Anyone else invited to the GRE General Test research project?

[Cesare]

"As a registered examinee for the GRE General Test, we are pleased to offer you web-based practice for the quantitative section of the GRE General Test.  This practice is offered as part of a research project that is conducted by Educational Testing Service of Princeton, NJ."

I took the first practice test and got 3 wrong. You can go back and examine the questions, but no explanations are provided. It is probably a bit unethical for me to post the questions, but I also think admissions based off of a standardized test is unethical..., so if anyone can enlighten me on how these can be solved I would be very grateful. 

9) The rectangular solid above is made up of eight cubes of the same size, each of which has exactly one face painted blue. What is the greatest fraction of the total surface area of the solid that could be blue?

11) If r and s are positive integers, each greater than 1, and if , what is the least possible value of  ?

 

17) A distillate flows into an empty 64-gallon drum at spout A and out of the drum at spout B. If the rate of flow through A is 2 gallons per hour, how many gallons per hour must flow out at spout B so that the drum is full in exactly 96 hours?

 

[Sol_Barber]

Hey - I try the first one - this comes to my mind:

 

The rectangular would look like two rows of 4 cubes or one line of 8 cubes, right?

 

OOOO                               or                   OOOOOOOO

OOOO

 

So there are 8 cube sides on the front and 8 on the back; there are 4 cube sides along the top and the bottom and two cube sides on each short side. That makes 28 cube sides. These cube surfaces add up to the surface area of the rectangular - 8 of these 28 could be blue - so max fraction 8/28 - 2/7. 2 parts of 7 of the surface area could be blue max

 

Or there are 8+8+8+8+1+1 cube sides which add up to 34 cube sides which make up the surface area and so to 8/34 or 4/17.

 

Or am I following some crooked logic?

[Cesare]

Hey - I try the first one - this comes to my mind:

 

The rectangular would look like two rows of 4 cubes or one line of 8 cubes, right?

 

OOOO                               or                   OOOOOOOO

OOOO

 

So there are 8 cube sides on the front and 8 on the back; there are 4 cube sides along the top and the bottom and two cube sides on each short side. That makes 28 cube sides. These cube surfaces add up to the surface area of the rectangular - 8 of these 28 could be blue - so max fraction 8/28 - 2/7. 2 parts of 7 of the surface area could be blue max

 

 

According to ETS, the right answer is 2/7. Thank you so much for this. 

[Sol_Barber]

I guess the second one is 14(s)+12® is 26. Because they are both primes and all other solutions would be fractions and no integers, or?

[Cesare]

I guess the second one is 14(s)+12® is 26. Because they are both primes and all other solutions would be fractions and no integers, or?

 

That is correct as well. Thanks! 

[Sol_Barber]

And the third one: since the drum would be filled in 32h without a leak and it should take 96h - three times as long - it would take that long if only one third would go in, or if 2/3 would leak out - so 2/3 of 2 should leak out - 1 and 1/3.

But just an idea

[Cesare]

And the third one: since the drum would be filled in 32h without a leak and it should take 96h - three times as long - it would take that long if only one third would go in, or if 2/3 would leak out - so 2/3 of 2 should leak out - 1 and 1/3.

But just an idea

Well, you are very impressive. Many thanks. 

[Sol_Barber]

Your welcome and best of luck with the GRE.