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2015 AMC 10B Problem 5

2015 AMC 10B Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Logic Puzzles · difficulty 1 of 5 · about 2 min · position P1-10

David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$ -person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$ th place. Who finished in $8$ th place?

A)
$\text{David}$
B)
$\text{Hikmet}$
C)
$\text{Jack}$
D)
$\text{Rand}$
E)
$\text{Todd}$

Answer

Key insight

Marta's 6th place anchors a chain: Jack 5th, Todd 3rd, Rand 2nd, then Hikmet is 6 behind Rand in 8th.

Solution

Only one clue gives an absolute position, so start there and follow the chain of relative clues. "Ahead" means a smaller place number, "behind" a larger one.

Marta is $6$ th.
Marta is $1$ behind Jack, so Jack is $5$ th.
Jack is $2$ behind Todd, so Todd is $3$ rd.
Todd is $1$ behind Rand, so Rand is $2$ nd.
Rand is $6$ ahead of Hikmet, so Hikmet is $2 + 6 = 8$ th.
David is $2$ behind Hikmet, so David is $10$ th.

The $8$ th-place finisher is Hikmet: $\boxed{\textbf{(B)}\ \text{Hikmet}}$ .

Why this works

Relative clues form a chain; one absolute fact pins the whole chain down. Find the anchor, then translate each "ahead by $k$ " into subtract $k$ and each "behind by $k$ " into add $k$ . The six unnamed runners just fill the gaps and can be ignored.

The trap

Reversing 'ahead' and 'behind' (ahead means a smaller place number), which sends the chain the wrong way.

Common mistakes

Reversing 'ahead' and 'behind' (ahead means a smaller place number), which sends the chain the wrong way.
Stopping at David ( $10$ th) or Rand and misreading which person the question asks about.

Techniques

Set up the equation/formula and compute; no special trick needed