursday and Dave’s audition on Friday. Finally, Carol’s audition will
be the second audition on Thursday. The final schedule is “C and E on
Thursday, D on Friday, B and F on Saturday and G on Sunday”. Only choice B
is consistent with this schedule, so “B” is the correct choice.
3 Since only one audition can take place on Friday, it cannot be Carol’s (Cx);
and, of course, it cannot be Bob’s (B = Sat). Any of the other four actors
could audition on Friday as indicated in the following schedules:
E/F on Thu, D on Fri, B on Sat, C/G on Sun
C/F on Thu, E on Fri, B on Sat, D/G on Sun
C/E on Thu, E on Fri, B on Sat, D/G on Sun
E/F on Thu, G on Fri, B on Sat, C/D on Sun
So the correct choice is D.
4 The only schedule that fulfils the conditions is “F on Thu, E/G on Fri, B on Sat,
and C/D on Sun”. Only choice E is consistent with this schedule.
5 Since Ed and Bob’s auditions are both taking place on Saturday, eliminate
choices A and C. Since Carole must audition on the same day as another
actor, eliminate B. Finally, since Dave’s audition must take place on Sunday
(E < D), Frank’s audition must take place on Thursday and Grace’s audition on
Friday (F < G). Eliminate choice D. The complete schedule is: “C/F on Thu, G
on Fri, B/F on Sat, and D on Sun.”
Solutions 6-10:
Let A, B, C, D, H, L represents professor names.
C=February, A<H, B<<D and D<<B
6 Only choice E is there, which does not violate any of the conditions, so is the
correct choice.
7 With C serving in February and L in May, the only consecutive months
available for B and D are March and April. Then since A must serve is before H
in June. There are two possible schedules, depending on the order of B and D,
so the correct choice is B.
8 If L serves in April, the consecutive months available for B and D are May and
June; so choices C and D could be true. Since A must serve before H, choices
A and B must be true, only choice E cannot be true.
9 Since A must serve before H does, A cannot serve in June. Can A serve in
April? No, because then, D would serve in June (B<<D or D<<B), and again A
would not precede H. The latest that A could serve in March, which could
occur in the final order: L, C, A, D, B and H.
10 The only professors that can serve in January are A and L, so, one of them
must serve in January, and neither serves in February. So choice A cannot be
true.
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Analytical
Below are some Question with Explaination, try to solve it.Six actors ---- Bob, Carol, Dave Ed, Frank, and Grace audition for a part in an
off-Broadway play. The auditions will take place over four consecutive days,
starting on a Thursday. Each actor will have one audition; the days on which
the different actors will audition must conform to the following conditions.
i. At least one audition will take place each day.
ii. No more than two auditions will take place on any day.
iii. No more than three auditions will take place on any two
consecutive days.
iv. Bob’s audition must take place on Saturday.
v. Carol’s audition must take place on the same day as another
audition.
vi. Frank’s auditions must take place on the day before Grace’s
audition.
vii. Dave’s audition must take place on a day after Ed’s audition.
1 If only one audition takes place on Thursday which actor could have that
audition?
(A) Bob (B) Carol (C) Dave (D) Frank (E) Grace
2 If Bob’s and Frank’s auditions are on the same day, which of the following
must be true
(A) Dave’s audition will take place on Thursday
(B) Dave’s audition will take place on Friday
(C) Grace’s audition will take place on Thursday
(D) Carol’s audition will take place on Sunday
(E) Ed’s audition will take place on Sunday
3 If the director decides to hold two auditions on Thursday and two on
Sunday, how many actors would be eligible to audition on Friday?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
4 If Ed and Grace have their auditions on the same day which of the following
must be true?
(A) Ed’s audition will take place on Thursday.
(B) Frank’s audition will take place on Friday.
(C) Carol’s audition will take place on Saturday.
(D) Grace’s audition will take place on Saturday.
(E) Carol’s audition will take place on Sunday.
5 If Ed’s audition is on Saturday, which of the following actors cannot
audition on the same day as any other actor?
(A) Bob
(B) Carol
(C) Ed
(D) Frank
(E) Gr
ace
Questions 6-10:
During the first half of the year, from January through June, the chairperson of
the mathematics department will be on sabbatical. The dean of the college has
asked each of the six professors in the department --- Arkes, Borofsky, Chang,
Denture, Hobbes, and Lee--- to serve as acting chairperson during one ofthose months. The mathematicians can decide the order in which they will
serve, subject only to the following criteria established by the dean.
i. Chang will serve as chairperson in February.
ii. Arkes will serve as chairperson before Hobbes does.
iii. Borofsky and Dexter will serve as chairpersons in consecutive
months.
6 Which of the following professors could serve as chairperson in January?
(A) Borodfsky (B) Chang (C) Dexter (D) Hobbes (E) Lee
7 In how many ways can the schedule be made up if Lee has to serve as
chairperson in May?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 6
8 If Lee serves in April, all of the following could be true EXCEPT
(A) Arkes serves in January
(B) Hobbes serves in march
(C) Borofsky serves in may
(D) Borofsky serves in June
(E) Hobbes serves in June
9 If Borofsky serves in May, what is the latest month in which Arkes could
serve?
(A) January
(B) February
(C) March
(D) April
(E) June
10 Which of the following CANNOT be true?
(A) Arkes and Lee serve in consecutive months.
(B) Lee and Hobbes serve in consecutive months.
(C) Hobbes and Dexter serve in consecutive months.
(D) Arkes and Chang serve in consecutive months.
(E) Borofsky and Chang serve in consecutive months.
Solutions 1-5:
First express each of the conditions symbolically:
B, C, D, E, F, and G: 1 audition each
Days: Thu, Fri, Sat, Sun
Each day: 1 or 2 auditions
2 consecutive days: 2 or 3 auditions
B=Sat Cx F<G E<D
1 A violates the condition that Bob’s audition will take place on Saturday
(B=Sat). B violates the condition that Carol’s audition cannot be the only
audition on a particular day (Cx). Choices C and E are impossible. Since
Dave’s audition must take place on a day after Ed’s audition (E<D) and Grace’s
audition must take place on a day after Frank’s audition (F<G) neither can
take place on Thursday. Only choice D does not violate any of the given
conditions, so this is the correct answer.
2 The condition that Bob’s and Frank’s auditions are on the same day completely
determines the schedule. They must take place on Saturday (B=Sat). To
avoid having more than three auditions on two consecutive days, there can be
only one audition on Friday and one on Sunday, which means there will be two
on Thursday. Since Frank must have to precede Grace (F<G), Grace’s audition
will take place on Sunday. Since Ed must precede Dave, Ed’s audition will take
place on Thursday and Dave’s audition on Friday. Finally, Carol’s audition will
be the second audition on Thursday. The final schedule is “C and E on
Thursday, D on Friday, B and F on Saturday and G on Sunday”. Only choice B
is consistent with this schedule, so “B” is the correct choice.
3 Since only one audition can take place on Friday, it cannot be Carol’s (Cx);
and, of course, it cannot be Bob’s (B = Sat). Any of the other four actors
could audition on Friday as indicated in the following schedules:
E/F on Thu, D on Fri, B on Sat, C/G on Sun
C/F on Thu, E on Fri, B on Sat, D/G on Sun
C/E on Thu, E on Fri, B on Sat, D/G on Sun
E/F on Thu, G on Fri, B on Sat, C/D on Sun
So the correct choice is D.
4 The only schedule that fulfils the conditions is “F on Thu, E/G on Fri, B on Sat,
and C/D on Sun”. Only choice E is consistent with this schedule.
5 Since Ed and Bob’s auditions are both taking place on Saturday, eliminate
choices A and C. Since Carole must audition on the same day as another
actor, eliminate B. Finally, since Dave’s audition must take place on Sunday
(E < D), Frank’s audition must take place on Thursday and Grace’s audition on
Friday (F < G). Eliminate choice D. The complete schedule is: “C/F on Thu, G
on Fri, B/F on Sat, and D on Sun.”
Solutions 6-10:
Let A, B, C, D, H, L represents professor names.
C=February, A<H, B<<D and D<<B
6 Only choice E is there, which does not violate any of the conditions, so is the
correct choice.
7 With C serving in February and L in May, the only consecutive months
available for B and D are March and April. Then since A must serve is before H
in June. There are two possible schedules, depending on the order of B and D,
so the correct choice is B.
8 If L serves in April, the consecutive months available for B and D are May and
June; so choices C and D could be true. Since A must serve before H, choices
A and B must be true, only choice E cannot be true.
9 Since A must serve before H does, A cannot serve in June. Can A serve in
April? No, because then, D would serve in June (B<<D or D<<B), and again A
would not precede H. The latest that A could serve in March, which could
occur in the final order: L, C, A, D, B and H.
10 The only professors that can serve in January are A and L, so, one of them
must serve in January, and neither serves in February. So choice A cannot be
true.
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