I don't recall the problem exactly, but my question is
If * is defined as a*b = a+b-ab
is a*(b*c) = (a*b)*c ?
Can I use the associative property here? I am a bit confused because of the '-' sign in the operation.
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- RonPurewal
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Re: can I use the associative property here
first off, i've taken the liberty of moving this problem to the general math folder, where it belongs. (the gmatprep math folder is for ACTUAL GMATPREP MATH QUESTIONS only; even if this is a piece of such a problem, it doesn't belong here out of context.)
heh, i wouldn't apply the associative property by default if i were you; the entire crux of the problem is to figure out whether it applies.
the best way is just to grind out the algebra:
left hand side = a * (b + c - bc)
= (a) + (b + c - bc) - a(b + c - bc)
= a + b + c - ab - ac - bc + abc (notice the SYMMETRY in this expression - you can switch any or all of a, b, c to your heart's content and the expression doesn't change a bit. in fact, that alone, along with the observation that the original expression is commutative, is enough to prove that the associative property is going to work, although that's difficult to see without lots of practice doing these sorts of things.)
right hand side = (a + b - ab) * c
= (a + b - ab) + c - (a + b - ab)c
= a + b + c - ab - ac - bc + abc
they're the same, so, yeah, this strange operation turns out to be associative. still, i would NOT take such things for granted.
that didn't take that long. still, if your algebra skills (or the degree of thoroughness with which you apply those skills) are dicey, you could always plug in 2-3 different sets of numbers for a, b, c, notice that you get the same numbers from the left- and right-hand expressions every time, and conclude that they must be the same. after 2-3 sets of randomly chosen numbers, such conclusions are usually on pretty solid ground.
cutlass Wrote:I don't recall the problem exactly, but my question is
If * is defined as a*b = a+b-ab
is a*(b*c) = (a*b)*c ?
Can I use the associative property here? I am a bit confused because of the '-' sign in the operation.
heh, i wouldn't apply the associative property by default if i were you; the entire crux of the problem is to figure out whether it applies.
the best way is just to grind out the algebra:
left hand side = a * (b + c - bc)
= (a) + (b + c - bc) - a(b + c - bc)
= a + b + c - ab - ac - bc + abc (notice the SYMMETRY in this expression - you can switch any or all of a, b, c to your heart's content and the expression doesn't change a bit. in fact, that alone, along with the observation that the original expression is commutative, is enough to prove that the associative property is going to work, although that's difficult to see without lots of practice doing these sorts of things.)
right hand side = (a + b - ab) * c
= (a + b - ab) + c - (a + b - ab)c
= a + b + c - ab - ac - bc + abc
they're the same, so, yeah, this strange operation turns out to be associative. still, i would NOT take such things for granted.
that didn't take that long. still, if your algebra skills (or the degree of thoroughness with which you apply those skills) are dicey, you could always plug in 2-3 different sets of numbers for a, b, c, notice that you get the same numbers from the left- and right-hand expressions every time, and conclude that they must be the same. after 2-3 sets of randomly chosen numbers, such conclusions are usually on pretty solid ground.
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