AMC 10 · 2021 · #8

Grade 8 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

fraction-decimal-conversiondigit-decompositionlinear-equations-one-var convert-to-algebraguess-and-check ↑ Prerequisites: fraction-decimal-conversion

📏 Medium solution 💡 2 insights

Problem

A student should have multiplied 66 by the repeating decimal 1.ab where a and b are digits. Instead the student used the terminating decimal 1.ab (just two digits after the point). The wrong answer turned out to be 0.5 less than the correct one. Find the two-digit number ab.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

Only five candidate values {15, 30, 45, 60, 75} — Tool #6 (Guess and Check) plugs each into the repeating-vs-terminating gap and tests whether 66 × (gap) = 0.5. Tool #3 (Eliminate) discards every N that misses 0.5. Tool #11 (Work Backwards) is the verification: starting from the required gap $\frac{0.5}{66}$ = $\frac{1}{132}$ , run the gap formula in reverse to recover N in one step — this confirms the unique answer without sweeping all five choices.

1STEP 1

Let N be the two-digit block. The terminating 1.ab is 1 + $\frac{N}{100}$ ; the repeating 1.ab is 1 + $\frac{N}{99}$ .

1.ab = 1 +

\frac{N}{100}

, 1.ab = 1 +

\frac{N}{99}

2STEP 2

Subtract; the 1s cancel and $\frac{N}{99}$ - $\frac{N}{100}$ = $\frac{N}{9900}$ over the common denominator 9900.

1.ab - 1.ab =

\frac{N}{99}

\frac{N}{100}

\frac{100N - 99N}{9900}

\frac{N}{9900}

3STEP 3

Multiply the gap by 66: 66· $\frac{N}{9900}$ = $\frac{N}{150}$ , since 9900 = 150 × 66.

66 ·

\frac{N}{9900}

\frac{66 N}{9900}

\frac{N}{150}

4STEP 4

The problem's gap is 0.5, so $\frac{N}{150}$ = 0.5 and N = 75.

\frac{N}{150}

= 0.5 → N = 150 · 0.5 = 75

5STEP 5

Check N=75: correct 66·(1+ $\frac{75}{99}$ ) minus wrong 66·1.75, gap = $\frac{49.5}{99}$ = 0.5 ✓.

66 · 1.75 - 66 · 1.75 = 0.5 ✓

6STEP 6

N = 75 matches choice (E).

N = 75 → (E)

Answer

Each given choice corresponds to a different gap

\frac{N}{150}

which would have to equal 0.5. Only N = 75 works — the other four choices give gaps

\frac{15}{150}

= 0.1,

\frac{30}{150}

= 0.2,

\frac{45}{150}

= 0.3,

\frac{60}{150}

= 0.4 — none of which is 0.5. The unique value is (E). The size also makes sense: a missed two-digit repeat is small, so the resulting product gap of 0.5 on a multiplier near 1 is plausible.

💡Key takeaway

This AMC 10 problem only needs Grade 8 repeating-decimal-to-fraction conversion you already know — turn 1.ab = 1 + $\frac{N}{99}$ and 1.ab = 1 + $\frac{N}{100}$ , find their gap $\frac{N}{9900}$ , multiply by 66 to get $\frac{N}{150}$ = 0.5, and N = 75.

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