I have a couple questions that I know the solutions to, but I can't for the life of me figure out the process involved at arriving at those solutions...

1) If 18*sqrt(18) = r*sqrt(t), where t and r are positive integers and r > t, which of the following could be the value of r*t? (solution is 108)

2) The eggs in a certain basket are either white or brown. If the ration of the number of white egges to the number of brown eggs is (2/3), each of the following could be the number of eggs in the basket EXCEPT:

a) 10
b) 12 (this is the answer)
c) 15
d) 30
e) 60

I thought about proportionalities with problem 1 and got no where, and with problem 2 I thought I had it nailed until I checked the answer.

As always, any advice is greatly appreciated




REPLY (from Denevo - thanks again)

these are not university-level algebra questions.

for (1):

since everything in sight is positive we can square both sides without fear.

thus from:

18√18 = r√t, we have:

(324)(18) = r2t

183 = 5832 = r2t.

clearly t < 18, or else r2t > t3 > 5832. so t is some divisor of 18: 1,2,3,6,or 9.

if t = 1, r = √(5832), which is not an integer.
if t = 2, r = √(2916) = 54 <--this works ( (18)3/2 = (9)(18)2, which has square root 3*18 = 54).
if t = 3, r = √(1944), not an integer
if t = 6, r = √(972), not an integer
if t = 9, r = √(648), not an integer

(look at the prime factorization of 18 cubed)

so the only case where r and t are integers with r > t is t = 2, r = 54, hence rt = 108.

(2) 2x + 3x = 5x, the number of eggs in the basket must be a multiple of 5. 12 is not a multiple of 5.



MY FOLLOW UP QUESTION

This is for question 1. (I feel a bit sheepish for posting question 2 once I realized it )

From Denevo's reply:

"since everything in sight is positive we can square both sides without fear.

thus from:

√18 = r√t, we have:

(324)(18) = r2t

183 = 5832 = r2t.

clearly t < 18, or else r2t > t3 > 5832. so t is some divisor of 18: 1,2,3,6,or 9."

I think where I'm getting hung up in the explanation is the "clearly t < 18, or else r2t > t3 > 5832. so t is some divisor of 18: 1,2,3,6,or 9."



I understand that if t > r that the result would be > 5832. The part I'm having trouble with is how to indentify that t is in fact less than 18 specifically. How do I know it's 18 and not some other integer? In other words, by saying this, it seems like saying that means you understand that r would be = 18, thus t < 18. From the problem, all I know is that r > t, and that the product of r*sqrt(18) is equivalent to the product of r*sqrt(t), and that r doesn't equal 18 nor does t equal sqrt(18). It seems like this is also confirmed by the solution that r = 54 and t = 2.

I certainly hope the follow question I'm asking here is making sense. Also, is there a link/website or some section of math one would practice in a book/class for a problem of this nature? It's important to me to understand the "why" of it instead of memorizing "how" to do a particular problem type.

Thanks.