Three positive numbers x, y, and z have the following relationships y

Question

Three positive numbers x, y, and z have the following relationships y = x + 2 and z = y + 2. When the median of x, y, and z is subtracted from the product of the smallest number and the median, the result is 0. What is the value of the largest number?

(A) –2
(B) \(\pi\)
(C) 5
(D) 8
(E) 21/2

Source: Nova GMAT
Difficulty Level: 600

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Official Answer

C

(A) –2 
(B)  \pi 
(C) 5 
(D) 8 
(E) 21/2

Source:   Nova GMAT  
Difficulty Level:   600

amanvermagmat · Answer

y is 2 more than x and z is 2 more than y. So the three numbers are:
x, x+2, x+4

Their median = x+2
Product of smallest number and median = x(x+2)

Given: x(x+2) - (x+2) = 0
Taking (x+2) common, (x+2) (x-1) = 0
So either x=-2 or x=1

But we are given that these are positive numbers. So x=1
Largest number = x+4 = 1+4 =5

Hence  C

quantumliner · Answer

Given:

y = x+2
z = y+2 = x+4

So there are three positive numbers: x, y and z or x, x+2 and x+4
Clearly the median of the three numbers is x+2

Now

x(x+2) - (x+2) = 0
(x+2)(x-1)=0
x=-2 or 2. As x is a positive integer, -2 is ruled out.

Largest number is x+4 = 5

Answer is (C) 5

Luckisnoexcuse · Answer

Answer replacement Technique
Median-Median*x=0
Hence x should be 1 (All x,y,z are positive)

Z=5 -> Y=3, X=1
C is the answer

generis · Answer

Method 1

y = x + 2, and 
z = y + 2, so we have

An arithmetic sequence x, y, z with a common difference of two. Median is y

Product of median and smallest is xy, then subtract y

So xy - y = 0, and

y (x - 1) = 0

y = 0, or (x - 1) =0

The numbers are positive, so y can't equal 0.

(x - 1) = 0
 x = 1

Count up by twos. y = 3,  z = 5

Method 2  - backsolve

(A) -2. Prompt says numbers are positive. Reject.

(B)  \pi  - is ridiculous. Don't strike it yet, but move on.

(C) 5. If that's z, then y is 3 and x is 1. (xy) = 3. Subtract y, middle number. 3 - 3 = 0. Correct.

Now strike B, which, come to think of it, given Nova's reputation, is also hilarious.

Having just done the math for C, answers D and E are too big. (Think 8, 6, 4, where  xy  is 24. If 8 is too big, 21/2 or 10.5 is worse.)

Answer C

gracie · Answer

if y=x+2, then y>x
if z=y+2, then z>y
hence, z>y>x
if xy-y=0, then x=1
if x=1, then y=3
if y=3, then z=5
5
C

EMPOWERgmatRichC · Answer

Hi rohan2345,

This question can be solved by solved in a variety of ways, including Algebraically, by TESTing THE ANSWERS or by spotting a hidden Number Property so that you can avoid most of the math altogether...

We're told that X, Y and Z are all POSITIVE numbers and that Y = X+2 and Z = Y+2. This means that the numbers are 'evenly spaced terms that differ by 2' (such as 1, 3, 5 or 2, 4, 6, etc.) and that X<Y<Z. Next, we're told that when the MEDIAN of the group is subtracted from the product of the smallest number and the median, the result is 0. This can be translated into the following equation:

(X)(MEDIAN) - (MEDIAN) = 0.

Take a good look at that equation. We already know that both X and the median are POSITIVE numbers. What MUST X equal? If you don't 'see' it yet, try manipulating the equation a bit...

(X)(MEDIAN) - (MEDIAN) = 0
(X)(MEDIAN) = (MEDIAN) 
X = 1

Since X is 1, we can quickly figure out that Y=3 and Z=5.

Final Answer:  C

GMAT assassins aren't born, they're made,
Rich

bumpbot · Answer

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Three positive numbers x, y, and z have the following relationships y

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