AMC 8 · 2001 · #2
Grade 4 arithmeticnumber-theoryProblem
I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Two whole numbers have product $24$ and sum $11$. Find the larger of the two numbers.
Givens: The two numbers are whole numbers (non-negative integers); Their product is $24$; Their sum is $11$; Answer choices: (A) $3$, (B) $4$, (C) $6$, (D) $8$, (E) $12$
Unknowns: The larger of the two whole numbers
Understand
Restated: Two whole numbers have product $24$ and sum $11$. Find the larger of the two numbers.
Givens: The two numbers are whole numbers (non-negative integers); Their product is $24$; Their sum is $11$; Answer choices: (A) $3$, (B) $4$, (C) $6$, (D) $8$, (E) $12$
Plan
Primary tool: #2 Make a List
Secondary: #6 Guess and Check
Because the product is $24$, the two numbers must be a factor pair of $24$. There are only four positive whole-number factor pairs, so Tool #2 (Make a List) lets us write them all down in a few seconds. For each pair, Tool #6 (Guess and Check) does the rest: just add the two numbers and see whether the sum is $11$. No algebra is needed — the short list does all the work.
Execute — Answer: D
4.OA.B.4 Step 1 - List all positive whole-number factor pairs of $24$.
- Start with $1$ and walk up through the divisors of $24$.
💡 Grade 4 factor-pair work: list divisors in order so no pair is missed.
4.OA.A.3 Step 2 Add the two numbers in each pair and compare each sum to $11$.
💡 Only one row meets both conditions, so that pair must be the answer.
4.OA.A.3 Step 3 - The matching pair is $3$ and $8$.
- The question asks for the larger number.
💡 Once the right pair is found, picking the larger value is a final reading-the-question check.
4.OA.B.4 List all positive whole-number factor pairs of $24$. Start with $1$ and walk up 4.OA.A.3 Add the two numbers in each pair and compare each sum to $11$. 4.OA.A.3 The matching pair is $3$ and $8$. The question asks for the larger number. Review
Reasonableness: Verify both conditions on the chosen pair: $3 \times 8 = 24$ matches the product, and $3 + 8 = 11$ matches the sum. Both hold, so $8$ is correct. Choice (E) $12$ can be ruled out quickly because its partner under product $24$ is $2$, and $2 + 12 = 14 \ne 11$. Choices (A) $3$ and (D) $8$ name the two members of the right pair; the question asks for the larger, so (D) wins over (A).
Alternative: Tool #6 (Guess and Check) on the answer choices: try each choice as the larger number and see whether the partner forced by sum $11$ multiplies to $24$. (A) $3$: partner $8$, $3 \times 8 = 24$ — works, but $3$ is the smaller. (D) $8$: partner $3$, same pair, and $8$ is the larger. Same answer (D), reached by checking the choices instead of listing factor pairs.
CCSS standards used (min grade 4)
4.OA.B.4Find all factor pairs for a whole number in the range 1-100; recognize that a whole number is a multiple of each of its factors (Listing the four factor pairs of $24$: $(1,24), (2,12), (3,8), (4,6)$.)4.OA.A.3Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations (Adding the two numbers in each factor pair, comparing each sum to $11$, and then selecting the larger member of the winning pair.)
⭐ When a problem gives both the product and the sum of two whole numbers, list the factor pairs of the product first — there are usually only a handful — and pick the one whose sum matches. Here the factor pair $(3, 8)$ adds to $11$, so the larger number is $8$, choice (D).
⭐ When a problem gives both the product and the sum of two whole numbers, list the factor pairs of the product first — there are usually only a handful — and pick the one whose sum matches. Here the factor pair $(3, 8)$ adds to $11$, so the larger number is $8$, choice (D).