The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give 24, while the other two multiply to 30. What is the sum of the ages of Jonie's four cousins?
Try it yourself first — the explanation is most useful after you’ve attempted it.
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Toolkit + CCSS Solution
Understand
Restated: Jonie has four cousins whose ages are four different single-digit positive integers (so each age is in $\{1, 2, \dots, 9\}$). One pair of ages multiplies to $24$, the other pair multiplies to $30$. Find the sum of the four ages.
Givens: Four ages are distinct positive integers in $\{1, 2, \dots, 9\}$; Two ages multiply to $24$; The other two ages multiply to $30$; Answer choices: (A) $21$, (B) $22$, (C) $23$, (D) $24$, (E) $25$
Unknowns: The sum of the four ages
Understand
Restated: Jonie has four cousins whose ages are four different single-digit positive integers (so each age is in $\{1, 2, \dots, 9\}$). One pair of ages multiplies to $24$, the other pair multiplies to $30$. Find the sum of the four ages.
Givens: Four ages are distinct positive integers in $\{1, 2, \dots, 9\}$; Two ages multiply to $24$; The other two ages multiply to $30$; Answer choices: (A) $21$, (B) $22$, (C) $23$, (D) $24$, (E) $25$
Plan
Primary tool: #2 Make a Systematic List
Secondary: #3 Eliminate Possibilities
Tool #2 (Systematic List) — list every single-digit factor pair of $24$ in increasing order, then every single-digit factor pair of $30$. The set of pairs is tiny, so listing finishes quickly. Tool #3 (Eliminate) handles the "distinct" constraint: combine each $24$-pair with each $30$-pair, drop any combination that repeats an age, and the survivor is forced.
Execute — Answer: B
#2 Make a Systematic List 4.OA.B.4Step 1
List the single-digit factor pairs of $24$.
Walk from the smallest factor up: $1 \times 24$ (drop, $24$ is two digits), $2 \times 12$ (drop), $3 \times 8$ (keep), $4 \times 6$ (keep).
So the $24$-pair is $\{3, 8\}$ or $\{4, 6\}$.
$$24 = 3 \times 8 = 4 \times 6$$
💡 List factor pairs in order — Grade 4 "find all factor pairs of a whole number".
💡 Add four small whole numbers — Grade 2 within-$100$ addition.
[1]
#2 4.OA.B.4List the single-digit factor pairs of $24$. Walk from the smallest factor up: $1
[2]
#2 4.OA.B.4List the single-digit factor pairs of $30$. $1 \times 30$ (drop), $2 \times 15$
[3]
#3 5.OA.B.3Combine the $24$-pair with $\{5, 6\}$ and drop the case with a repeat. Trying $\
[4]
#2 2.NBT.B.5Add the four ages. $3 + 5 + 6 + 8 = 22$, which is choice (B).
Review
Reasonableness: Double-check the two products: $3 \times 8 = 24$ ✓ and $5 \times 6 = 30$ ✓. All four ages $\{3, 5, 6, 8\}$ are distinct single-digit positive integers ✓. Sum $22$ lies right in the middle of the answer choices $21$–$25$, which is plausible for four ages around $5$–$6$ each.
Alternative: Tool #6 (Guess and Check) — guess the sum directly. The smallest plausible pair is $\{3, 8\}$ summing $11$, and $\{5, 6\}$ summing $11$, so the total is $11 + 11 = 22$ without listing all cases. Same answer (B).
CCSS standards used (min grade 5)
2.NBT.B.5 Fluently add and subtract within 100 (Computing $3 + 5 + 6 + 8 = 22$ at the end.)
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Listing every single-digit factor pair of $24$ and of $30$.)
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships (Combining each $24$-pair with the $30$-pair and applying the "all distinct" rule to pick the valid set.)
⭐ This AMC 10 problem only needs Grade 5 "list and compare factor pairs" — $24 = 3 \times 8$ or $4 \times 6$, $30 = 5 \times 6$, and the "all distinct" rule forces $\{3, 5, 6, 8\}$ with sum $22$!
⭐ This AMC 10 problem only needs Grade 5 "list and compare factor pairs" — $24 = 3 \times 8$ or $4 \times 6$, $30 = 5 \times 6$, and the "all distinct" rule forces $\{3, 5, 6, 8\}$ with sum $22$!