Competition · AMC preparation · step 4 of 4
AMC 8 · 2005 · #12
Grade 6 arithmeticPick an answer.
AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The phrase "each day, six more than the previous day" is the textbook signal for Tool #5 (Look for a Pattern) — the five counts form an evenly-spaced list, going up by 6 each step. With five terms spaced evenly, Tool #11 (Find an Invariant) catches a quiet but powerful fact: the middle term (May 3) is always the average of all five terms. That invariant turns the sum directly into the middle-day count, and from there May 5 is just two steps of +6 away. Picking Tool #5 plus Tool #11 keeps the work at Grade 5-6 arithmetic and avoids Tool #13 (Set Up an Equation) entirely.
List the five days
Start at May 1 and add 6 each day to get five counts that climb by the same step — an evenly spaced list.
Writing out the evenly-spaced list is the Grade 4 "generate a pattern from a rule" move. The constant gap of 6 is the whole pattern.
4.OA.C.5Look For A PatternFind the middle day
For an evenly-spaced list with an odd count of terms, the middle equals the average, so May 3 = 100 ÷ 5 = 20.
In an evenly-spaced list, every step above the middle is matched by an equal step below, so the middle term is the average — a Grade 6 mean idea.
May 3, the middle day, holds exactly the whole of 100 bananas split into five equal shares — the total divided by five.
▸ Why?
The whole 100 is just the five daily counts added together, and those five counts add up to exactly five times the May 3 count.
▸ Why?
Adding the five days gives back the whole because the days are separate pieces with none counted twice and none left out.
▸ Why?
Each day above May 3 stands as far above it as its mirror day below stands beneath it, so the extra bananas on the high days exactly refill the missing bananas on the low days, leaving all five counts equal to May 3.
▸ Why?
May 4 is 6 above May 3 while May 2 is 6 below, and May 5 is 12 above while May 1 is 12 below, because the list climbs by the same 6 every single day.
▸ Why?
Stepping one day forward always adds 6 and stepping one day back takes that same 6 away, so equal numbers of steps up and down land equal distances above and below May 3.
▸ Why?
Taking the surplus off a high day to fill the shortfall on its mirror low day cancels exactly, because adding an amount and then removing the same amount changes nothing.
▸ Why?
Five days now each counting as one May 3 is five equal groups of the May 3 count, which is five times that count.
▸ Why?
Because the total is five times the May 3 count, cutting the total into five equal shares undoes that multiplying and hands back one May 3 count.
Step forward to May 5
May 5 is two days past May 3, so add 6 twice: 20 + 6 + 6 = 32, which is choice (D).
Adding two more 6s keeps walking along the same arithmetic pattern — just continuing the rule from the middle to the end.
5.NBT.B.5Look For A PatternFive days going up by the same 6 each time form an evenly-spaced list, and the middle day (May 3) is automatically the average 100 ÷ 5 = 20. From there, May 5 is just two +6 steps away, giving 32 — that one "middle equals mean" idea turns this AMC 8 problem into a Grade 6 mean exercise.
- List the five days
- Find the middle day
- Step forward to May 5
A parent dashboard for the family lives at sensimlab.com.