AMC 8 · 2010 · #22

Grade 6 arithmeticnumber-theory

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

digit-decompositionplace-valueunits-digit-trackinglinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: digit-decompositionplace-value

📏 Medium solution 💡 4 insights

Problem

Take a three-digit number whose hundreds digit is exactly 2 more than its units digit. Reverse its digits to make a new three-digit number. Subtract the reversed number from the original. What is the units digit of that difference?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

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.

1STEP 1

Start with the simplest legal number 200 (h=2, u=0); reverse it to 2 and subtract to get 198.

200 - 002 = 198

2STEP 2

Try a different number 553 (u=3, t=5, h=5); reverse to 355 and subtract to get 198 again.

553 - 355 = 198

3STEP 3

A third number 795 (u=5, t=9, h=7) reverses to 597, and subtracting gives 198 once more.

795 - 597 = 198

4STEP 4

In place-value form the tens cancel, leaving 99(h - u) = 99 · 2 = 198, so the tens digit never matters.

(100h + 10t + u) - (100u + 10t + h) = 99h - 99u = 99(h - u) = 99 · 2 = 198

5STEP 5

Read the ones place of 198 to get the units digit 8, which is choice (E).

Units digit of 198 = 8 → (E)

Answer

Across three very different starting numbers (200, 553, 795), the difference came out to exactly 198 each time, and the algebra step 99(h - u) explains why: the constraint h - u = 2 forces the difference to be 99 × 2 = 198 no matter what the tens digit is. The units digit of 198 is 8, which matches choice (E) — and (E) is the only choice that could come from a multiple of 99 ending in 8, so the answer is consistent.

💡Key takeaway

Trying two or three concrete examples first (Tool #9) often cracks an "abstract digits" problem before you ever need algebra!

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