If x is an integer between 0 and 10 is y less than the average of x an

Question

If x is an integer between 0 and 10 is y less than the average of x and 10 ?

(1)  10-y < y - \frac{x +10}{2}

(2) y is 4 times as large as x

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GMATinsight · Answer

Question: Is y < (x+10)/2? Statement 1 : 10-y < y - \frac{x +10}{2} i.e. 2y > x+30/2 i.e. y > \frac{x+30}{4} at x = 1, y > 7.75 i.e. y > (x+10)/2 at x = 9, y > 9.75 i.e. y > (x+10)/2 i.e. Always NO to question SUFFICIENT Statement 2 : y is 4 times as large as x i.e. y = 4x i.e. We need to compare 4x and (x+10)/2 at x = 1, y = 4 and (x+10)/2= 5.5 i.e .y < (x+10)/2 at x = 2, y = 8 and (x+10)/2= 6 i.e .y > (x+10)/2 NOT SUFFICIENT Answer: Option A

nick1816 · Answer

Statement 1-

10-y < y - \frac{x +10}{2}

2y > 10+ \frac{x +10}{2}

y> \frac{10}{2} + \frac{x +10}{4}

y> \frac{10}{2} + \frac{x}{4} + \frac{10}{4}

y>\frac{10}{2} +\frac{ x}{2} - \frac{x}{4} +\frac{10}{4}

y>\frac{x +10}{2} + \frac{10-x}{4}

Since  0≤x≤10 ,  \frac{10-x}{4}  is always greater than or equal to 0

y> +non-negative number

Sufficient

Statement 2-  y=4x

1.  x=1; y=4;   \frac{x+10}{2} = 5  (Yes)

2.  x=2; y=8;  \frac{x+10}{2} = 6  (No)

Insufficient

GMATWhizTeam · Answer

Solution Step 1: Analyse Question Stem • x is an integer. • 0 < x < 10 ⟹ 1 ≤ x ≤9 • We need to find if y is less than the average of x and 10 o Now, average of x and 10 = \frac{x+10}{2}  Therefore, \frac{1 + 10}{2 }≤ \frac{x + 10}{2} ≤ \frac{9+ 10}{2} ⟹ 5.5 ≤ \frac{x + 10}{2} ≤ 9.5 Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE Statement 1 : 10−y \frac{1}{2}*maximum\space of\space \frac{x + 10}{2} ……….(ii) • So, from (i) and (ii), we can write, o \frac{1}{2}*\frac{x+10}{2} + \frac{1}{2}*\frac{x+10}{2} < y ⟹ \frac{x+10}{2} < y Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E. Statement 2: y is 4 times as large as x • We know, 1 ≤ x ≤9 ⟹ 4 ≤ 4x= y ≤36 • Thus, when x = 1, y = 4 and \frac{x+10}{2} = 5.5 > 4 o Here y is greater than the average of x and 10 • However, when x = 9, y = 36 and \frac{x+10}{2} = 9.5 < 36 o Here y is less than the average of x and 10. • Results of the above two cases are contradictory. Hence, statement 2 is not sufficient. Thus, the correct answer is Option A.

sambitspm · Answer

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