AMC 8 · 2011 · #21

Easy mode Grade 4

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Problem

A bunch of students try to guess Norb's age. Their guesses are:
$24,\ 28,\ 30,\ 32,\ 36,\ 38,\ 41,\ 44,\ 47,\ 49.$

Norb says three things about his real age:

At least half of those guesses are too low.
Exactly two of the guesses are off by $1$ (one is $1$ above his age, the other is $1$ below).
His age is a prime number.

How old is Norb?

$\textbf{(A) }29\qquad\textbf{(B) }31\qquad\textbf{(C) }37\qquad\textbf{(D) }43\qquad\textbf{(E) }48$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Ten students guess Norb's age: $24, 28, 30, 32, 36, 38, 41, 44, 47, 49$. Norb tells them three facts about the right answer: (1) at least half of the guesses are too low, (2) exactly two guesses are off by one (one above and one below his age), and (3) his age is a prime number. Find Norb's age.

Givens: Guesses: $\{24, 28, 30, 32, 36, 38, 41, 44, 47, 49\}$ (ten numbers); At least $5$ of the $10$ guesses are strictly less than Norb's age; Exactly two guesses differ from Norb's age by $1$; Norb's age is a prime number; Answer choices: (A) $29$, (B) $31$, (C) $37$, (D) $43$, (E) $48$

Unknowns: Norb's age (one of the five answer choices)

Understand

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #2 Make a Systematic List

Three clean conditions slice through a finite list — perfect setup for Tool #3 (Eliminate Possibilities). Tool #2 (Systematic List) builds the short candidate set from the "off by one" clue: scan the guess list for pairs that differ by exactly $2$, and the age sits squarely in the middle of each such pair. Then the "at least half too low" condition kills the candidates that are too small, and the "prime" condition finishes off whatever remains. No algebra, just three filters.

Execute — Answer: C

#3 Eliminate Possibilities 3.OA.A.3 Step 1

Turn "at least half" into a number.
Half of the $10$ guesses is $5$, so at least $5$ guesses are strictly less than Norb's age.
Look at the sorted list and count up $5$ from the bottom: $24, 28, 30, 32, 36$.
For all five of these to be too low, Norb's age must be greater than $36$.

$$\tfrac{1}{2} \times 10 = 5 \;\Rightarrow\; \text{age} > 36$$

💡 Computing half of $10$ and counting the first five values is Grade 3 multiplication/division reasoning.

#2 Make a Systematic List 4.OA.A.3 Step 2

Apply the "off by one" clue with a Systematic List.
If Norb's age is $x$, then both $x-1$ and $x+1$ must appear in the guess list.
Scan the list for two guesses that differ by exactly $2$ — the age sits exactly in the middle.
Pairs in the list with difference $2$ are $(28, 30)$, $(30, 32)$, $(36, 38)$, and $(47, 49)$.

$$\{(28,30),\ (30,32),\ (36,38),\ (47,49)\} \;\Rightarrow\; x \in \{29,\ 31,\ 37,\ 48\}$$

💡 Listing every "$x-1$ and $x+1$ both in the list" candidate turns the clue into a finite checklist — the core move of Tool #2.

#3 Eliminate Possibilities 4.NBT.A.2 Step 3

Combine the two clues.
The candidate ages from the "off by one" pairs are $\{29, 31, 37, 48\}$, but the "at least half too low" clue requires age $> 36$.
Cross out $29$ and $31$.
Two candidates survive.

$$\{29,\ 31,\ 37,\ 48\} \cap \{x : x > 36\} = \{37,\ 48\}$$

💡 Comparing multi-digit whole numbers to a threshold is Grade 4 place-value reasoning.

#3 Eliminate Possibilities 4.OA.B.4 Step 4

Use the prime clue to break the tie.
Check the two survivors: $37$ has no divisors other than $1$ and $37$, so it is prime.
$48 = 2 \times 24$, so $48$ is composite.
Only $37$ is prime.

$$37 \text{ prime},\quad 48 = 2 \times 24 \text{ composite}$$

💡 Identifying a number as prime or composite within $100$ is exactly the Grade 4 factor-pair standard.

#3 Eliminate Possibilities 4.OA.A.3 Step 5

Read off the answer.
Norb is $37$ years old, which is choice $\textbf{(C)}$.

$$\text{age} = 37 \;\Rightarrow\; \textbf{(C)}$$

💡 After three filters leave one candidate, that candidate is forced — the cleanest finish for Tool #3.

[1] #3 3.OA.A.3 Turn "at least half" into a number. Half of the $10$ guesses is $5$, so at least

[2] #2 4.OA.A.3 Apply the "off by one" clue with a Systematic List. If Norb's age is $x$, then b

[3] #3 4.NBT.A.2 Combine the two clues. The candidate ages from the "off by one" pairs are ${29,

[4] #3 4.OA.B.4 Use the prime clue to break the tie. Check the two survivors: $37$ has no diviso

[5] #3 4.OA.A.3 Read off the answer. Norb is $37$ years old, which is choice $\textbf{(C)}$.

Review

Reasonableness: Verify all three clues against $37$. (1) Too-low guesses: $24, 28, 30, 32, 36$ — that's $5$ out of $10$, satisfying "at least half." (2) Off by one: $37 - 1 = 36$ and $37 + 1 = 38$ are both in the list — exactly two guesses off by one. (3) $37$ is prime. All three clues check out, and none of the other answer choices does — for instance, $43$ would need $42$ and $44$ both in the list, but $42$ isn't there; $29$ and $31$ both fail the "more than $36$" condition.

Alternative: Tool #6 (Guess and Check) directly on the five answer choices. Test each: $29$ — only $4$ guesses are below it ($24, 28$), fails "at least half too low." $31$ — same problem, only $4$ guesses below. $37$ — $5$ guesses below, $36$ and $38$ in list, prime. Works. $43$ — $42$ not in list, fails "off by one." $48$ — $47$ and $49$ in list and $7$ guesses below, but $48$ is not prime. Only $37$ passes all three tests.

CCSS standards used (min grade 4)

3.OA.A.3 Solve multiplication and division word problems within 100 (Translating "at least half of $10$ guesses" into "at least $5$ guesses" via the basic computation $\tfrac{1}{2} \times 10 = 5$.)
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Coordinating the three separate clues (too low, off by one, prime) as a multi-step filtering plan and reading the surviving candidate as the answer.)
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols (Comparing the candidate ages $\{29, 31, 37, 48\}$ against the threshold $36$ to eliminate $29$ and $31$.)
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Checking that $37$ is prime and $48 = 2 \times 24$ is composite, which breaks the tie between the two surviving candidates.)

⭐ This AMC 8 problem only needs Grade 4 prime-or-composite checking and a couple of "is it bigger than $36$?" comparisons you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 4)

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