AMC 8 · 2010 · #4

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

mean-median-mode-rangefraction-arithmetic identify-subproblems ↑ Prerequisites: mean-median-mode-rangemulti-digit-arithmetic

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Problem

What is the sum of the mean, median, and mode of the numbers $2,3,0,3,1,4,0,3$ ?

Pick an answer.

(A)

6.5

(B)

(C)

7.5

(D)

8.5

(E)

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Toolkit + CCSS Solution

Understand

Restated: Take the data set $2, 3, 0, 3, 1, 4, 0, 3$. Compute its mean, median, and mode, then add the three results together.

Givens: Data set: $2, 3, 0, 3, 1, 4, 0, 3$ ($8$ values); Three summary statistics to compute: mean, median, mode; Answer choices: (A) $6.5$, (B) $7$, (C) $7.5$, (D) $8.5$, (E) $9$

Unknowns: The single value $\text{mean} + \text{median} + \text{mode}$

Understand

Restated: Take the data set $2, 3, 0, 3, 1, 4, 0, 3$. Compute its mean, median, and mode, then add the three results together.

Givens: Data set: $2, 3, 0, 3, 1, 4, 0, 3$ ($8$ values); Three summary statistics to compute: mean, median, mode; Answer choices: (A) $6.5$, (B) $7$, (C) $7.5$, (D) $8.5$, (E) $9$

Plan

Primary tool: #2 Make a Systematic List

Secondary: #7 Identify Subproblems

The data is given in a jumbled order, so the first move is Tool #2 (Make a Systematic List): sort the eight numbers from smallest to largest. That single ordered list makes the median and mode obvious by inspection and keeps us from miscounting duplicates. Tool #7 (Identify Subproblems) is the natural frame for the whole problem because the question is really three small problems — find mean, find median, find mode — followed by one easy addition.

Execute — Answer: C

#2 Make a Systematic List 5.MD.B.2 Step 1

Sort the data from smallest to largest so the middle values and any repeats are easy to spot.

$$2, 3, 0, 3, 1, 4, 0, 3 \;\Rightarrow\; 0, 0, 1, 2, 3, 3, 3, 4$$

💡 Lining up data on a number line — exactly what a Grade 5 line plot does — makes the order easy to read.

#7 Identify Subproblems 6.SP.B.5 Step 2

Find the mode: scan the sorted list for the value that appears most often.
$0$ appears twice, $3$ appears three times, every other value appears once.
So the mode is $3$.

$$\text{mode} = 3$$

💡 Picking the most frequent value is the simplest Grade 6 measure-of-center reading.

#7 Identify Subproblems 6.SP.B.5 Step 3

Find the median.
With $8$ values (an even count) the median is the average of the $4$th and $5$th terms in the sorted list.
Those are $2$ and $3$.

$$\text{median} = \dfrac{2 + 3}{2} = 2.5$$

💡 Averaging the two middle values when the count is even is part of the Grade 6 "summarize a data set" standard.

#7 Identify Subproblems 6.SP.B.5 Step 4

Find the mean: add every value and divide by the count $8$.

$$\text{mean} = \dfrac{0+0+1+2+3+3+3+4}{8} = \dfrac{16}{8} = 2$$

💡 Sum-over-count is the textbook Grade 6 definition of the mean.

#7 Identify Subproblems 5.NBT.B.7 Step 5

Add the three results to answer the question.

$$\text{mean} + \text{median} + \text{mode} = 2 + 2.5 + 3 = 7.5 \;\Rightarrow\; \textbf{(C)}$$

💡 Adding whole numbers and a decimal to the tenths is a Grade 5 decimal-arithmetic skill.

[1] #2 5.MD.B.2 Sort the data from smallest to largest so the middle values and any repeats are

[2] #7 6.SP.B.5 Find the mode: scan the sorted list for the value that appears most often. $0$ a

[3] #7 6.SP.B.5 Find the median. With $8$ values (an even count) the median is the average of th

[4] #7 6.SP.B.5 Find the mean: add every value and divide by the count $8$.

[5] #7 5.NBT.B.7 Add the three results to answer the question.

Review

Reasonableness: The eight numbers all sit between $0$ and $4$, so each measure of center has to land in that range. We got mean $= 2$, median $= 2.5$, mode $= 3$ — all in $[0, 4]$ and clustered where the data piles up (three $3$s pull the mode and median upward, the two $0$s drag the mean down). Their sum $7.5$ lies between $0 + 0 + 0 = 0$ and $4 + 4 + 4 = 12$, and matches choice (C).

Alternative: Tool #1 (Draw a Diagram) via a dot plot: place dots above $0, 0, 1, 2, 3, 3, 3, 4$ on a number line. The tallest stack ($3$) is the mode by eye. The center of mass — counted by sliding from each end toward the middle — sits between the $4$th and $5$th dots, giving median $2.5$. The mean ($2$) is the balance point you can verify by checking that the total "distance" of dots left of $2$ equals the total distance of dots right of $2$. Same three numbers, same sum $7.5$.

CCSS standards used (min grade 6)

5.MD.B.2 Make a line plot to display a data set and solve problems using the data (Sorting the eight values from smallest to largest, the same ordering step a Grade 5 line plot requires.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Adding $2 + 2.5 + 3 = 7.5$ at the final step — whole numbers plus a tenths decimal.)
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures (Computing the three measures of center — mean, median, and mode — of the eight-value data set.)

⭐ This AMC 8 problem only needs Grade 6 measures of center — mean, median, and mode — you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

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