AMC 10 · 2016 · #4
Grade 6 arithmeticPick an answer.
The mean condition is the strongest one: it is a single equation that pins down x exactly. Tool #13 (Convert to Algebra) turns the sentence "the mean equals x" into the equation (540+x)/7 = x, which solves in one move. The median and mode conditions are then used with Tool #2 (Make a Systematic List) — sort the seven numbers — and Tool #3 (Eliminate Possibilities) to confirm the same x works, so we know the answer is consistent rather than a coincidence.
Add the six known numbers
The known values add to 540.
Bundle everything you already know into one number so the unknown stands alone.
4.NBT.B.4Convert To AlgebraTurn "mean equals x" into an equation
The mean gives one clean equation.
"Mean equals x" is just a balance: the total must be 7 copies of x.
Saying the mean equals a value is saying the total is that many copies of it.
▸ Why?
An average is a total shared out over a count, so multiplying it back by the count returns the total.
▸ Why?
That total is the listed numbers added together, so the unknown one is the leftover.
Check the median by sorting
Sorting confirms the middle value.
Line the numbers up in order and the middle one is the median.
6.SP.B.5Make A Systematic ListCheck the mode and read the answer
The repeat check gives 90, choice (D).
The number that shows up most often is the mode; here 90 appears twice and nothing else repeats.
6.SP.B.5Eliminate PossibilitiesWhen a problem says the mean equals some unknown, set the total equal to that many copies of the unknown — then sort the list to double-check the median and mode.
- Add the six known numbers
- Turn "mean equals x" into an equation
- Check the median by sorting
- Check the mode and read the answer