AMC 8 · 2016 · #3

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

mean-median-mode-rangelinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticmean-median-mode-range

📏 Short solution 💡 2 insights

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Problem

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.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

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.

1STEP 1

Name the unknown — let x stand for the missing fourth score, so the average becomes an equation.

x = fourth score

2STEP 2

Run the mean backwards: the four scores must total the average times four, so their sum is 280.

sum of all four scores = 70 × 4 = 280

3STEP 3

Add the three known scores; together they already make up 240 of that total.

70 + 80 + 90 = 240

4STEP 4

The fourth score is the leftover: subtract 240 from 280 to get 40.

x = 280 - 240 = 40 → (A)

Answer

Plug the answer back in:

\frac{70 + 80 + 90 + 40}{4}

\frac{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 × 3 = +30 surplus. That gives 70 - 30 = 40, matching choice (A).

💡Key takeaway

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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