Quant Problem

[sukh]

Each card in a deck has an integer written on it, and the integers
on each of the 12 cards in the deck are consecutive. In a certain
game, the number points awarded for each turn is determined by
drawing two cards and multiplying the numbers shown on the
cards. If the points awarded in three turns are 40, 72, and 60,
all of the following could be the smallest numbered card in the
deck EXCEPT:
(A) -1
(B) 0
(C) 4
(D) 5
(E) 6

Answer E

Eight machines, each working at the same constant rate,
together can complete a certain job in 9 days. How many
additional machines, each working at the same constant rate,
will be needed to complete the job in 6 days?
(A) 1
(B) 3
(C) 4
(D) 6
(E) 12

Answer C


A certain academic department consists of 3 senior professors
and 6 junior professors. How many di¤erent committees of 3
professors can be formed in which at least one member of the
committee is a senior professor? (Two groups are considered
di¤erent if at least one group member is di¤erent.)
(A) 168
(B) 127
(C) 66
(D) 64
(E) 36

[cans]

Each card in a deck has an integer written on it, and the integers
on each of the 12 cards in the deck are consecutive. In a certain
game, the number points awarded for each turn is determined by
drawing two cards and multiplying the numbers shown on the
cards. If the points awarded in three turns are 40, 72, and 60,
all of the following could be the smallest numbered card in the
deck EXCEPT:
(A) -1
(B) 0
(C) 4
(D) 5
(E) 6

40 = 1*40, 2*20, 4*10, 5*8,
only 4*10 or 5*8 combinations are possible.
so minimum can't be 6. (we require atleast 4 or 5)
IMO E

[sukh]

solve the other 2 plz

[cans]

Eight machines, each working at the same constant rate,
together can complete a certain job in 9 days. How many
additional machines, each working at the same constant rate,
will be needed to complete the job in 6 days?
(A) 1
(B) 3
(C) 4
(D) 6
(E) 12

8 machines 9 days. 1 machine 72 days.
to complete work in 6 days, 12 machines
IMO E

[Geva@EconomistGMAT]

A certain academic department consists of 3 senior professors
and 6 junior professors. How many di¤erent committees of 3
professors can be formed in which at least one member of the
committee is a senior professor? (Two groups are considered
di¤erent if at least one group member is di¤erent.)
(A) 168
(B) 127
(C) 66
(D) 64
(E) 36

At least questions call for total number of options - forbidden options.

total number of options are the total number of ways of choosing 3 out of 9 people, regardless of senior/junior limitations. In other words: 9C3 = 9!/6!3! = 9*8*7/3! = 3*4*7 = 84

Forbidden options: at least one senior could mean
one senior (and 2 junior), OR
two seniors (and one junior), OR
all three Seniors.

The only forbidden scenario is therefore zero Seniors (and 3 Juniors). Calculate this scenario, and subtract from total number of options: the number of ways to choose 3 juniors out of only the 6 juniors is 6C3 = 6!/3!3! = 6*5*4/3! = 5*4=20.

So the final answer is total number of options - forbidden options = 84-20 = 64 - answer is D.

[sukh]

2nd one answer is 4 they said we have to subtract 9 from 12 .
3rd , the answer is correct but why those options are forbidden
(Two groups are considered different if at least one group member is different )
Why are we calculating for 3 seniors and not 2 and 1 seniors

[Geva@EconomistGMAT]

2nd one answer is 4 they said we have to subtract 9 from 12 .
3rd , the answer is correct but why those options are forbidden
(Two groups are considered different if at least one group member is different )
Why are we calculating for 3 seniors and not 2 and 1 seniors

Let's try this presentation.

At least one senior means on 3 commitee members allows:

1S, 2J
OR

2S, 1J

OR

3S, 0J

You can calculate these 3 scenarios separately, but you can also focus on the only forbidden scenario:

0S, 3J.

0 seniors is forbidden because we want at least one senior. Take that out of the total, and you have the equivalent of finding the 3 "good" scenarios.

[Geva@EconomistGMAT]

Eight machines, each working at the same constant rate,
together can complete a certain job in 9 days. How many
additional machines, each working at the same constant rate,
will be needed to complete the job in 6 days?
(A) 1
(B) 3
(C) 4
(D) 6
(E) 12

Answer C

In order to avoid confusion here, plug in for the work.
Let say that the work is 72 "tables" to be assembled.
8 machines take 9 days to make 72 tables, so in 1 day, these 8 machines do 72/9=8 "tables" --> each machine works at a rate of 1 "table" an hour.

We want to be able to do the job in 6 days. To make 72 tables in 6 days, we need to reach a combined rate of 72/6 =12 "tables" a day - so we need 12 machines. Since we have 8 machines already, we need 4 extra machines - which is what the question asked.