AMC 10 · 2002 · #3
Grade 4 arithmeticPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Adding nine large numbers and long-dividing by 9 is slow and error-prone. Instead, notice the shared structure with Tool #5 (Look for a Pattern): every number is a run of 9s, so each equals 9 times a run of 1s. Tool #7 (Identify Subproblems) then splits the work — factor the 9 out so it cancels the divide-by-9, then add the leftover runs of 1s column by column. Finally Tool #3 (Eliminate Possibilities) reads off which listed digit is missing, since four of the five choices actually appear in M.
Read the pattern of the set
Every number is a run of 9s, one nine longer each time: nine numbers in all, the k-th being k nines.
Spotting that the numbers are just growing runs of the same digit turns a messy list into one clean rule.
4.OA.C.5Look For A PatternRewrite each run of 9s
A run of 9s is nine times the matching run of 1s: 9 = 9×1, 99 = 9×11, 999 = 9×111.
A block of nines is exactly nine copies of the same block of ones.
4.NBT.B.5Identify SubproblemsFactor out the 9 and cancel
Pull the shared 9 out of the sum; it cancels the divide-by-9, leaving only 1 + 11 + … + 111111111.
Pulling out the shared factor 9 lets it wipe out the divide-by-9, so the hard arithmetic disappears.
Pulling out the shared factor lets it wipe out the divisor, so the hard arithmetic disappears.
▸ Why?
A factor shared by every term can be lifted out of the sum, leaving something simple inside.
▸ Why?
Dividing by that same factor undoes the multiplication, so it leaves without being computed.
Add the runs of 1s by columns
Stack them and add down: the columns give 9, 8, 7, …, 1 with no carrying, so M = 123456789.
Each column just counts how many of the numbers are long enough to reach that place, so the totals march 9,8,7,…,1.
4.NBT.B.4Look For A PatternA row of nines is just nine times a row of ones, so factoring out the 9 cancels the divide-by-9 and the answer 123456789 falls out with no messy arithmetic.
- Read the pattern of the set
- Rewrite each run of 9s
- Factor out the 9 and cancel
- Add the runs of 1s by columns