AMC 10 · 2024 · #12

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

mean-median-mode-rangemulti-digit-arithmeticgraph-reading identify-subproblems ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Short solution 💡 2 insights 📊 Diagram

Problem

Zelda's score on August 1 is 1700 points. On each of the next 5 days a bar chart gives the change from the previous day: +80, -90, -10, +60, -40. Find her average score over the six days, August 1 through August 6.

Pick an answer.

(A)

1700

(B)

1702

(C)

1703

(D)

1713

(E)

1715

AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

The data arrives one day at a time and the rule "next score = previous score + change" is purely sequential, so Tool #2 (Make a Systematic List) is the cleanest fit: walk down the days in order, write each score, then apply the mean formula to the finished list. With only six entries there is no need to reach for algebra (Tool #13) or a shortcut formula — keeping the list visible also makes the answer easy to audit.

1STEP 1

Start from Aug 1 = 1700 and add each signed change in order: 1700, 1780, 1690, 1680, 1740, 1700.

Day & change & score ; Aug 1 & - & 1700 ; Aug 2 & +80 & 1780 ; Aug 3 & -90 & 1690 ; Aug 4 & -10 & 1680 ; Aug 5 & +60 & 1740 ; Aug 6 & -40 & 1700

2STEP 2

Add the six scores, pairing them to stay tidy, for a total of 10290.

3400 + 3520 + 3370 = 10290

3STEP 3

Divide the total by the 6 days: 10290 ÷ 6 = 1715, choice (E).

\frac{10290}{6}

= 1715 → (E)

Answer

1715

Sanity check the magnitude. All six scores sit between 1680 and 1780, so their average must land in that band — and 1715 does. A second check: the net change from Aug 1 to Aug 6 is 80 - 90 - 10 + 60 - 40 = 0, so Aug 1 and Aug 6 both equal 1700, which the list confirms. Finally, 1715 × 6 = 10290 recovers the total, closing the loop.

💡Key takeaway

When data comes from a chart, list the values first and average second. Pair the addends to keep the partial sums tidy, and use the net change as a quick sanity check on your list.

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