AMC 10 · 2021 · #5

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 a decimal whose two digits after the point repeat forever. Instead the student used the terminating decimal with just those two digits. The wrong answer came out 0.5 less than the correct one. Find the two-digit number.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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 0.5/66 = 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

Write both as fractions

Write both decimals as fractions.

1.ab = 1 + N/100, 1.ab = 1 + N/99

2STEP 2

Compute the difference

The difference has denominator 9900.

1.ab - 1.ab = N/99 - N/100 = (100N - 99N)/9900 = N/9900

3STEP 3

Multiply by sixty-six

Multiplying cancels most of the denominator.

66 · N/9900 = (66 N)/9900 = N/150

4STEP 4

Solve the equation

Set it equal to 0.5 and solve.

N/150 = 0.5 → N = 150 · 0.5 = 75

5STEP 5

Check it directly

Plug it back in to confirm.

66 · 1.75 - 66 · 1.75 = 0.5 ✓

6STEP 6

Match the choice

The answer is 75.

N = 75 → (E)

Answer

Each given choice corresponds to a different gap N/150 which would have to equal 0.5. Only N = 75 works — the other four choices give gaps 15/150 = 0.1, 30/150 = 0.2, 45/150 = 0.3, 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 12 problem only needs Grade 8 repeating-decimal-to-fraction conversion you already know — turn 1.ab = 1 + N/99 and 1.ab = 1 + N/100, find their gap N/9900, multiply by 66 to get N/150 = 0.5, and N = 75.