AMC 10 · 2020 · #12

Grade 8 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

exponentsdecimal-arithmeticdigit-countingestimation identify-subproblemseasier-related-problem ↑ Prerequisites: exponentsdecimal-arithmetic

📏 Medium solution 💡 2 insights

Problem

Write

\frac{1}{20²⁰}

as a decimal. After the decimal point comes a run of 0's, then a 9, then more digits. How many 0's are in that opening run?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Subproblems): split the denominator 20²⁰ into 2²⁰ · 10²⁰ and handle each piece separately — the 10²⁰ part contributes exactly 20 zeros, the 2²⁰ part is what we still need to convert to a power-of-10 shift. Tool #9 (Easier Problem): replace the unwieldy 2²⁰ with the small fact 2¹⁰ = 1024 to find the digit count of 2²⁰. Tool #3 (Eliminate): only one answer choice matches once we count digits.

1STEP 1

Rewrite the denominator as 2²⁰ · 10²⁰ — the 10²⁰ just shifts the decimal, the 2²⁰ sets the first nonzero digit.

20²⁰ = (2 · 10)²⁰ = 2²⁰ · 10²⁰

2STEP 2

Square the easy fact 2¹⁰ = 1024: 2²⁰ = 1,048,576, a 7-digit number.

2²⁰ = 1024² = 1,048,576 (7 digits)

3STEP 3

Since $\frac{1}{1,048,576}$ is just under $\frac{1}{10⁶}$ , $\frac{1}{2²⁰}$ ≈ 9.5 · 10⁻⁷ — its 9 sits at the 7th decimal place.

\frac{1}{2²⁰} = \frac{1}{1,048,576}

≈ 9.537 · 10⁻⁷

4STEP 4

Multiplying by 10⁻²⁰ shifts everything 20 places right, so the 9 moves from the 7th to the 27th decimal place.

\frac{1}{20²⁰}

≈ 9.537 · 10⁻²⁷

5STEP 5

With the first nonzero digit at place 27, the zeros before it number 27 - 1 = 26.

leading zeros = 27 - 1 = 26

6STEP 6

Only choice (D) equals 26; the others miss the digit count or the shift by a place or two.

26 → (D)

Answer

Quick estimate: 20²⁰ = (2 · 10)²⁰ = 2²⁰ · 10²⁰ ≈ 10⁶ · 10²⁰ = 10²⁶, so

\frac{1}{20²⁰}

≈ 10⁻²⁶. That puts the first nonzero digit near the 26th or 27th decimal place — which means 25, 26, or 27 zeros. Refining the estimate (2²⁰ is just over 10⁶, not equal), the first nonzero digit slides to the 27th place, giving 26 zeros. Consistent with (D).

💡Key takeaway

This AMC 10 problem only needs Grade 8 scientific notation you already know! Split 20²⁰ = 2²⁰ · 10²⁰. The 10²⁰ part contributes 20 zeros for free. The 2²⁰ = 1,048,576 part has 7 digits, so $\frac{1}{2²⁰}$ ≈ 9.5 · 10⁻⁷ — 9 sits at the 7th decimal place. Shift 20 more places right and the 9 lands at the 27th place, leaving 27 - 1 = 26 leading zeros, answer (D).

More like this