AMC 10 · 2025 · #4
Grade 8 number-theoryThe value of the two-digit number a b in base seven equals the value of the two-digit number b a in base nine. What is a+b?
Pick an answer.
AMC 10 2025 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: A two-digit numeral $\underline{a}\,\underline{b}$ read in base seven has the same value as the two-digit numeral $\underline{b}\,\underline{a}$ read in base nine. Find the sum $a+b$ of its two digits.
Givens: $\underline{a}\,\underline{b}$ is a two-digit number in base seven; $\underline{b}\,\underline{a}$ is the same two digits reversed, read in base nine; The base-seven value equals the base-nine value; Answer choices: (A) $7$, (B) $9$, (C) $10$, (D) $11$, (E) $14$
Unknowns: The two digits $a$ and $b$, and their sum $a+b$
Understand
Restated: A two-digit numeral $\underline{a}\,\underline{b}$ read in base seven has the same value as the two-digit numeral $\underline{b}\,\underline{a}$ read in base nine. Find the sum $a+b$ of its two digits.
Givens: $\underline{a}\,\underline{b}$ is a two-digit number in base seven; $\underline{b}\,\underline{a}$ is the same two digits reversed, read in base nine; The base-seven value equals the base-nine value; Answer choices: (A) $7$, (B) $9$, (C) $10$, (D) $11$, (E) $14$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #3 Eliminate Possibilities, #6 Guess and Check
The digits $a$ and $b$ are the unknowns, so Tool #4 (Introduce a Variable) names them and turns the place-value meaning of each numeral into an equation. Solving that equation only pins down the ratio of the digits, so Tool #6 (Guess and Check) tests the small digit pairs that fit, and Tool #3 (Eliminate Possibilities) uses the base-seven digit limit ($a,b\le 6$) to throw out the larger pair that also fits the ratio but is an illegal digit.
Execute — Answer: A
6.EE.B.6 Step 1 Write each numeral by place value
- In a two-digit numeral the left digit sits in the 'base' place and the right digit in the 'ones' place.
- So in base seven the left digit is worth seven each: $\underline{a}\,\underline{b}$ means $7a+b$.
- In base nine the left digit is worth nine each: the reversed numeral $\underline{b}\,\underline{a}$ means $9b+a$.
💡 A two-digit number is just the base times the first digit plus the second digit.
8.EE.C.7 Step 2 Set the values equal and simplify
- The two values are equal, so $7a+b=9b+a$.
- Collect the $a$ terms on one side and the $b$ terms on the other: subtract $a$ and $b$ from both sides to get $6a=8b$.
- Divide both sides by $2$ to reach $3a=4b$.
💡 Move like terms across the equals sign to shrink the relationship down to its simplest form.
6.RP.A.1 Step 3 Find the digits and add
- The equation $3a=4b$ says the digits are in the ratio $a:b=4:3$.
- So $a$ must be a multiple of $4$ and $b$ the matching multiple of $3$: the pairs are $(4,3)$, then $(8,6)$, and so on.
- But base-seven digits can be at most $6$, so $(8,6)$ is illegal and is thrown out — that trap would give the false sum $14$ in choice (E).
- The only legal pair is $a=4,\;b=3$.
- Check it: $7(4)+3=31$ and $9(3)+4=31$, equal as required.
- Their sum is $a+b=4+3=7$, which is choice (A).
💡 A ratio fixes the digits up to scaling, and the digit limit keeps only the smallest valid scale.
6.EE.B.6 In a two-digit numeral the left digit sits in the 'base' place and the right dig 8.EE.C.7 The two values are equal, so $7a+b=9b+a$. Collect the $a$ terms on one side and 6.RP.A.1 The equation $3a=4b$ says the digits are in the ratio $a:b=4:3$. So $a$ must be Review
Reasonableness: Both numerals evaluate to $31$, so the setup is consistent: $7\cdot4+3=31$ and $9\cdot3+4=31$. The digits $4$ and $3$ are each below $7$, so they are legal in base seven (and base nine), and the leading digit $4$ is nonzero, so $\underline{a}\,\underline{b}$ really is a two-digit number. The sum $4+3=7$ matches (A). The larger ratio pair $(8,6)$ would give $8+6=14$, exactly the trap in (E), but $8$ is not a base-seven digit, so it is correctly rejected.
Alternative: Skip the ratio and guess-and-check straight from $6a=8b$: since the smallest whole-number pair making $6a=8b$ true is $a=4,\,b=3$ (as $6\cdot4=8\cdot3=24$), and any larger pair exceeds the digit limit, you land on $(4,3)$ and $a+b=7$ without writing the ratio explicitly.
CCSS standards used (min grade 8)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Naming the digits $a,b$ and writing each numeral by place value as $7a+b$ and $9b+a$.)8.EE.C.7Solve linear equations in one variable (Rearranging $7a+b=9b+a$ by collecting like terms across the equals sign down to $3a=4b$.)6.RP.A.1Understand the concept of a ratio and use ratio language (Reading $3a=4b$ as the ratio $a:b=4:3$ to list the candidate digit pairs before applying the digit limit.)
⭐ A two-digit number in any base is just the base times the first digit plus the second digit, so writing that out turns a base puzzle into a simple equation.
⭐ A two-digit number in any base is just the base times the first digit plus the second digit, so writing that out turns a base puzzle into a simple equation.
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