AMC 8 · 2013 · #13

Grade 5 number-theoryarithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

digit-decompositionplace-valuemultiples convert-to-algebrasystematic-enumeration ↑ Prerequisites: place-valuemulti-digit-arithmetic

📏 Medium solution 💡 3 insights

📘 View easy version →

Problem

Clara added up several test scores but, on one score, she accidentally swapped the tens digit and the units digit (for example, she wrote 72 as 27). Because of that single swap, her total came out different from the true total. Which of the five answer choices could that difference be?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

Instead of jumping into algebra, try small concrete cases first (Tool #9). Reverse the digits of a few two-digit numbers — 72 → 27, 63 → 36, 41 → 14 — and write down each difference. Tool #5 (Look for a Pattern) spots that every difference is a multiple of 9. Tool #3 (Eliminate Possibilities) then sweeps the five answer choices and keeps only the multiple of 9. This path uses Grade 4 multiples / place-value reasoning instead of Grade 6 algebra.

1STEP 1

Work a few cases by hand: 72 - 27 = 45, 63 - 36 = 27, 41 - 14 = 27.

72 - 27 = 45, 63 - 36 = 27, 41 - 14 = 27

2STEP 2

Every difference — 45, 27, 27 — is a multiple of 9; one more case, 85 - 58 = 27, confirms it.

45 = 9 × 5, 27 = 9 × 3, 85 - 58 = 27 = 9 × 3

3STEP 3

Place value proves it: (10t + u) - (10u + t) = 9(t - u), always a multiple of 9.

(10t + u) - (10u + t) = 9t - 9u = 9(t - u)

4STEP 4

Test the five choices against the 9-rule; only 45 is a multiple of 9 (4+5=9).

45 = 9 × 5 ✓; 46, 47, 48, 49 not divisible by 9

5STEP 5

Check it's reachable: 9|t - u| = 45 needs |t - u| = 5, e.g. 72 - 27 = 45.

9 × |t - u| = 45 → |t - u| = 5; 72 - 27 = 45 → (A)

Answer

Every test case we tried — 72 → 27, 63 → 36, 41 → 14, 85 → 58 — gave a difference that is a multiple of 9. The algebraic shortcut 9(t-u) explains why this must be true. Among the five choices, only 45 is a multiple of 9, and we exhibited a real swap (72 ⇔ 27) that produces exactly 45. The reasoning and the example both point to (A).

💡Key takeaway

Swapping the two digits of a number always changes it by a multiple of 9 — a Grade 5 place-value fact, not an AMC mystery!

More like this