2025 AMC 12B Problem 4

Below is the professionally curated solution for Problem 4 of the 2025 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.

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Concepts:number base linear equation

Difficulty rating: 1200

The value of the two-digit number $\underline{a}\,\underline{b}$ in base seven equals the value of the two-digit number $\underline{b}\,\underline{a}$ in base nine. What is $a + b?$

$7$

$9$

$10$

$11$

$14$

Solution:

Setting $7a + b = 9b + a$ gives $6a = 8b,$ so $3a = 4b.$ The digits are $a = 4, b = 3,$ which check out since $43_7 = 31 = 34_9.$ Hence $a + b = 7.$

Thus, the correct answer is A.

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