AMC 10 · 2012 · #6
Grade 6 algebraPick an answer.
Tool #4 (Introduce a Variable) turns the words into three clean pair-sum equations by naming the numbers a ≤ b ≤ c. Tool #15 (Organize Information in More Ways) is what makes it easy: order the pair sums so each sum is matched to the pair it came from. Then Tool #7 (Identify Subproblems) splits the work into two small steps — first find the total of all three numbers, then peel off the middle one — instead of solving a full three-way system.
Name the numbers, match the sums
Three names match the three pair sums.
Giving the unknown numbers names lets you write the word clues as equations you can work with.
6.EE.B.6Introduce A VariableAdd all three pair sums
Adding all three counts everything twice.
Summing all the pair sums counts each number twice, so it equals double the grand total.
Summing all three pair sums counts each number twice, so the result is double the grand total.
▸ Why?
Each number belongs to exactly two of the three pairs, so summing over pairs visits it twice.
▸ Why?
The grand total is the three numbers added together, so halving the doubled version recovers it.
Peel off the middle number
Subtracting one pair sum leaves 7, choice (D).
Taking the total of all three and removing the two that flank the middle leaves the middle number alone.
4.NBT.B.4Identify SubproblemsAdd all the pair sums to get double the total, then subtract the pair that skips the middle number to find it.
- Name the numbers, match the sums
- Add all three pair sums
- Peel off the middle number