2024 AMC 10A Problem 12

P11-15~2 minPrint

Statistics & Data Set up the equation/formula and compute; no special trick needed

Zelda played the Adventures of Math game on August 1 and scored $1,700$ points. She continued to play daily over the next $5$ days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was $1,700 + 80 = 1,780$ points.) What was Zelda's average score in points over the $6$ days?

Show solution without answering

Key insight

Turn the day-to-day changes into actual scores with a running total. Then average how far each score is above 1700 instead of adding six big numbers.

Solution

The chart shows how much the score changed from the day before, not the score itself. So first work out each day's actual score.

Reading the bars for August 2 through 6, the changes are $+80$ , $-90$ , $-10$ , $+60$ , $-40$ .
Keep a running total starting from $1700$ :

Aug 1	Aug 2	Aug 3	Aug 4	Aug 5	Aug 6
$1700$	$1780$	$1690$	$1680$	$1740$	$1700$

Adding six four-digit numbers is slow, so look only at how far each score is from $1700$ : $0, +80, -10, -20, +40, 0$ . These add to $90$ .
Share that $90$ over the $6$ days: $90 \div 6 = 15$ . So on average the score is $15$ above $1700$ .

The average score is $1700 + 15 = 1715$ .

The answer is $\boxed{\textbf{(E)}\ 1715}$ .

Why this works

A chart of changes has to be turned into real values by adding up as you go, before you can average. When all the values are near one round number, average only the amounts above or below that number and add it back at the end; the arithmetic stays small.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

The bars give the day-to-day changes $+80, -90, -10, +60, -40$ for August 2 through 6. Accumulate them starting from $1700$ :

Aug 1	Aug 2	Aug 3	Aug 4	Aug 5	Aug 6
1700	1780	1690	1680	1740	1700

Rather than adding six four-digit numbers, add the amounts above $1700$ : $0 + 80 - 10 - 20 + 40 + 0 = 90$ . Spread over $6$ days that is $15$ per day, so the average score is $1700 + 15 = 1715$ .

The answer is $\boxed{\textbf{(E)}\ 1715}$ .

Review in the AoPS books

Prealgebra, Chapter 13: Data and Statistics