AMC 8 · 2016 · #3
Grade 6 arithmeticProblem
Four students take an exam. Three of their scores are 70,80, and 90. If the average of their four scores is 70, then what is the remaining score?
Pick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Four students took an exam. Three of the scores are $70$, $80$, and $90$. The average of all four scores is $70$. Find the fourth score.
Givens: Three known scores: $70$, $80$, $90$; Total number of students $= 4$; Average of all four scores $= 70$; Answer choices: (A) $40$, (B) $50$, (C) $55$, (D) $60$, (E) $70$
Unknowns: The fourth (missing) score
Understand
Restated: Four students took an exam. Three of the scores are $70$, $80$, and $90$. The average of all four scores is $70$. Find the fourth score.
Givens: Three known scores: $70$, $80$, $90$; Total number of students $= 4$; Average of all four scores $= 70$; Answer choices: (A) $40$, (B) $50$, (C) $55$, (D) $60$, (E) $70$
Plan
Primary tool: #14 Work Backwards
Secondary: #2 Use a Variable
The average is already given, so we can run the mean formula in reverse: instead of dividing a known sum by $4$, multiply the target average by $4$ to recover the required total. That is Tool #14 (Work Backwards). Once we know the total the four scores must add to, the fourth score is just the total minus the three known scores. Tool #2 (Use a Variable) lets us write the missing score as $x$ so the relationship becomes a clean one-line equation.
Execute — Answer: A
6.EE.B.6 Step 1 - Name the unknown.
- Let $x$ be the fourth student's score so we can write the average as an equation.
💡 Using a letter for the unknown is the Grade 6 "use variables to represent numbers" move.
6.SP.B.5 Step 2 - Work backwards from the average to the total.
- The mean of $4$ numbers equals their sum divided by $4$, so the sum must equal the mean times $4$.
💡 Reversing "divide by $4$" with "multiply by $4$" is exactly the Tool #14 idea — start from the answer and undo the operation.
4.NBT.B.4 Step 3 Add the three known scores to find how much of the total is already accounted for.
💡 A straightforward multi-digit addition, well within Grade 4 fluency.
6.EE.B.7 Step 4 - The fourth score is whatever is left after the three known scores fill in.
- Subtract to isolate $x$.
💡 Solving $240 + x = 280$ in one step is the Grade 6 one-variable equation skill.
6.EE.B.6 Name the unknown. Let $x$ be the fourth student's score so we can write the aver 6.SP.B.5 Work backwards from the average to the total. The mean of $4$ numbers equals the 4.NBT.B.4 Add the three known scores to find how much of the total is already accounted fo 6.EE.B.7 The fourth score is whatever is left after the three known scores fill in. Subtr Review
Reasonableness: Plug the answer back in: $\dfrac{70 + 80 + 90 + 40}{4} = \dfrac{280}{4} = 70$, which is exactly the given average. The fourth score also lines up with intuition — the three known scores ($70$, $80$, $90$) average to $80$, which is $10$ above the target mean of $70$, so the fourth score must be $30$ below the mean to cancel that $+10 \times 3 = +30$ surplus. That gives $70 - 30 = 40$, matching choice (A).
Alternative: Tool #3 (Look for a Pattern) — balance the deviations from the mean. Compared with the target average $70$: $70$ contributes $0$, $80$ contributes $+10$, $90$ contributes $+20$, total surplus $+30$. The fourth score must contribute $-30$, so it is $70 - 30 = 40$. Same answer (A) without ever computing the sum $280$.
CCSS standards used (min grade 6)
4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Adding the three known scores $70 + 80 + 90 = 240$ and subtracting $280 - 240 = 40$.)6.EE.B.6Use variables to represent numbers and write expressions when solving real-world problems (Letting $x$ stand for the unknown fourth score so the average relationship can be written as an equation.)6.EE.B.7Solve real-world and mathematical problems by writing and solving equations of the form $x + p = q$ (Solving $240 + x = 280$ to get $x = 40$.)6.SP.B.5Summarize numerical data sets by giving quantitative measures of center (mean) (Using the definition of the arithmetic mean ($\text{sum} \div \text{count}$) in reverse to recover the required total $70 \times 4 = 280$.)
⭐ Whenever the average is given, multiply it by the count to get the total — then the missing number is just one subtraction away.
⭐ Whenever the average is given, multiply it by the count to get the total — then the missing number is just one subtraction away.
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