AMC 10 · 2010 · #17

Grade 6 logic

Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination

mean-median-mode-rangelogical-deductionlinear-equations-one-var bound-inequality-then-enumerate ↑ Prerequisites: mean-median-mode-range

📏 Long solution 💡 3 insights

Problem

Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$ th and $64$ th, respectively. How many schools are in the city?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2010 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.

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Toolkit + CCSS Solution

Understand

Restated: Every school sends exactly 3 students, so the total number of students is 3 times the number of schools. All scores are different, so everyone has a unique rank. Andrea has the median score of everyone, and she is the top scorer on her own team. Her teammates finished 37th and 64th. Find how many schools there are.

Givens: Each school sends exactly 3 students.; All scores are distinct, so ranks 1st, 2nd, 3rd, ... are all unique.; Andrea's score is the median of all students.; Andrea is the highest scorer on her team.; Her teammates Beth and Carla placed 37th and 64th.

Unknowns: The number of schools in the city.

Understand

Plan

Primary tool: #4 Introduce a Variable

Secondary: #14 Extreme Principle, #3 Eliminate Possibilities

Name the number of schools with a letter so the total count and Andrea's median rank become simple expressions. The two teammate placements act as boundaries: they squeeze the median rank from above and below (Extreme Principle). Because the total must be odd and the schools count is a whole number, only one choice survives (Eliminate Possibilities).

Execute — Answer: B

#4 Introduce a Variable 6.EE.A.2 Step 1

Name the number of schools

Let n be the number of schools.
Since each school sends 3 students, the total number of students is 3n.

$$\text{total students} = 3n$$

💡 One letter for the thing you want lets every other quantity be written in terms of it.

#4 Introduce a Variable 6.SP.B.5 Step 2

Locate the median rank

The median is one real student's score, so the students cannot split evenly into two halves.
That means 3n must be odd, which forces n to be odd.
When 3n is odd, the median sits exactly in the middle at rank (3n+1)/2, counting 1st as the highest score.

$$\text{Andrea's rank} = \frac{3n+1}{2}, \quad 3n \text{ odd} \Rightarrow n \text{ odd}$$

💡 A single middle value only exists when the list has an odd number of entries.

#14 Extreme Principle 6.EE.B.8 Step 3

Use Beth to bound n above

Andrea is the highest scorer on her team, and Beth placed 37th.
So Andrea's rank must be better than 37th, meaning her rank number is at most 36.
Set the median rank at most 36 and solve for n.

$$\frac{3n+1}{2} \le 36 \;\Rightarrow\; 3n \le 71 \;\Rightarrow\; n \le 23$$

💡 Beating a known rank puts a ceiling on where the median can sit.

#4 Introduce a Variable 6.EE.B.8 Step 4

Use Carla to bound n below

Carla placed 64th, so a 64th position must exist.
That means there are at least 64 students, so 3n is at least 64.
Dividing by 3, n is at least 22 (since 3 times 21 is only 63).

$$3n \ge 64 \;\Rightarrow\; n \ge 22$$

💡 A place number that actually happened proves at least that many people showed up.

#3 Eliminate Possibilities 6.EE.B.5 Step 5

Keep only the value that fits

Now n must satisfy 22 <= n <= 23 and also be odd.
Between 22 and 23, only 23 is odd, so n = 23.

$$22 \le n \le 23, \; n \text{ odd} \;\Rightarrow\; n = 23$$

💡 When a whole-number range holds just one value with the right parity, that value is forced.

#3 Eliminate Possibilities 6.SP.A.3 Step 6

Confirm the count

With 23 schools there are 3 times 23 = 69 students, an odd number, so the median is the person at rank (69+1)/2 = 35th.
Andrea at 35th is ahead of Beth at 37th and Carla at 64th, so she really is her team's best and everything holds.
The number of schools is 23, which is answer (B).

$$3 \cdot 23 = 69, \quad \frac{69+1}{2} = 35$$

💡 Plugging the winner back in should make every clue in the story come out true.

[1] #4 6.EE.A.2 Let n be the number of schools. Since each school sends 3 students, the total nu

[2] #4 6.SP.B.5 The median is one real student's score, so the students cannot split evenly into

[3] #14 6.EE.B.8 Andrea is the highest scorer on her team, and Beth placed 37th. So Andrea's rank

[4] #4 6.EE.B.8 Carla placed 64th, so a 64th position must exist. That means there are at least

[5] #3 6.EE.B.5 Now n must satisfy 22 <= n <= 23 and also be odd. Between 22 and 23, only 23 is

[6] #3 6.SP.A.3 With 23 schools there are 3 times 23 = 69 students, an odd number, so the median

Review

Reasonableness: Check the answer against every clue. 23 schools give 69 students, an odd count, so a single median exists at rank 35. Andrea at 35th beats both teammates (37th and 64th), so she is her team's top scorer, and 64th place exists because 69 is at least 64. All conditions are met, and 23 is one of the listed choices.

Alternative: Instead of algebra, test the choices directly. For each option compute 3n and its middle rank (3n+1)/2: n=25 gives 75 students with median rank 38, but then Beth at 37th would beat Andrea, contradicting the story; n=22 or 24 give an even 66 or 72 students, so no single person is the median. Only n=23 gives an odd total with the median (35th) ahead of both teammates.

CCSS standards used (min grade 6)

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Letting n be the number of schools and writing the total as 3n and the median rank as (3n+1)/2.)
6.SP.B.5 Summarize numerical data sets in relation to their context, including the median (Placing the median at the middle rank and reasoning that a single median needs an odd number of students.)
6.EE.B.8 Write an inequality to represent a constraint in a real-world problem (Turning 'Andrea beats 37th' and '64th place exists' into the bounds n <= 23 and n >= 22.)
6.EE.B.5 Understand solving an inequality as finding the values that make it true (Narrowing the range 22 <= n <= 23 with n odd down to the single value n = 23.)
6.SP.A.3 Recognize that a measure of center summarizes a data set with a single number (Verifying that with 69 students the median is exactly the 35th-ranked person.)

⭐ Name the unknown, turn each clue into a squeeze from above and below, then keep only the whole number with the right parity.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Name the number of schools

Locate the median rank

Use Beth to bound n above

Use Carla to bound n below

Keep only the value that fits

Confirm the count

Review

CCSS standards used (min grade 6)

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