AMC 10 · 2009 · #16
Grade 7 arithmeticLet a, b, c, and d be real numbers with ∣a−b∣=2, ∣b−c∣=3, and ∣c−d∣=4. What is the sum of all possible values of ∣a−d∣?
Pick an answer.
AMC 10 2009 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: Four real numbers a, b, c, d sit on a number line. Neighbors are a fixed distance apart: a and b are 2 apart, b and c are 3 apart, c and d are 4 apart. The gap from a to d, measured as |a - d|, can land on several different numbers depending on which way each neighbor jumps. Add up every distinct value that |a - d| can take.
Givens: |a - b| = 2; |b - c| = 3; |c - d| = 4; a, b, c, d are real numbers
Unknowns: The sum of all distinct possible values of |a - d|
Understand
Restated: Four real numbers a, b, c, d sit on a number line. Neighbors are a fixed distance apart: a and b are 2 apart, b and c are 3 apart, c and d are 4 apart. The gap from a to d, measured as |a - d|, can land on several different numbers depending on which way each neighbor jumps. Add up every distinct value that |a - d| can take.
Givens: |a - b| = 2; |b - c| = 3; |c - d| = 4; a, b, c, d are real numbers
Plan
Primary tool: #2 Make a Systematic List
Secondary: #4 Introduce a Variable, #3 Eliminate Possibilities
The distance from a to d is built from three separate jumps, and each jump can point either forward or backward. That is a small, finite set of choices: two directions for each of three jumps, so eight cases in total. The clean move is to write a - d as one signed sum of the three jumps, then list all eight sign patterns without missing or repeating any. Once the list is complete, take absolute values, throw out duplicates, and add what remains.
Execute — Answer: D
6.EE.B.6 Step 1 Chain the gaps together
- The gap from a to d can be rebuilt out of the three neighbor gaps.
- Write a - d as (a - b) + (b - c) + (c - d); the middle terms b and c cancel in pairs, leaving exactly a - d.
- So instead of tracking a and d directly, track the three known differences and add them.
💡 Walking a to b to c to d and adding each step lands you exactly at the a-to-d gap.
6.NS.C.7 Step 2 Turn each absolute value into a sign choice
- An absolute value only fixes the size of a difference, not its direction.
- So |a - b| = 2 means a - b is either +2 or -2; likewise b - c is +3 or -3, and c - d is +4 or -4.
- Substituting into the chain, a - d equals plus-or-minus 2, plus-or-minus 3, plus-or-minus 4, with each sign chosen independently.
💡 Absolute value tells you how far, not which way, so each gap gets a free choice of sign.
7.NS.A.1 Step 3 List all eight sign patterns
- Three independent plus-or-minus choices make 2 x 2 x 2 = 8 cases.
- Go through them in order so none is missed.
- +2+3+4 = 9.
- +2+3-4 = 1.
- +2-3+4 = 3.
- +2-3-4 = -5.
- -2+3+4 = 5.
- -2+3-4 = -3.
- -2-3+4 = -1.
- -2-3-4 = -9.
- These eight results are the possible values of a - d before taking absolute value.
💡 Two directions for each of three jumps gives a tidy list of eight signed totals.
7.NS.A.3 Step 4 Take absolute values and drop duplicates
- Now apply |a - d| to each result: 9, 1, 3, 5, 5, 3, 1, 9.
- Every positive value appears twice, once from a sign pattern and once from its exact opposite, which is why the list pairs up.
- The distinct sizes are 1, 3, 5, and 9.
- Adding these distinct values gives 1 + 3 + 5 + 9 = 18.
- So the sum of all possible values of |a - d| is 18, which is answer (D).
💡 Flipping every sign flips the total's sign but not its size, so distinct sizes come in mirror pairs.
6.EE.B.6 The gap from a to d can be rebuilt out of the three neighbor gaps. Write a - d a 6.NS.C.7 An absolute value only fixes the size of a difference, not its direction. So |a 7.NS.A.1 Three independent plus-or-minus choices make 2 x 2 x 2 = 8 cases. Go through the 7.NS.A.3 Now apply |a - d| to each result: 9, 1, 3, 5, 5, 3, 1, 9. Every positive value a Review
Reasonableness: The largest |a - d| happens when all three jumps point the same way: 2 + 3 + 4 = 9, so no value can exceed 9, and the four distinct sizes 1, 3, 5, 9 all sit at or below it. They also all share the parity of 2 + 3 + 4 = 9 (odd), which is why only odd sizes appear and even choices like the answer 12 or 24 never show up as a single |a - d|. Summing the genuine distinct values 1, 3, 5, 9 gives 18, matching (D).
Alternative: Fix a - b = 2 and b - c = 3 without loss of generality since adding a constant to all four numbers or reflecting the line does not change any gap. Then a - c = 5, and c - d = plus-or-minus 4 gives a - d = 5 + 4 = 9 or a - d = 5 - 4 = 1. Repeating with b - c = -3 gives a - c = -1, so a - d = -1 plus-or-minus 4 = 3 or -5, i.e. sizes 3 and 5. The four sizes 9, 1, 3, 5 again sum to 18.
CCSS standards used (min grade 7)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Rewriting a - d as the telescoping sum (a - b) + (b - c) + (c - d) so the known gaps can be combined.)6.NS.C.7Understand ordering and absolute value of rational numbers (Reading each condition like |a - b| = 2 as the two signed possibilities a - b = +2 or -2.)7.NS.A.1Apply and extend understanding of addition and subtraction to rational numbers (Adding the signed jumps across all eight patterns of the form plus-or-minus 2 plus-or-minus 3 plus-or-minus 4.)7.NS.A.3Solve real-world problems involving the four operations with rational numbers (Taking absolute values, removing duplicate sizes, and summing the distinct values to 18.)
⭐ When distances chain together, each hop can go two ways, so list every sign pattern, take absolute values, and add only the different sizes.
⭐ When distances chain together, each hop can go two ways, so list every sign pattern, take absolute values, and add only the different sizes.
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