Fractional composition of a pencil: 1/8 of a pencil is black; half of the remaining part is white. The rest is blue, and this blue section measures 3 1/2 cm. What is the full length of the pencil?

Introduction / Context:
This is a fraction-of-a-whole problem where the whole is the length of a pencil. Parts of the pencil are colored differently by fractional proportions. By expressing each colored segment as a fraction of the total and using the given absolute length for one segment, we can determine the entire length.

Given Data / Assumptions:

• Black part = 1/8 of the pencil.
• Remaining after black = 1 − 1/8 = 7/8.
• White part = 1/2 of the remaining = 1/2 × 7/8 = 7/16.
• Blue part = Remaining after black and white = 7/8 − 7/16 = 7/16.
• Blue length = 3 1/2 cm = 3.5 cm.

Concept / Approach:
Let total length be L. Since blue part equals 7/16 of L and we know its numerical length, set (7/16)L = 3.5 and solve for L. This ratio-to-absolute conversion is standard in fraction word problems.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 6 cm, 7 cm, 9 cm: When multiplied by 7/16 do not yield 3.5 cm, violating the given blue segment length.

Common Pitfalls:

• Treating “half of the remaining” as “half of the total.”
• Misreading 3 1/2 cm as 31/2 cm instead of 3.5 cm.

Final Answer: