The total volume of a wooden wine container and a plastic wine contain

This question is a part of Are You Up For the Challenge: 700 Level Questions collection.

Concept: the arithmetic mean of 2 values is simply the midpoint of the 2 values on the number line

Let S = volume of steel
Let P = volume of plastic
Let W = volume of wooden

Given: W + P > 2S

Is: W > S ?

Volumes are positive values — so we have a positive value constraint on all variables

Given can be written as:

S < (W + P) / 2

Which means that S is LESS THAN the MIDPOINT of W and P on the number line. That means we have two possible cases:

Case 1: W < P

[ ————- W —————(midpoint)—————P]

S must fall to the LEFT of the midpoint on the number line.
We can not be sure whether W or S is the greater value (ie, S could call between W and the midpoint OR S could fall to the left of W

CASE 2: P < W

[————-P —————-(midpoint) —————-W]

S must fall to the LEFT of the midpoint on the number line.
In this case, we are SURE that the value of S will be less than the value of W since

S < (w + p)/2 < W

Rephrased question: Is P < W?

S1: “volume of plastic is less than volume of steel”

P < S

It can NOT be the case in which W is also less than S because if we had either:

W < P < S
Or
P < W < S

S would fall to the RIGHT of each value on the number line, meaning that S would be GREATER THAN the Midpoint of W, in violation of the given condition

The only way the given condition can be satisfied and P < S is in case 2 above:

P < S < (midpoint of P and W) < W

So YES: P < W
S1 sufficient

S2: “volume of plastic is smaller than volume of wooden”

P < W

Directly answers the rephrased question

*D*
Each statement is sufficient



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