Problem 3 · 2016 AMC 8
Easy
Arithmetic & Operations
sum-constraint
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?
Show answer
Answer: A — 40.
Show hints
Hint 1 of 2
An average is just the total shared out equally — so multiplying the average back by the count un-shares it and hands you the TOTAL of all four scores.
Still stuck? Show hint 2 →
Hint 2 of 2
Once you know what all four must add to, the missing score is whatever's left after subtracting the three you already know.
Show solution
Approach: average × count rebuilds the total; the unknown is the leftover
- The average is the total split 4 ways, so the total = average × count = 70 × 4 = 280.
- Three scores are 70 + 80 + 90 = 240, so the missing score is 280 − 240 = 40.
- Sanity check: two of the known scores (80, 90) sit ABOVE the average of 70, so something must sit well below to pull it back — a low score like 40 fits, an answer of 70 wouldn't.
- You'll see this again as: "find the missing value given the mean" — always convert the average into a total first, then the unknown is just total minus the rest.
Another way — balance around the average:
- Measure each score as a deviation from the target average 70: 70 is 0, 80 is +10, 90 is +20. The known scores are +30 total above 70.
- For the average to land exactly on 70, the deviations must cancel, so the last score must be 30 BELOW 70: 70 − 30 = 40.
Mark:
· log in to save