Competition · AMC preparation · step 4 of 4
AMC 8 · 2010 · #22
Grade 6 arithmeticnumber-theoryPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem describes a whole family of three-digit numbers, not just one. Instead of fighting with abstract digits, Tool #9 (Easier Problem) says: pick a couple of concrete numbers that fit the rule and just do the subtraction. Tool #5 (Pattern) then checks whether the units digit comes out the same every time — if it does, that is the answer regardless of which specific number we picked. Tool #13 (Algebra) is held in reserve to confirm the pattern using place-value expansion (100h + 10t + u), but only after the simpler tools have already revealed the answer.
Try the simplest example
Start with the simplest legal number 200 (h=2, u=0); reverse it to 2 and subtract to get 198.
Replacing the abstract digits with the smallest legal choice is the Grade 4 "use what you can compute" move — a concrete three-digit subtraction.
4.NBT.B.4Solve An Easier Related ProblemTry a second example
Try a different number 553 (u=3, t=5, h=5); reverse to 355 and subtract to get 198 again.
Same answer again. Tool #5 (Pattern) suggests the difference is always 198 whenever h - u = 2, no matter what the tens digit is — a Grade 5 "analyze patterns and relationships" observation.
5.OA.B.3Look For A PatternTry a third example
A third number 795 (u=5, t=9, h=7) reverses to 597, and subtracting gives 198 once more.
Three different originals, identical difference of 198. The pattern is real.
5.OA.B.3Look For A PatternCheck the pattern with algebra
In place-value form the tens cancel, leaving 99(h - u) = 99 · 2 = 198, so the tens digit never matters.
Writing each number with letters for digits is Grade 6 expression-writing, and it explains why t never matters: the tens digits subtract to 0.
Subtracting the reversed number from the original leaves exactly 99(h-u), so the fixed gap h-u=2 makes the difference 99 · 2 no matter what the tens digit is.
▸ Why?
Write each number by what its digits are worth — original =100h+10t+u, reversed =100u+10t+h — then subtract place by place: the two middle 10t parts cancel and the outer parts leave 99h-99u=99(h-u).
▸ Why?
A digit's worth depends on its place — the hundreds digit counts hundreds, the tens digit counts tens, the units digit counts ones — so 100h+10t+u really is that number's value.
▸ Why?
Both numbers carry the very same tens part 10t, and taking a quantity away from an equal quantity leaves nothing, so the tens digit drops out of the difference entirely.
▸ Why?
The leftover 99h-99u is 99 groups of h less 99 groups of u, which is 99 groups of (h-u) — the shared factor 99 can be pulled out front.
Read off the units digit
Read the ones place of 198 to get the units digit 8, which is choice (E).
Identifying the ones place of a multi-digit number is a Grade 4 place-value skill.
4.NBT.A.2Look For A PatternTrying two or three concrete examples first (Tool #9) often cracks an "abstract digits" problem before you ever need algebra!
- Try the simplest example
- Try a second example
- Try a third example
- Check the pattern with algebra
- Read off the units digit
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