Math Problem 10

In 1980, the cost of p pounds of potatoes was d dollars. In 1990, the cost of 2p pounds of potatoes was dollars. By what percent did the price of potatoes decrease from 1980 to 1990?

(A) 25% (B) 50% (C) 75% (D) 100% (E) 400%

Solution for GRE Analytical Problem 2:

We’ll name countries for their first letters: B, C, D, E, F, G. Now we shall make some preliminary conclusion from the conditions.

1. At least one film will be shown each day.
2. No more than three films will be shown on any two consecutive days.

 

Having 6 films and 4 days, we’ve got a math problem here:
In how many ways can we get 6 out of 4 positive integers, with any two consecutive terms not bigger than 3?
If the first number is 2, then the second must be 1. The third and the forth numbers can be 1+2 or 2+1.
So, we have two ways for now: 2+1+1+2 and 2+1+2+1.
And if the first number is 1, the second number can’t 1 (otherwise the sum of the third and the fourths numbers would be 4). So, only one possibility is there: 1+2+1+2

Now we have three possible patterns for showing films during the festival:
1+2+1+2
2+1+1+2
2+1+2+1
They’ll be useful during our further solution

3. The Belgian film must be shown on Saturday.

 

It’s time to start filling the table:

Thu	Fri	Sat	Sun
		B

4. The Canadian film must be shown on the same day as another film.

C+X

5. The French film must be shown on a day before the German film is shown.

F – G

6. The Danish film must be shown on a day after the English film is shown.

E – D

IMPORTANT Note: phrase “on a day after” shows that any number of days can pass between English and Danish films. To indicate two consecutive days phrases like “on a day immediately after”, “on a next day” etc.

And one more to do in the preparatory phase is to make a restriction table:

Thu	Fri	Sat	Sun
G D			F E

Now we may proceed to answering questions.

Question I

B cant’ be shown on Thu, for it has already been scheduled to Sat. C must be accompanied by another film, so, it can’t be shown on Thu, either. D and G are in our restriction table for Thu. So, only F remaining.

Answer (D) France

Question II

If F is on Sat, the G is on Sun. The only pattern with 2 films on Sat is 2+1+2+1, so, C must be on Thu, so must E. And В will be shown on Fri

Thu	Fri	Sat	Sun
C E	D	B F	G

Now we can see, that (B) The film from Denmark will be shown on Friday is the only possible answer.

Answer (B)

Question III

With two films on Thu and two on Sun, we have 2+1+1+2 pattern. So, C can’t be shown on Fri. Neither does B, because it is shown on Sat. And any of other four films may be shown on Friday.
D:

Thu	Fri	Sat	Sun
E F	D	B	G C

E:

Thu	Fri	Sat	Sun
F C	E	B	D G

F:

Thu	Fri	Sat	Sun
E C	F	B	D G

G:

Thu	Fri	Sat	Sun
F E	G	B	D C

Answer (D) 4

Question IV

If the English and German films are shown on the same day, we have

F – E+G –D

Thus, E+G can’t be shown neither on Thu, nor on Sun. These films can’t be shown on Sat, because in that case we would have three films then. So, E+G must be on Fri and we have 1+2+1+2 pattern.

Thu	Fri	Sat	Sun
	E G	B

F must be on Thu, D and C – on Sun

Thu	Fri	Sat	Sun
F	E G	B	C D

That’s why answer (E) “he film from Canada will be shown on Sunday” is the only true possibility.

Answer (E)

Question V

If the film from England is shown on Saturday, we have 2+1+2+1 pattern.

Thu	Fri	Sat	Sun
		B E

Sunday slot must be taken by D.

Thu	Fri	Sat	Sun
		B E	D

C must be shown on Thursday:

Thu	Fri	Sat	Sun
C		B E	D

At last, F and G will take Thu and Fri slots.

Thu	Fri	Sat	Sun
C F	G	B E	D

Now we can see, that the film from Germany cannot be shown on the same day as any other film.

Answer (E) The film from Germany.

Analytical problem 2

The Fairfield Foreign Film Festival plans to show six films—one each from Belgium, Canada, Denmark, England, France, and Germany. The festival will open on a Thursday and close three days later on a Sunday. Each film will be shown once during the four days of the festival.

The order in which the films will be shown must adhere to the following conditions:

1. At least one film will be shown each day.
2. No more than three films will be shown on any two consecutive days.
3. The Belgian film must be shown on Saturday.
4. The Canadian film must be shown on the same day as another film.
5. The French film must be shown on a day before the German film is shown.
6. The Danish film must be shown on a day after the English film is shown.

Question I
If only one film is shown on Thursday, it could be the entry from which of the following countries?
(A) Belgium
(B) Canada
(C) Denmark
(D) France
(E) Germany

Question II
If the Belgian and French films are shown on the same day, which of the following must be true?
(A) The film from Denmark will be shown on
Thursday.
(B) The film from Denmark will be shown on
Friday.
(C) The film from Germany will be shown on
Thursday.
(D) The film from Canada will be shown on
Sunday.
(E) The film from England will be shown on
Sunday.

Question III
If the director of the festival decides to show two films on Thursday and two on Sunday, how many films would be eligible to be shown on Friday?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Question IV
If the English and German films are shown on the same day, which of the following must be true?
(A) The film from England will be shown on
Thursday.
(B) The film from France will be shown on Friday.
(C) The film from Canada will be shown on
Saturday.
(D) The film from Germany will be shown on
Saturday.
(E) The film from Canada will be shown on
Sunday.

Question V
If the film from England is shown on Saturday, which of the following films cannot be shown on the same day as any other film?
(A) The film from Belgium.
(B) The film from Canada.
(C) The film from England.
(D) The film from France.
(E) The film from Germany.

Solution for GRE Math Problem 9:

"ratio of the number of insects in a given population having characteristic X to the number of insects in the population not having characteristic X is 5:3" means that of population has characteristics X, and of the population has not characteristics X.

To find, what proportion of the total insect population are male insects having the characteristic X, we must multiply the fraction of X-insects in the whole population by the fraction of males in X-insects.

Answer: D

Analytical problem 1

Dr. William Weinberg is a psychiatrist who has to schedule one therapy session for each of six patients— Bernstein, Cox, Dawson, Edson, Friedman, and Grant. Each session will take place on the same day. The sessions will be scheduled at 9:00, 10:00, 11:00, 1:00, 2:00, and 3:00, with a lunch break at noon.

In making up the schedule for the session, Dr. Weinberg will conform to the following conditions:

• Friedman will have the first session in the afternoon.
• Dawson's session must be earlier in the day than Edson's.
• Bernstein's session must be earlier in the day than either Edson's or Friedman's.
• Cox's session must immediately precede Grant's session.

Question I. If Bernstein is only available at 10:00, in how many ways can the sessions be scheduled?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 6

Question II. If Friedman switches to a morning session, and Cox takes the 1:00 spot vacated by Friedman, which of the following must be true?

(A) Bernstein has the first session of the day.

(B) Bernstein or Friedman has the second session

of the day.

(C) Bernstein or Dawson has the second session of

the day.

(D) Dawson or Friedman has the third session of

the day.

(E) Dawson or Grant has the last session of the

day.

Question III. If Grant has the last session of the day, each of the following could be true EXCEPT

(A) Bernstein has the first session of the day.

(B) Dawson has the first session of the day.

(C) Dawson has his session at 10:00.

(D) Edson has her session at 10:00.

(E) Cox has his session at 2:00.

Question IV. If Cox's session is earlier in the day than Friedman's, which of the following must be true?

(A) Bernstein has the first session of the day.

(B) Cox has the first session of the day.

(C) Bernstein has a session at 11:00.

(D) Grant has a session at 11:00.

(E) Dawson has a session at 2:00.

Solution for GRE Math Problem 4:
Number 2n has all the prive factors, which number n has. Additionaly, as number n is odd, prime number 2 is not among its prime factors, but number 2n has 2 among it prime factors. So, the number of prime factors of 2n is bigger than the number of prime factors of n.

Answer: B