Two numbers are deemed similar if the positive difference be

Question

Two numbers are deemed similar if the positive difference between the two is no more than 20 percent of either number. If x, y, and z are integers such that x and y are similar, y and z are similar, and x+y+z=455, what is the smallest possible value of x?

A. 91
B. 105
C. 120
D. 125
E. 134

A. 91
B. 105
C. 120
D. 125
E. 134

ankushsambare · Answer

Assume the numbers are ordered as:
x ≤ y ≤ z
Two numbers are “similar” if their positive difference is at most 20% of either number.
For x and y:
y − x ≤ 20% of x
y − x ≤ 0.2x
y ≤ 1.2x
Similarly, for y and z:
z − y ≤ 0.2y
z ≤ 1.2y
To make x as small as possible, make y and z as large as allowed.
So take:
y = 1.2x
z = 1.2y = 1.44x
Now use the sum condition:
x + y + z = 455
Substitute:
x + 1.2x + 1.44x = 455
3.64x = 455
x = 455 / 3.64 = 125
Check:
x = 125
y = 150
z = 180
Differences:
150 − 125 = 25, which is 20% of 125
180 − 150 = 30, which is 20% of 150
All conditions are satisfied.
 Answer: 125 (Choice D)

officiisnobis · Answer

x+1.2x+1.44x=455
3.64x = 455
x = 125

MarkGMATMentor · Answer

To find the smallest or greatest value of a variable (i.e., a min / max problem), you generally must find the smallest or greatest values of other variables.

Here, to make x as small as possible, we want y and z to take up as much of the 455 as possible; that is, we want y and z to be the greatest values possible. To do that, we want the differences between the three integers to be as large as possible. So  we want the differences to be 20% in each case.

But the prompt tells us that the positive differences can be no more than 20% of  either  number. 20% of the larger number will be more than 20% of the smaller number, so  we   must use 20% of the smaller number.

Since x and y are similar, we can say that y is 20% greater than x, or  y = 1.2x = 6/5 x . (Fractions are generally easier than decimals to compute without a calculator.)

Likewise, since y and z are similar, we can say that z is 20% greater than y, or z = 1.2y = 6/5 y. Since y = 6/5 x, z = 6/5 (6/5 x), or  z = 36/25 x .

Now substitute for y and z in the given equation and solve:
 x + y + z = 455 (given) 
x + 6/5 x + 36/25 x = 455 --> substitute for y and z
25/25 x + 30/25 x + 36/25 x = 455 --> convert to the common denominator
91/25 x = 455 --> combine like terms
x = 455 * 25/91 --> isolate x by multiplying by 25/91
x = 5 * 25 --> cancel the common factor (91) before multiplying
 x = 125

The answer is D.

To minimize or maximize one value, you must usually minimize or maximize other values.  In this problem, you must maximize y and z in order to minimize x, and you do that by making the differences between the variables as large as possible.

Two numbers are deemed similar if the positive difference be

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Two numbers are deemed similar if the positive difference be

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