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x is the average of three different integers, y is the average of minimum and maximum integers among the three. If x is two more than y, what is the difference between x and the third integer, which is neither minimum nor maximum?
- a)0
- b)1
- c)2
- d)4
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
x is the average of three different integers, y is the average of mini...
Assume a, b and c are the numbers. a > b > c
a + b + c = 3x
a + c = 3x – b
2y = 3 x – b
2(x – 2) = 3x – b
2x – 4 = 3x – b
x – b = - 4
So, the difference between x and b is 4.
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Community Answer
x is the average of three different integers, y is the average of mini...
To solve this problem, let's assume the three integers are a, b, and c, where a < b="" />< />
Given that x is the average of the three integers, we can write the equation:
x = (a + b + c) / 3
And y is the average of the minimum and maximum integers, so we have:
y = (a + c) / 2
It is also given that x is two more than y, so we can write another equation:
x = y + 2
We can now solve these equations to find the values of x, y, and the three integers.
Solving the equation x = (a + b + c) / 3 for a + b + c, we have:
a + b + c = 3x
Substituting this value in the equation y = (a + c) / 2, we get:
y = (3x - c) / 2
Substituting the value of y in the equation x = y + 2, we can solve for x:
x = (3x - c) / 2 + 2
2x = 3x - c + 4
x - 3x = 4 - c
-2x = 4 - c
2x = c - 4
x = (c - 4) / 2
Now, substituting this value of x in the equation y = (3x - c) / 2, we have:
y = (3(c - 4) / 2 - c) / 2
y = (3c - 12 - 2c) / 4
y = (c - 12) / 4
Since y is the average of the minimum and maximum integers, we can write the equation:
y = (a + c) / 2
Substituting the value of y, we get:
(c - 12) / 4 = (a + c) / 2
2(c - 12) = 4(a + c)
2c - 24 = 4a + 4c
-2c + 24 = 4a
a = (24 - 2c) / 4
a = (12 - c) / 2
Since a < b="" />< c,="" we="" can="" conclude="" that="" a="(12" -="" c)="" 2,="" b="(c" -="" 4)="" 2,="" and="" c="" is="" the="" maximum="" />
Now, we need to find the difference between x and the third integer, which is neither the minimum nor the maximum.
The third integer can be represented as (a + b + c) - (a + c) = b, since a and c cancel out.
So, the difference between x and the third integer is:
x - b = [(c - 4) / 2 - (12 - c) / 2]
Simplifying this expression, we get:
x - b = (c - 4 - 12 + c) / 2
x - b = (2c - 16) / 2
x - b = (c - 8)
Therefore, the difference between x and the third integer is (c - 8), which is equal to 4. Hence, the correct answer is option D)
Given that x is the average of the three integers, we can write the equation:
x = (a + b + c) / 3
And y is the average of the minimum and maximum integers, so we have:
y = (a + c) / 2
It is also given that x is two more than y, so we can write another equation:
x = y + 2
We can now solve these equations to find the values of x, y, and the three integers.
Solving the equation x = (a + b + c) / 3 for a + b + c, we have:
a + b + c = 3x
Substituting this value in the equation y = (a + c) / 2, we get:
y = (3x - c) / 2
Substituting the value of y in the equation x = y + 2, we can solve for x:
x = (3x - c) / 2 + 2
2x = 3x - c + 4
x - 3x = 4 - c
-2x = 4 - c
2x = c - 4
x = (c - 4) / 2
Now, substituting this value of x in the equation y = (3x - c) / 2, we have:
y = (3(c - 4) / 2 - c) / 2
y = (3c - 12 - 2c) / 4
y = (c - 12) / 4
Since y is the average of the minimum and maximum integers, we can write the equation:
y = (a + c) / 2
Substituting the value of y, we get:
(c - 12) / 4 = (a + c) / 2
2(c - 12) = 4(a + c)
2c - 24 = 4a + 4c
-2c + 24 = 4a
a = (24 - 2c) / 4
a = (12 - c) / 2
Since a < b="" />< c,="" we="" can="" conclude="" that="" a="(12" -="" c)="" 2,="" b="(c" -="" 4)="" 2,="" and="" c="" is="" the="" maximum="" />
Now, we need to find the difference between x and the third integer, which is neither the minimum nor the maximum.
The third integer can be represented as (a + b + c) - (a + c) = b, since a and c cancel out.
So, the difference between x and the third integer is:
x - b = [(c - 4) / 2 - (12 - c) / 2]
Simplifying this expression, we get:
x - b = (c - 4 - 12 + c) / 2
x - b = (2c - 16) / 2
x - b = (c - 8)
Therefore, the difference between x and the third integer is (c - 8), which is equal to 4. Hence, the correct answer is option D)
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