x and y are integers from 0 to 170, both inclusive. For how many order

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Competition Mode Question

x and y are integers from 0 to 170, both inclusive. For how many ordered pairs (x,y) is the average (arithmetic mean) of x and y equal to 60 ?

A. 60 
B. 118
C. 119
D. 120
E. 121

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M37-98

Bunuel · Accepted Answer

Official Solution:

x  and  y  are integers from 0 to 170, both inclusive. For how many ordered pairs  (x,y)  is the average (arithmetic mean) of  x  and  y  equal to 60 ?

A.  60 
B.  118 
C.  119 
D.  120 
E.  121

The average (arithmetic mean) of  x  and  y  is 60 means that the sum of  x  and  y  is 120. 
 
 x+y=120 .
 
 x  and  y  are integers from 0 to 170, both inclusive, so  (x,y)  can be:
 
 (0,120) ;
 
 (1,119) ;
 
 (2,118) ;
 
 (3,117) ;
 
 ... 
 
 (120,0) .
 
So, there are total of 121 such pairs.

Answer: E

MARVEL13 · Answer

X can take values total 60 values ranging from 0 to 59 for which y can take values from 120 to 61 respectively. 
Now as ordered pairs is mentioned. We need to double the value. So 120.

D

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eakabuah · Answer

We are given that x and y are integers from 0 to 170 inclusive, and we are to determine the number of ordered pairs of x and y whereby the average of x and y equals 60.
(x+y)/2=60 hence x+y=120 and x=120-y
We have an arithmetic progression with a common difference of 1,
So, the ordered pairs of (x,y) are: (0,120), (1,119), ... ,(120,0)
Focusing on the x values, number of terms = number of ordered pairs = (120-0)+1 = 121.

The answer is E

Leadership · Answer

If the arithmetic mean of X and Y is 60, we can say that X + Y - 120
We can start from X=0 and Y=120 .. taking all the values by doing +1 in X and -1 in Y
so, X = 0 to 59 and Y = 120 to 61 .. in total we have 60 increasing pairs but question ask us to find total ordered pairs, hence 60 will be decreasing pairs.

120 is the answer.

CareerGeek · Answer

x and y are integers from 0 to 170, both inclusive. For how many ordered pairs (x,y) is the average (arithmetic mean) of x and y equal to 60 ?

A. 60
B. 118
C. 119
D. 120
E. 121

--> Average of x & y = 60
--> Sum of x & y = 120
Possible ordered pairs (x, y) = {(0, 120), (1, 119), (2, 118), . . . . . . . . . (120, 0)} = 121

Option E

exc4libur · Answer

x+y/2=60
x+y=120
x=0, y=120;
x=1, y=110;
…
x=120, y=0.

range 0 to 120 inclusive: 120-0+1=121

Ans (E)

Archit3110 · Answer

from given info we can say that
x+y= 120
so max value of x= 120 ,y=0 ; 119,1 ; 118,2 ... so on total 120 such pairs would be there , and unique pairs would be 60 
as 120,0 is same as 0,120
IMO A:

x and y are integers from 0 to 170, both inclusive. For how many ordered pairs (x,y) is the average (arithmetic mean) of x and y equal to 60 ?

A. 60
B. 118
C. 119
D. 120
E. 121

ArunSharma12 · Answer

(0,120)(1,119),(2,118)...(59,61),(60,60),(61,59)...(120,0).
total of 121 pairs.

gurmukh · Answer

What if three numbers are to be chosen x,y and z whose average is 60 from the given numbers?

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unraveled · Answer

x and y are integers from 0 to 170, both inclusive. For how many ordered pairs (x,y) is the average (arithmetic mean) of x and y equal to 60 ?

A. 60
B. 118
C. 119
D. 120
E. 121

\frac{x + y}{2} = 60  OR x + y = 120
n(x,y) = number of (x,y) pairs

Possibilities are: x and y 
0 + 120 ----> 1st
1 + 119
2 + 118
....
....
120 + 0 -----> Last i.e. 121st

Total n(x,y) = 121

Answer E.

lacktutor · Answer

x and y are integers from 0 to 170, both inclusive. For how many ordered pairs (x,y) is the average (arithmetic mean) of x and y equal to 60 ?

\frac{x+y}{2} = 60 
-->  x +y = 120

if x=0, then y should be 120 --> (0,120)
if x=1, then y should be 119--> (1,119)
if x=2, then y should be 118 --> (2,118)
....
if x=59, then y should be 61 --> (59,61)
if x=60, then y should be 60 --> (60,60)
if x=61, then y should be 59 --> (61,59)
.....
if x=119, then y should be 1 --> (119,1)
if x=120, then y should be 0 --> (120,0)
-------------------------------------------
As you see, 
--> in the first  60 cases , the value of x increases and the value of y decreases
--> in the last  60 cases , the value of y increases and the value of x decreases
Only in  one case,  the value of x is equal to y (60,60)
In total, there are  121 pairs (x,y ), in which the average of x and y is equal to 60.

Answer (E)

prathyushaR · Answer

arithmetic mean of x and y is (x+y)/2=60
thus we need pairs of (x,y) where x+y=120
as x increases from 0,1,2..so on..
y decreases from 120,119,118..so on to keep the sum at 120
and the ordered pair (0,120) is different to the pair(120,0)
thus the number of ordered pairs is same as no. of distinct values for x or y 
0 to 120 inclusive thus we get a total of 121 ordered pairs.
answer : E

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x and y are integers from 0 to 170, both inclusive. For how many order

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