AMC 10 · 2014 · #6
Grade 7 number-theoryalgebraPick an answer.
The words "reversing its digits" only become usable once the digits are separate objects, so Tool #4 names them a and b and Tool #13 turns place value into algebra: N = 10a+b and M = 10b+a. Every phrase in the problem then becomes an expression — the difference is 9(a-b), the digit sum is a+b, and the quantity asked for is 11(a+b). That reduces the whole puzzle to one linear equation. One equation with two unknowns is normally hopeless, so Tool #3 supplies the missing pressure: a and b are digits, so only whole numbers from a short list are allowed, and the equation 2a = 7b leaves exactly one of them standing. Tool #6 then does the part that is easy to skip — plugging the candidate back into the original sentence to confirm such a number really exists, rather than only that it would have to look this way. Tool #16 covers the wording gap by re-running the argument with the roles of the two numbers swapped.
Name the digits
Two digits describe both numbers.
Once the digits have names, "reverse the digits" is no longer an action — it is just a second expression built from the same two letters.
6.EE.B.6Introduce A VariableTranslate the sentence
The sentence becomes one clean equation.
Reversing moves each digit between the tens place and the ones place, a swing of 10 - 1 = 9 per unit of digit difference.
Reversing moves each digit between the tens place and the ones place, a swing of nine per unit of digit difference.
▸ Why?
A two-digit number is its tens digit weighted by ten plus its ones digit, so swapping swaps those weights.
▸ Why?
Both numbers carry the same two digits, so everything shared cancels and only the weighting is left.
Reduce to 2a = 7b
It reduces to a small ratio.
One equation cannot fix two unknowns, but it can lock their ratio — and a ratio plus "they must be digits" is enough.
7.EE.B.4Convert To AlgebraForce the digits
Divisibility forces both digits.
A 7 on one side of an equation has to be matched by a 7 on the other side, and 2 cannot supply it.
6.NS.B.4Eliminate PossibilitiesCheck that it really works
Substituting back confirms it.
Deriving what a solution must look like and showing a solution exists are two different jobs — the second one takes a real number, not an equation.
4.NBT.B.4Guess And CheckHandle the other orientation
The other orientation gives the same pair.
A number and its reverse form one unordered pair, so which one you subtract from which cannot change their sum.
7.EE.B.4Change Focus Count The ComplementAnswer the question asked
Their total is 99, choice (D).
Adding a number to its reverse gives each digit one turn in the tens place and one in the ones place, which is 11 copies of the digit sum.
1.NBT.B.2Convert To AlgebraReversing a two-digit number only trades the tens and the ones, so the gap is always 9 times the digit difference and the total is always 11 times the digit sum — write those two facts down and the puzzle turns into one small equation.
- Name the digits
- Translate the sentence
- Reduce to 2a = 7b
- Force the digits
- Check that it really works
- Handle the other orientation
- Answer the question asked