The rational number that lies exactly halfway between a/b and c/d on a number line is x = (ad + bc)/2bd or 1/2(a/b + c/d).
To calculate the mid-value of two numbers x and y, find the average of these two numbers. I.e., (x + y)/2
Thus, the average of these two numbers is the mid-value between them.
It is given that,
a/b and c/d are the two rational numbers on a number line.
Consider the required rational number that lies exactly halfway between a/b and c/d as 'x'.
Then, the mid-value of these two rational numbers is
(a/b + c/d)/2
⇒ (ad + bc)/2bd ...(i)
To know that the obtained rational number is exactly halfway between a/b and c/d, consider the distance from a/b to x and c/d to x.
So, the distance from a/b to x is - "x - a/b" (x > a/b) and
the distance from x to c/d is - "c/d - x"
From the given condition, the above-obtained distances are equal
⇒ x - a/b = c/d - x
⇒ x + x = a/b + c/d
⇒ 2x = (ad + bc)/bd
⇒ x = (ad + bc)/2bd ...(ii)
From (i) and (ii), it is concluded that the required rational number is
x = (ad + bc)/2bd
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Question: CHALLENGE: Let a/b and c/d be two distinct rational numbers. Find the rational number that lies exactly halfway between a/b and c/d on a number line.
