Competition · AMC preparation · step 4 of 4

AMC 8 · 2005 · #6

Grade 5 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration
place-valuedigit-constraintsinterval-arithmeticsystematic-enumeration systematic-enumerationcasework ↑ Prerequisites: place-value
📏 Medium solution 💡 3 insights
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Problem
d is a single digit (one of 0,1,2,…,9). For how many of those values does 2.00d5 ＞ 2.005?

Pick an answer.

(A)
0
(B)
4
(C)
5
(D)
6
(E)
10

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Count Carefully

The question asks "for how many values," so the goal is to count digits that pass a test (Tool #13). The test itself is the decimal comparison 2.00d5 ＞ 2.005. To run the test cleanly, split the 10 digits into three cases by how d compares to 5 in the thousandths place (Tool #7). The case d=5 needs an extra look at the ten-thousandths place, so isolating it as its own subproblem keeps the reasoning honest.

1STEP 1

Line up the decimals

Line up the place values: rewrite 2.005 as 2.0050 so it has the same four decimal places as 2.00d5.

2.00d5 vs. 2.0050
2STEP 2

Check the matching places

The ones, tenths, and hundredths all match, so the whole comparison is decided at the thousandths place.

ones = ones, tenths = tenths, hundredths = hundredths → decide at thousandths
3STEP 3

Test digits above 5

Case A, d ＞ 5: 2.00d5 already wins at the thousandths place, so d = 6, 7, 8, 9 all work — four values.

d ∈ {6,7,8,9} → 2.00d5 ＞ 2.0050 ✓
4STEP 4

Test digits below 5

Case B, d ＜ 5: 2.00d5 loses at the thousandths place, so d = 0, 1, 2, 3, 4 all fail.

d ∈ {0,1,2,3,4} → 2.00d5 ＜ 2.0050 ×
5STEP 5

Test the digit 5

Case C, d = 5: the thousandths tie, so 2.0055 beats 2.0050 at the ten-thousandths place — so d = 5 counts.

d = 5 → 2.0055 ＞ 2.0050 ✓
6STEP 6

Count the winning digits

Collect the winners 5, 6, 7, 8, 9 — that is five digits, so the answer is (C).

|{5, 6, 7, 8, 9}| = 5 → (C)
Answer
5
Spot-check with the boundary digit d = 5: 2.0055 vs. 2.005 = 2.0050. The extra 5 in the ten-thousandths place clearly makes 2.0055 bigger, confirming d = 5 counts. Spot-check d = 4: 2.0045 vs. 2.0050. At the thousandths place 4 ＜ 5, so 2.0045 ＜ 2.0050 — correctly excluded. The winning set {5,6,7,8,9} has 5 elements, matching answer (C). Also, intuitively, about half of the 10 digits should be larger than the cutoff, so 5 feels right.
💡Key takeaway

Stack the decimals, find the first place where they differ, and count the digits that come out on top — that turns a tricky-looking inequality into a quick Grade 5 check.

  • Line up the decimals
  • Check the matching places
  • Test digits above 5
  • Test digits below 5
  • Test the digit 5
  • Count the winning digits

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