AMC 8 · 2004 · #9
Grade 6 arithmeticProblem
The average of the five numbers in a list is . The average of the first two
numbers is . What is the average of the last three numbers?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: A list has five numbers. Their average is $54$. The first two of them have average $48$. What is the average of the remaining three numbers?
Givens: There are $5$ numbers in the list; Average of all $5$ numbers is $54$; Average of the first $2$ numbers is $48$; Answer choices: (A) $55$, (B) $56$, (C) $57$, (D) $58$, (E) $59$
Unknowns: The average of the last $3$ numbers
Understand
Restated: A list has five numbers. Their average is $54$. The first two of them have average $48$. What is the average of the remaining three numbers?
Givens: There are $5$ numbers in the list; Average of all $5$ numbers is $54$; Average of the first $2$ numbers is $48$; Answer choices: (A) $55$, (B) $56$, (C) $57$, (D) $58$, (E) $59$
Plan
Primary tool: #16 Change Focus
We are given averages but asked for another average. Tool #16 (Change Focus) says: stop looking at averages and look at totals instead. Each average comes with a count, so multiplying gives a sum — and sums split and combine cleanly across groups, while averages do not. Once we have the total of all $5$ numbers and the total of the first $2$, subtracting gives the total of the last $3$, and dividing by $3$ gives the average we want.
Execute — Answer: D
6.SP.B.5 Step 1 Switch from "average of all $5$" to "total of all $5$" by using average $\times$ count $=$ sum.
💡 The Grade 6 mean formula reverses easily: if you know the mean and the count, the sum is just their product.
6.SP.B.5 Step 2 Do the same switch for the first two numbers: their average times their count gives their total.
💡 Same reversal of the mean formula, now with count $2$ instead of $5$.
4.OA.A.3 Step 3 The five numbers split into the first two and the last three with no overlap, so the total of the last three is the grand total minus the first-two total.
💡 Totals across disjoint groups add up — the same part-whole reasoning used in Grade 4 multi-step word problems.
6.SP.B.5 Step 4 Now convert back from total to average for the last three by dividing the total by the count $3$.
💡 Reverse the reversal: once we have the sum and the count, dividing recovers the mean.
6.SP.B.5 Switch from "average of all $5$" to "total of all $5$" by using average $\times$ 6.SP.B.5 Do the same switch for the first two numbers: their average times their count gi 4.OA.A.3 The five numbers split into the first two and the last three with no overlap, so 6.SP.B.5 Now convert back from total to average for the last three by dividing the total Review
Reasonableness: Quick sanity check. The first two numbers average $48$, which is $6$ below the overall average of $54$. So those two numbers are $2 \times 6 = 12$ below where they would need to be to make the overall average. The remaining three numbers have to make up that deficit of $12$, so on average each of them must sit $12 \div 3 = 4$ above $54$, giving $54 + 4 = 58$. That matches answer (D).
Alternative: Tool #4 (Introduce a Variable): let $x$ be the desired average. Then the last three numbers total $3x$, the first two total $96$, and the whole list totals $270$. So $96 + 3x = 270$, giving $3x = 174$ and $x = 58$. Same answer (D), arrived at by naming the unknown directly.
CCSS standards used (min grade 6)
6.SP.B.5Summarize numerical data sets, including reporting the number of observations and measures of center (Using the mean formula (sum $\div$ count) in both directions: total $= 54 \times 5 = 270$ and total $= 48 \times 2 = 96$ to get sums from averages, then $174 \div 3 = 58$ to get the answer's average back from a sum.)4.OA.A.3Solve multi-step word problems using the four operations (Splitting the grand total into two disjoint groups and subtracting: total of last 3 $= 270 - 96 = 174$.)
⭐ When a problem gives you several averages and asks for another, switch to totals first — totals add and subtract cleanly across groups, and you can always divide at the end to get the average you wanted.
⭐ When a problem gives you several averages and asks for another, switch to totals first — totals add and subtract cleanly across groups, and you can always divide at the end to get the average you wanted.