Out of five distinct integral numbers A, B, C, D and E, C is the average of A and D. The number B is greater than C but less than D, and B is also the average of A and E. Which letter represents the middle value when the five numbers are arranged in increasing order?

Introduction / Context:
This is a logical reasoning and algebra question involving five integral numbers A, B, C, D and E. Several relationships are given: C is the average of A and D, B lies strictly between C and D, and B is also the average of A and E. We are asked to identify which letter corresponds to the middle value when these five numbers are arranged in ascending order.

Given Data / Assumptions:

• C is the average of A and D: C = (A + D) / 2.
• B is greater than C and less than D: C < B < D.
• B is also the average of A and E: B = (A + E) / 2.
• All five numbers are integral and distinct.
• We must find which one is the middle number when the sequence is sorted.

Concept / Approach:
From C = (A + D) / 2, C is the midpoint between A and D on the number line, so A < C < D. Also, C < B < D places B to the right of C but still left of D. From B = (A + E) / 2, B is the midpoint between A and E, so A < B < E. Putting these inequalities together helps us order A, C, B, D and E and identify the central value.

Step-by-Step Solution:
Step 1: From C = (A + D) / 2, we know C lies exactly between A and D. Thus, A < C < D (since A and D are distinct and C is their average). Step 2: From C < B < D, B is strictly between C and D, so we have A < C < B < D. Step 3: From B = (A + E) / 2, B lies between A and E. Thus, A < B < E. Step 4: Combine the ordering information: We already have A < C < B < D. Since A < B < E and B is greater than C, it forces E to be to the right of B, and usually to the right of or equal to D for many integer configurations. Therefore, a consistent order is often A, C, B, D, E (or A, C, B, E, D), but in all valid arrangements, B is in the central third position. Step 5: Hence, B is the middle number when all five are sorted in increasing order.
Verification / Alternative check:
Take an example: Let A = 0, D = 10. Then C = (0 + 10) / 2 = 5. Let B = 7 (so that 5 < 7 < 10). From B = (A + E) / 2 = (0 + E) / 2, we get E = 14. Now the five numbers are 0 (A), 5 (C), 7 (B), 10 (D), 14 (E). In sorted order: 0, 5, 7, 10, 14 – the middle one is 7, which corresponds to B.
Why Other Options Are Wrong:
A is the smallest in most configurations since both C and B are averages involving A and larger numbers. C, being the average of A and D, must lie strictly between them but is always to the left of B because B is given to be greater than C. D and E are on the higher side and cannot consistently occupy the middle position once B is placed between C and D and also between A and E.
Common Pitfalls:
Some students may assume that C is the middle number merely because it is an average, ignoring the extra information about B and E. Others may not test with a concrete numerical example, which often makes the ordering much clearer.
Final Answer:
The middle value in the ordered sequence is represented by B.