Question

If a, b, c are three distinct natural numbers, all less than 100, such that \( |a - b| + |b - c| = |c - a| \), then the maximum possible value of b is ________

Hint:

The expression \(|x-y| + |y-z| = |x-z|\) is a standard identity which holds if and only if y lies between x and z (inclusive). This is a useful property to remember.

Answer 1

Correct Answer: 98

Step 1: Understanding the Concept:
The given equation is a property related to the triangle inequality. The triangle inequality states that for any two numbers x and y, \(|x| + |y| \ge |x+y|\). The equality holds if and only if x and y have the same sign (or one of them is zero).
Step 2: Key Formula or Approach:
1. Let \(x = a-b\) and \(y = b-c\). Then \(x+y = (a-b)+(b-c) = a-c\). 2. The given equation is \(|x| + |y| = |-(a-c)| = |a-c| = |x+y|\). 3. This implies that x and y must have the same sign. 4. Analyze the two cases: both positive or both negative. 5. Use the constraints on a, b, and c to maximize b.
Step 3: Detailed Explanation:
The condition \(|a-b| + |b-c| = |a-c|\) implies that b must lie between a and c. Case 1: \(a-b>0\) and \(b-c>0\) This means \(a>b\) and \(b>c\), which can be written as \(a>b>c\).
Case 2: \(a-b<0\) and \(b-c<0\) This means \(a<b\) and \(b<c\), which can be written as \(a<b<c\).
In both cases, b is the middle value of the three distinct natural numbers.
We are given that a, b, and c are natural numbers less than 100. This means \(1 \le a, b, c \le 99\).
We want to find the maximum possible value of b.
To maximize b, we should use the ordering \(a<b<c\). Since c is a natural number and \(c \le 99\), and b must be strictly less than c, the maximum possible value for c is 99. If we set \(c = 99\), then the largest possible integer value for b such that \(b<c\) is \(b = 98\).
We also need to find a value for 'a' such that \(a<b\). We can choose any natural number for 'a' from 1 to 97. For instance, we can choose \(a=97\). So, the set \(a=97, b=98, c=99\) satisfies all conditions:

They are distinct natural numbers less than 100.
The condition is satisfied: \(|97-98| + |98-99| = |-1| + |-1| = 1+1=2\). And \(|99-97| = 2\). So \(2=2\).
Thus, the maximum possible value for b is 98.
Step 4: Final Answer:
The maximum possible value of b is 98.