AMC 10 · 2015 · #5
Easy mode Grade 2Twelve people ran a race. Six of them are David, Hikmet, Jack, Marta, Rand, and Todd. Here is how they placed:
- 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 came in 6th.
Who came in 8th?
Pick an answer.
AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Six named runners — David, Hikmet, Jack, Marta, Rand, Todd — placed somewhere in a 12-person race. Each clue ties two runners' finishing places together, and Marta's place is given outright. Find who took 8th place.
Givens: 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 6th place.
Unknowns: The runner who finished in 8th place.
Understand
Restated: Six named runners — David, Hikmet, Jack, Marta, Rand, Todd — placed somewhere in a 12-person race. Each clue ties two runners' finishing places together, and Marta's place is given outright. Find who took 8th place.
Givens: 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 6th place.
Plan
Primary tool: #11 Work Backwards
Secondary: #4 Introduce a Variable
Only one place is known directly — Marta's 6th. So start from that fixed fact and use the clues that connect each runner to an already-known place, one link at a time, until 8th place is filled.
Execute — Answer: B
1.OA.A.1 Step 1 Anchor on Marta
- Marta finished in 6th place, and she finished 1 place behind Jack.
- Behind means a bigger number, so Jack's place is 1 less than Marta's.
💡 If you are right behind someone, they are one spot in front of you.
1.OA.A.1 Step 2 Find Todd from Jack
Jack finished 2 places behind Todd, so Todd is 2 places ahead of Jack: Todd's place is 2 less than Jack's.
💡 Two places behind someone puts them two spots in front.
1.OA.A.1 Step 3 Find Rand from Todd
Todd finished 1 place behind Rand, so Rand is 1 place ahead of Todd.
💡 Back up one place from Todd to land on Rand.
2.OA.A.1 Step 4 Find Hikmet from Rand
Rand finished 6 places ahead of Hikmet, so Hikmet is 6 places behind Rand: add 6 to Rand's place.
💡 Six places back from 2nd lands squarely on 8th.
2.OA.A.1 Step 5 Read off 8th place
- Hikmet's place came out to 8.
- (David would be 8 + 2 = 10th, so he is not in 8th.) The runner in 8th place is Hikmet, choice (B).
💡 The chain of clues drops you exactly on the place the question asked for.
1.OA.A.1 Marta finished in 6th place, and she finished 1 place behind Jack. Behind means 1.OA.A.1 Jack finished 2 places behind Todd, so Todd is 2 places ahead of Jack: Todd's pl 1.OA.A.1 Todd finished 1 place behind Rand, so Rand is 1 place ahead of Todd. 2.OA.A.1 Rand finished 6 places ahead of Hikmet, so Hikmet is 6 places behind Rand: add 6 2.OA.A.1 Hikmet's place came out to 8. (David would be 8 + 2 = 10th, so he is not in 8th. Review
Reasonableness: Lay the solved places in order: Rand 2nd, Todd 3rd, Jack 5th, Marta 6th, Hikmet 8th, David 10th. Every original clue checks out — Rand (2) is 6 ahead of Hikmet (8), Marta (6) is 1 behind Jack (5), David (10) is 2 behind Hikmet (8), Jack (5) is 2 behind Todd (3), Todd (3) is 1 behind Rand (2), and Marta is 6th. All six places are distinct and within 1–12, so Hikmet in 8th is consistent.
Alternative: Instead of chaining backwards, introduce a variable for Rand's place r and write every runner in terms of it: Todd = r+1, Jack = r+3, Marta = r+4, Hikmet = r+6, David = r+8. Setting Marta = 6 gives r = 2, and Hikmet = r+6 = 8 — the same answer.
CCSS standards used (min grade 2)
1.OA.A.1Solve addition and subtraction word problems within 20 (Turning each "ahead/behind" clue into a single addition or subtraction on a place number.)2.OA.A.1Solve one- and two-step word problems using addition and subtraction within 100 (Chaining the linked clues together to carry the known place from Marta all the way to 8th.)
⭐ Start from the one place you know for sure, then hop along the clues one runner at a time.
⭐ Start from the one place you know for sure, then hop along the clues one runner at a time.
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