Doubt on geometry statements

vinni.k wrote:(1). A rectangular plot is half as long again as it is broad

I guess there won't be any 'again'.
Otherwise, this means length of the plot is half of its breadth, i.e. length = (breadth)/2

vinni.k wrote:(2).A rectangular garden is to be twice as long as it is wide.

This means length of the garden is twice its width, i.e. length = 2*(width)

vinni.k wrote:'again' is in the sentence. A bit confusing and shouldn't be in the sentence. Moreover the explanation that i have says L = x + (1/2)x = (3/2)x. I think it is wrong. What are your thoughts ?

I've never seen such phrase which is used to mean that in any maths problem.
A proper GMAT problem would have said "the length of the plot is 50% more than the width" or something like that.

"Half as long again" or "half again as long" is a colloquialism (mostly British, I believe, though I'm not entirely sure. It certainly sounds very old-fashioned to my American ears) that means "50% longer than." I have never seen this expression used on the GMAT, and I am nearly certain that you never would. Idiomatic language is only tested on SC, not on quant, and even then this type of idiom wouldn't be tested.

Also, I've never seen a GMAT problem that said "broad" rather than "wide." "Broad" wouldn't necessarily be incorrect, but it's just less commonly used in American English. (Also, the GMAT would probably never use "hectares"). My recommendation would be to stop studying from this source, as it clearly is not well steeped in GMAT language.

To your question about the solution... if length is 50% greater than the width (breadth), then:
L = 1.5W --> L = (3/2)W --> W = (2/3)L
L*W = 2/3
L*(2/3)L = 2/3
L^2 = 1
L = 1

So, that wasn't incorrect.