AMC 10 · 2004 · #15
Grade 6 arithmeticPick an answer.
The two ages are locked together by their digits, so Tool #4 (Introduce a Variable) is the natural start: call the tens digit of one age a and the ones digit b, and the reversal writes itself as 10a+b and 10b+a. The 'twice as old in five years' sentence then becomes a single equation. That equation, 8a=19b+5, has two digit-unknowns but only one equation — so instead of grinding algebra, Tool #6 (Guess and Check) walks b through 0,1,2,… and Tool #3 (Eliminate Possibilities) throws out every value that fails to leave a a single digit. Only one pair survives.
Name the two digits
Naming the two digits writes both ages with one pair of unknowns.
A two-digit number is just its tens digit times ten plus its ones digit, so swapping the digits swaps those two roles.
6.EE.B.6Introduce A VariableTurn the words into an equation
The five-years-later sentence becomes a single equation.
'Twice as old' is a multiply-by-two link between the two future ages, which is exactly what an equation records.
6.EE.B.7Introduce A VariableSimplify to one clean relation
Simplifying ties the two digits into one clean relation.
Collecting like terms boils the whole story down to one tidy link between a and b.
6.EE.B.7Introduce A VariableTest digit values for b
Only single digits are allowed, and just one pair fits: 31 and 13.
Since digits only run 0 through 9, a short march through the choices quickly leaves just one that keeps both digits legal.
4.OA.A.3Guess And CheckTake the difference
The difference is 18, matching nine times the digit gap, choice (B).
Reversing the digits trades tens for ones, so the gap is always nine times the digit difference.
Reversing the digits trades tens for ones, so the gap is always nine times the digit difference.
▸ Why?
A two-digit number is its tens digit weighted by ten plus its ones digit, so swapping them swaps those weights.
▸ Why?
Both numbers carry the same two digits, so everything shared cancels in the subtraction and only the weighting is left.
Write the two ages as 10a+b and 10b+a, turn 'twice as old in five years' into one equation, and testing single digits leaves only 31 and 13 — a difference of 18.
- Name the two digits
- Turn the words into an equation
- Simplify to one clean relation
- Test digit values for b
- Take the difference