AMC 10 · 2005 · #20
Grade 6 arithmeticPick an answer.
AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Listing all the numbers and adding them is hopeless by hand, so the winning idea is Tool #5 (Look for a Pattern): every digit is treated the same way, so across all the arrangements each digit lands in each place-value column equally often. Tool #16 (Change Focus) reframes the average — instead of averaging whole numbers, we average one place-value column at a time. Tool #15 (Organize Information) writes each number in place-value form so those columns are visible. Tool #7 (Identify Subproblems) then splits the work into 'find the average digit' and 'multiply by the place values', which are both easy.
Write a number by place value
Each number weights its five digits by place value 10000, 1000, 100, 10, 1 — so which digit sits in which column decides everything.
A number is just its digits, each multiplied by what its column is worth.
5.NBT.A.1Organize Information In More WaysAverage each place on its own
The average of a sum is the sum of the averages, so average each column on its own — write d_k for place k — and add the columns back up.
Averaging each column first gives the same total as averaging whole numbers, so we can handle one column at a time.
Averaging each column on its own gives the same total as averaging whole numbers.
▸ Why?
A number is its digits weighted by their places, so the total splits cleanly column by column.
▸ Why?
An average of a sum is the sum of the averages, since both are totals shared over the same count.
Every place has the same average digit
No column is special: each digit lands there equally often, so every column's average digit is the mean of 1, 3, 5, 7, 8, namely 4.8.
No digit prefers any column, so on average every column holds the same typical digit — the mean of the five.
6.SP.B.5Look For A PatternCombine the columns
Every column contributes 4.8 times its place value, so factor 4.8 out; what is left is 10000+1000+100+10+1 = 11111.
Every place carries the same average digit, so the place values simply add up to 11111.
6.EE.A.3Identify SubproblemsMultiply to finish
As whole numbers 48 × 11111 = 533328; restoring the one decimal place gives 4.8 × 11111 = 53332.8, which is choice (C).
Multiply as whole numbers first, then restore the single decimal place.
5.NBT.B.7Identify SubproblemsWhen every arrangement is equally likely, each place-value column just shows the average digit, so the average number is that one average digit times 11111.
- Write a number by place value
- Average each place on its own
- Every place has the same average digit
- Combine the columns
- Multiply to finish
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