- If x < y < z but x^2 > y^2 > z^2 > 0, which of the following must be positive?
(A) (x^3) * (y^4) * (z^5)
(B) (x^3) * (y^5) * (z^4)
(C) (x^4) * (y^3) * (z^5)
(D) (x^4) * (y^5) * (z^3)
(E) (x^5) * (y^4) * (z^3)
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Manhattan Challenge Problem
Source: Beat The GMAT — Problem Solving |
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GMATGuruNY
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Plug in combinations that satisfy the conditions in the question stem yet at the same time prove that the answers DON'T have to be positive.shankar.ashwin wrote:
- If x < y < z but x^2 > y^2 > z^2 > 0, which of the following must be positive?
(A) (x^3) * (y^4) * (z^5)
(B) (x^3) * (y^5) * (z^4)
(C) (x^4) * (y^3) * (z^5)
(D) (x^4) * (y^5) * (z^3)
(E) (x^5) * (y^4) * (z^3)
Let x=-3, y=-2, z=1.
-3 < -2 < 1, satisfying the condition that x<y<z.
(-3)² > (-2)² > 1², satisfying the condition that x² > y² > z² > 0.
When we plug into the answers, our only concern is the sign of the resulting product.
(A) (x^3) * (y^4) * (z^5) = (negative)(positive)(positive) = negative. Eliminate A.
(B) (x^3) * (y^5) * (z^4) = (negative)(negative)(positive) = positive. Hold onto B.
(C) (x^4) * (y^3) * (z^5) = (positive)(negative)(positive) = negative. Eliminate C.
(D) (x^4) * (y^5) * (z^3) = (positive)(negative)(positive) = negative. Eliminate D.
(E) (x^5) * (y^4) * (z^3) = (negative)(positive)(positive) = negative. Eliminate E.
The correct answer is B.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
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neelgandham
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If x < y < z
Possibilities
x = -ve, y = -ve z = -ve - (x=-5, y=-4, z=-3,- Satisfies the condition x^2 > y^2 > z^2 (25>16>9))
x = +ve, y = +ve z = +ve - (x=3, y=4, z=5,- Doesn't satisfy the equation x^2 > y^2 > z^2(9<16<25))
x = -ve, y = -ve z = +ve - (x=-5, y=-4, z=3 - Satisfies the condition x^2 > y^2 > z^2 (25>16>9))
x = -ve, y = +ve z = +ve - (x=-A, y=4, z=5 - Doesn't satisfy the equation y^2 > z^2(16<25))
Here we have two possibilities, let us test them against the options
which of the following must be positive?
(B) (x^3) * (y^5) * (z^4) - +ve
(C) (x^4) * (y^3) * (z^5) - +ve
(D) (x^4) * (y^5) * (z^3) - +ve
(E) (x^5) * (y^4) * (z^3) - +ve
(B) (x^3) * (y^5) * (z^4) - +ve
(C) (x^4) * (y^3) * (z^5) - -ve
(D) (x^4) * (y^5) * (z^3) - -ve
(E) (x^5) * (y^4) * (z^3) - -ve
Since the question reads, MUST be positive, the answer is B
Possibilities
x = -ve, y = -ve z = -ve - (x=-5, y=-4, z=-3,- Satisfies the condition x^2 > y^2 > z^2 (25>16>9))
x = +ve, y = +ve z = +ve - (x=3, y=4, z=5,- Doesn't satisfy the equation x^2 > y^2 > z^2(9<16<25))
x = -ve, y = -ve z = +ve - (x=-5, y=-4, z=3 - Satisfies the condition x^2 > y^2 > z^2 (25>16>9))
x = -ve, y = +ve z = +ve - (x=-A, y=4, z=5 - Doesn't satisfy the equation y^2 > z^2(16<25))
Here we have two possibilities, let us test them against the options
which of the following must be positive?
(A) (x^3) * (y^4) * (z^5) - +veIf x = -ve, y = -ve z = -ve
(B) (x^3) * (y^5) * (z^4) - +ve
(C) (x^4) * (y^3) * (z^5) - +ve
(D) (x^4) * (y^5) * (z^3) - +ve
(E) (x^5) * (y^4) * (z^3) - +ve
(A) (x^3) * (y^4) * (z^5) - -veIf x = -ve, y = -ve z = +ve
(B) (x^3) * (y^5) * (z^4) - +ve
(C) (x^4) * (y^3) * (z^5) - -ve
(D) (x^4) * (y^5) * (z^3) - -ve
(E) (x^5) * (y^4) * (z^3) - -ve
Since the question reads, MUST be positive, the answer is B
Anil Gandham
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"
I am sorry for such an inappropriate way of solving this question, but it is good for a lazy one (like me hehe)
I just take a quick look to the answer choices,and see that 4 out of 5 choices are identical.I mean:
(A) (x^3) * (y^4) * (z^5) and
(E) (x^5) * (y^4) * (z^3)
(C) (x^4) * (y^3) * (z^5) and
(D) (x^4) * (y^5) * (z^3)
are identical (in A and E x and z have odd exponents, and Ys have the same even exponent; in C and D Xes have the same even exponent 4, and y and z have odd exponents)
so, eliminating these choices we get that only (B) (x^3) * (y^5) * (z^4) is the answer.
I just take a quick look to the answer choices,and see that 4 out of 5 choices are identical.I mean:
(A) (x^3) * (y^4) * (z^5) and
(E) (x^5) * (y^4) * (z^3)
(C) (x^4) * (y^3) * (z^5) and
(D) (x^4) * (y^5) * (z^3)
are identical (in A and E x and z have odd exponents, and Ys have the same even exponent; in C and D Xes have the same even exponent 4, and y and z have odd exponents)
so, eliminating these choices we get that only (B) (x^3) * (y^5) * (z^4) is the answer.
0 ---- z^2 ---- y^2 ---- x^2 ----
we know that prime roots are +ve but need to find non-prime roots too (i.e. negative roots)
let us simplify, 2 ways are possible: either x>0 or x<0
a) x>0, all numbers are +ve ignore as we have more than one ans choice correct
b) x<0
~ y can be -ve or +ve
~ z can be -ve or +ve
Consider x=-3 always and y=(-2,2), z=(-1,1)??? and x<y<z Correctly noting z cannot be -ve/+ve, we state z is +ve only and if x=-ve (some -3), y=-ve (some -2) or +ve (some 2)??? but x<y<z, z=+ve (some 1), then y can be only -ve
now testing ans.choices: x=-ve,y=-ve,z=+ve
(A) (x^3) * (y^4) * (z^5) -> -*+*+=-
(B) (x^3) * (y^5) * (z^4) -> -*-*+=+ good choice
(C) (x^4) * (y^3) * (z^5) -> +*-*+=-
(D) (x^4) * (y^5) * (z^3) -> +*-*+=-
(E) (x^5) * (y^4) * (z^3) -> -*+*+=-
b
we know that prime roots are +ve but need to find non-prime roots too (i.e. negative roots)
let us simplify, 2 ways are possible: either x>0 or x<0
a) x>0, all numbers are +ve ignore as we have more than one ans choice correct
b) x<0
~ y can be -ve or +ve
~ z can be -ve or +ve
Consider x=-3 always and y=(-2,2), z=(-1,1)??? and x<y<z Correctly noting z cannot be -ve/+ve, we state z is +ve only and if x=-ve (some -3), y=-ve (some -2) or +ve (some 2)??? but x<y<z, z=+ve (some 1), then y can be only -ve
now testing ans.choices: x=-ve,y=-ve,z=+ve
(A) (x^3) * (y^4) * (z^5) -> -*+*+=-
(B) (x^3) * (y^5) * (z^4) -> -*-*+=+ good choice
(C) (x^4) * (y^3) * (z^5) -> +*-*+=-
(D) (x^4) * (y^5) * (z^3) -> +*-*+=-
(E) (x^5) * (y^4) * (z^3) -> -*+*+=-
b
shankar.ashwin wrote:
- If x < y < z but x^2 > y^2 > z^2 > 0, which of the following must be positive?
(A) (x^3) * (y^4) * (z^5)
(B) (x^3) * (y^5) * (z^4)
(C) (x^4) * (y^3) * (z^5)
(D) (x^4) * (y^5) * (z^3)
(E) (x^5) * (y^4) * (z^3)
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