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Three positive numbers x, y, and z have the following relationships y = x + 2 and z = y + 2. When the median of x, y, and z is subtracted from the product of the smallest number and the median, the result is 0. What is the value of the largest number?
- a)–2
- b)π
- c)5
- d)8
- e)21/2
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Three positive numbers x, y, and z have the following relationships y ...
Given: y = x + 2 z = y + 2
We need to find the value of the largest number, which is represented by z.
To find the median of x, y, and z, we need to arrange the numbers in ascending order:
The smallest number is x. The middle number is y. The largest number is z.
From the given relationships, we can substitute the values of y and z:
y = x + 2 z = (x + 2) + 2 = x + 4
The median is the middle number, y.
The product of the smallest number and the median is x * y.
The difference between the median and the product is y - (x * y).
According to the problem, this difference is 0:
y - (x * y) = 0
Substituting the value of y:
(x + 2) - (x * (x + 2)) = 0
Expanding and simplifying:
x + 2 - (x2 + 2x) = 0 x + 2 - x2 - 2x = 0 -x2 - x + 2 = 0
To solve this quadratic equation, we can factor it:
-(x - 1)(x + 2) = 0
Setting each factor equal to zero:
x - 1 = 0 or x + 2 = 0
Solving for x:
x = 1 or x = -2
Since the problem states that x, y, and z are positive numbers, we discard the negative solution x = -2.
Therefore, the value of x is 1.
Using the relationship y = x + 2:
y = 1 + 2 y = 3
Using the relationship z = y + 2:
z = 3 + 2 z = 5
The largest number is z, which is equal to 5.
Therefore, the answer is (C) 5.
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