AMC 10 · 2004 · #17
Grade 6 arithmeticPick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
Let a be the tens digit and b the ones digit of Jack's age: Jack is 10a+b, Bill is 10b+a, and a is bigger than b since Jack is older.
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
In five years both are 5 older: Jack is 10a+b+5, Bill is 10b+a+5, and 'twice as old' sets Jack's future age equal to 2 times Bill's.
'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
Expand the right side to 20b+2a+10 and collect like terms: the whole condition collapses to 8a=19b+5.
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
Digits run 0 to 9, so test b: b=0 and b=2 fail, b=1 gives 8a=24 so a=3, and larger b pushes a past 9. Jack is 31, Bill is 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
Their gap is 31-13=18, matching the shortcut (10a+b)-(10b+a)=9(a-b)=9(3-1). The answer is (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 swaps those weights.
▸ Why?
Both numbers carry the same two digits, so everything shared cancels 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