IPMAT Indore 2025 (SA) - 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 ______ | PYQs + Solutions

IPMAT Indore 2025

Algebra

Modulus

Medium

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 ______

Entered answer:

✅ Correct Answer: 98

To find the maximum value of

b

where

a

b

c

are distinct natural numbers less than 100 satisfying

|a - b| + |b - c| = |c - a|

First, let's analyze the equation

|a - b| + |b - c| = |c - a|

This equation is true if and only if

b

lies between

a

and

c

(either

a < b < c

c < b < a

To maximize

b

, we need to make it as close to 100 as possible while ensuring it satisfies our constraint.

Let's try

b = 98

with two possible arrangements:

Case 1: If

a < b < c

\newline

b = 98

\newline

c = 99

(largest possible value less than 100)

\newline

a = 1

(can be any number less than 98)

Verifying:

\newline

|a - b| + |b - c| = |1 - 98| + |98 - 99| = 97 + 1 = 98

\newline

|c - a| = |99 - 1| = 98

Case 2: If

c < b < a

\newline

b = 98

\newline

a = 99

(largest possible value less than 100)

\newline

c = 1

(can be any number less than 98)

Verifying:

\newline

|a - b| + |b - c| = |99 - 98| + |98 - 1| = 1 + 97 = 98

\newline

|c - a| = |1 - 99| = 98

Since both cases work with

b = 98

and we cannot choose a larger value for

b

(as we need distinct natural numbers less than 100), the maximum possible value of

b

is 98.