Of the total length of a pencil, one eighth is black, one half of the remaining length is white, and the remaining 3.5 cm is blue. What is the total length of the pencil?

Introduction / Context:
This question is a classic example of fraction and percentage style reasoning in quantitative aptitude. The pencil is divided into three coloured segments: black, white and blue. The fractions for the first two parts are given in terms of the total or remaining length, while the last part is given as an absolute length in centimetres. The goal is to translate these verbal conditions into equations and solve for the total length of the pencil.

Given Data / Assumptions:

Let the total length of the pencil be L cm.
Black portion = 1 / 8 of the total length L.
White portion = 1 / 2 of the remaining length after the black part is removed.
Blue portion equals the remaining length after black and white parts and is given to be 3.5 cm.
All parts together add up to the total length L.

Concept / Approach:
The idea is to apply fractional operations step by step. First we remove 1 / 8 of the total for the black part. Then from what is left, we remove half as the white part. Whatever still remains is the blue segment, whose length is given numerically. By expressing the remaining fractions of L after each stage, we can equate the last part to 3.5 cm and solve for L using basic algebra with fractions.

Step-by-Step Solution:
Let total length = L cm.Black part = L / 8.Length remaining after black part = L - L / 8 = (7L) / 8.White part = 1 / 2 of the remaining length = (1 / 2) * (7L / 8) = 7L / 16.Remaining length after black and white parts = (7L / 8) - (7L / 16) = 7L / 16.This remaining length is the blue part, which equals 3.5 cm.Therefore 7L / 16 = 3.5.So L = 3.5 * 16 / 7 = 8 cm.

Verification / Alternative check:
Check the result by reconstructing the pencil lengths. For L = 8 cm: black part = 8 / 8 = 1 cm, remaining = 7 cm. White part = half of 7 cm = 3.5 cm, remaining = 3.5 cm. This final remaining part is the blue section, which matches the given 3.5 cm. The sum of all parts is 1 + 3.5 + 3.5 = 8 cm, which is consistent with the assumed total length.

Why Other Options Are Wrong:

7 cm does not satisfy the condition; plugging L = 7 gives a different blue length than 3.5 cm.
9 cm produces fractional parts that do not match the given final blue portion of 3.5 cm.
11 cm likewise leads to a final remaining segment that is not equal to 3.5 cm.
10 cm also fails when the step by step fraction calculations are applied, so it cannot be the correct total length.

Common Pitfalls:
Learners often misinterpret phrases like one half of the remaining as one half of the total, which completely changes the fractions. Another common error is to forget that the blue part corresponds to the remaining length only after both black and white parts are removed. Some students also confuse 3.5 with 3 / 5 or interpret 3.5 as 3 / 2.2, which is incorrect. Careful reading and clear algebraic representation prevent these mistakes.

Final Answer:
After correctly applying the fractional conditions to the coloured parts of the pencil, the total length of the pencil is 8 cm.