AMC 8 · 2018 · #13

Grade 6 arithmeticalgebra

Archetype Small-Domain Constrained Search

mean-median-mode-rangelinear-equations-one-varbound-inequality-then-enumeratesystematic-enumeration bound-inequality-then-enumeratesystematic-enumerationconvert-to-algebra ↑ Prerequisites: linear-equations-one-varmulti-digit-arithmetic

📏 Medium solution 💡 3 insights

Problem

Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?

$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad \textbf{(E) }18$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Laila took $5$ tests, each scored on integers from $0$ to $100$. She got the SAME score on tests $1$ through $4$ (call it $x$) and a STRICTLY HIGHER score on test $5$ (call it $y$, with $y > x$). Her average across the $5$ tests was $82$. How many different integer values are possible for $y$?

Givens: $5$ tests, each integer score from $0$ to $100$ inclusive; Tests $1$-$4$ all received the same score $x$; Test $5$ received a higher score $y$, so $y > x$; The average of the $5$ scores is $82$; Answer choices: (A) $4$, (B) $5$, (C) $9$, (D) $10$, (E) $18$

Unknowns: The number of integer values of $y$ (the last-test score) consistent with all conditions

Understand

Plan

Primary tool: #6 Guess and Check

Secondary: #2 Make a Systematic List, #3 Eliminate Possibilities

Once we turn "average $= 82$" into the total $4x + y = 410$, the unknown $y$ is squeezed into a small window (just above $82$, at most $100$). That is a tiny, bounded set — perfect for Tool #6 (Guess and Check): try each candidate $y$ and see whether the forced $x = (410 - y)/4$ is a whole number that is less than $y$. Tool #2 (Systematic List) keeps the candidate scan in strict order (smallest $y$ first) so nothing is missed or repeated, and Tool #3 (Eliminate Possibilities) is how we drop any $y$ whose forced $x$ is not an integer or violates $y > x$.

Execute — Answer: A

#6 Guess and Check 6.SP.A.3 Step 1

Turn the average condition into a total.
The average of the five scores is $82$, so the sum of the five scores is $5 \times 82 = 410$.
Since the first four scores are all $x$ and the fifth is $y$, this gives $4x + y = 410$.

$$\dfrac{x + x + x + x + y}{5} = 82 \;\Longrightarrow\; 4x + y = 410$$

💡 Average $\times$ count $=$ total — the Grade 6 definition of mean as a single summary number lets us swap the average for an easy sum.

#3 Eliminate Possibilities 6.EE.B.8 Step 2

Find the window for $y$.
Because $y > x$, the last score must beat the average of $82$ (otherwise $x \ge 82$, forcing $4x \ge 328$ and $y \le 82 \le x$, contradiction).
Combined with the cap $y \le 100$, we get $83 \le y \le 100$.

$$y > 82 \text{ and } y \le 100 \;\Longrightarrow\; 83 \le y \le 100$$

💡 Writing inequalities like $y > 82$ and $y \le 100$ to trap a variable is the Grade 6 inequality-on-a-number-line idea.

#6 Guess and Check 4.OA.B.4 Step 3

For each candidate $y$ in that window, solve $4x = 410 - y$.
The fifth-test score $y$ is valid only when $410 - y$ is divisible by $4$ (so $x$ is a whole number) AND the resulting $x$ is less than $y$.
Divisibility by $4$ means $y$ must leave the same remainder mod $4$ as $410$.
Since $410 = 4 \cdot 102 + 2$, we need $y$ to leave remainder $2$ when divided by $4$.

$$4x = 410 - y \;\Longrightarrow\; y \text{ leaves remainder } 2 \text{ when divided by } 4$$

💡 Checking which numbers leave a particular remainder when divided by $4$ is just factor / multiple thinking from Grade 4.

#2 Make a Systematic List 4.OA.C.5 Step 4

List every $y$ from $83$ to $100$ that leaves remainder $2$ on division by $4$.
Starting at $86$ (which is $4 \cdot 21 + 2$) and adding $4$ each time: $86, 90, 94, 98$.
The next one, $102$, breaks the $\le 100$ cap.

$$y \in \{86,\; 90,\; 94,\; 98\}$$

💡 Generating numbers by the rule "start at $86$, add $4$" is exactly the Grade 4 shape-and-number pattern skill.

#6 Guess and Check 4.OA.A.3 Step 5

Verify each candidate by computing $x = (410 - y)/4$ and checking $y > x$.
All four pass, so all four are valid.
Count them.

$y = 86 \Rightarrow x = 81$, $y = 90 \Rightarrow x = 80$, $y = 94 \Rightarrow x = 79$, $y = 98 \Rightarrow x = 78$ — all integers with $y > x$. Count $= 4 \;\Rightarrow\; \textbf{(A)}$

💡 Plugging each candidate back into the equation and counting the survivors is multi-step word-problem reasoning from Grade 4.

[1] #6 6.SP.A.3 Turn the average condition into a total. The average of the five scores is $82$,

[2] #3 6.EE.B.8 Find the window for $y$. Because $y > x$, the last score must beat the average o

[3] #6 4.OA.B.4 For each candidate $y$ in that window, solve $4x = 410 - y$. The fifth-test scor

[4] #2 4.OA.C.5 List every $y$ from $83$ to $100$ that leaves remainder $2$ on division by $4$.

[5] #6 4.OA.A.3 Verify each candidate by computing $x = (410 - y)/4$ and checking $y > x$. All f

Review

Reasonableness: Sanity check the extremes. The smallest legal $y$ is $86$ with $x = 81$: average $= (4 \cdot 81 + 86)/5 = 410/5 = 82$ and $86 > 81$. The largest legal $y$ is $98$ with $x = 78$: average $= (4 \cdot 78 + 98)/5 = 410/5 = 82$ and $98 > 78$. Both ends work, and the four candidates step by $4$ (since each unit increase in $y$ must be matched by $\tfrac{1}{4}$ unit drop in $x$, which requires $y$ to change by multiples of $4$ to keep $x$ integer). Four values matches choice (A).

Alternative: Tool #11 (Work Backwards) starting from the boundaries: the largest possible $y$ is $100$, but $410 - 100 = 310$ is not divisible by $4$, so $y = 100$ fails. Step down by $1$ until divisibility works: $y = 98$ gives $x = 78$ — valid. Then jump down by $4$ each time ($94, 90, 86$). The next jump lands on $82$, which violates $y > x$ (would give $x = 82 = y$), so stop. Four values, answer (A).

CCSS standards used (min grade 6)

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number (Treating the average $82$ as a single summary of all five scores, which lets us replace it with the total $5 \times 82 = 410$ and write $4x + y = 410$.)
6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line (Trapping the last-test score with the two inequalities $y > 82$ (from $y > x$ and the average) and $y \le 100$ (problem cap) to bound the candidate set.)
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Recognizing which values of $y$ make $410 - y$ a multiple of $4$, so that $x = (410 - y)/4$ is a whole number.)
4.OA.C.5 Generate a number or shape pattern following a given rule (Listing the valid $y$ values by the rule "start at $86$, add $4$ each time, stop at $100$" to produce $86, 90, 94, 98$.)
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Verifying each candidate $y$ by computing the matching $x = (410 - y)/4$ with whole-number arithmetic and confirming $y > x$.)

⭐ This AMC 8 problem only needs Grade 6 mean-as-a-total reasoning plus a quick inequality you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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