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.

See also these GRE problems: