AMC 10 · 2002 · #2
Grade 6 arithmeticFor the nonzero numbers a, b, and c, define
D(a,b,c)=a+b+cabc
Find D(2,4,6).
Pick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A rule named $D$ takes three nonzero numbers and returns $D(a,b,c)=\dfrac{abc}{a+b+c}$ — the product of the three numbers divided by their sum. Compute the value of $D(2,4,6)$.
Givens: The definition $D(a,b,c)=\dfrac{abc}{a+b+c}$; The specific inputs $a=2$, $b=4$, $c=6$; Answer choices: (A) $1$, (B) $2$, (C) $4$, (D) $6$, (E) $24$
Unknowns: The single number that $D(2,4,6)$ equals
Understand
Restated: A rule named $D$ takes three nonzero numbers and returns $D(a,b,c)=\dfrac{abc}{a+b+c}$ — the product of the three numbers divided by their sum. Compute the value of $D(2,4,6)$.
Givens: The definition $D(a,b,c)=\dfrac{abc}{a+b+c}$; The specific inputs $a=2$, $b=4$, $c=6$; Answer choices: (A) $1$, (B) $2$, (C) $4$, (D) $6$, (E) $24$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #3 Eliminate Possibilities
The formula is one fraction built from two independent pieces — a product on top and a sum on the bottom — so Tool #7 (Identify Subproblems) says: compute the numerator and the denominator on their own, then divide. Tool #3 (Eliminate Possibilities) flags the built-in trap: $24$ is exactly the top-only product $2\cdot4\cdot6$, so a solver who forgets to divide lands on (E); keeping the denominator in view rules that out.
Execute — Answer: C
6.EE.A.2 Step 1 Substitute the inputs into the rule
- The letters $a$, $b$, $c$ are just placeholders.
- Replace them with the given numbers: $a=2$, $b=4$, $c=6$.
- The rule $D(a,b,c)=\dfrac{abc}{a+b+c}$ becomes $D(2,4,6)=\dfrac{2\cdot4\cdot6}{2+4+6}$.
💡 A formula is a fill-in-the-blank: put each number where its letter sits.
3.OA.C.7 Step 2 Compute the numerator (product)
- The top is the product $abc=2\cdot4\cdot6$.
- Multiply in steps: $2\cdot4=8$, then $8\cdot6=48$.
- So the numerator is $48$.
💡 Multiply two numbers at a time and carry the running product forward.
2.NBT.B.5 Step 3 Compute the denominator (sum)
- The bottom is the sum $a+b+c=2+4+6$.
- Add in steps: $2+4=6$, then $6+6=12$.
- So the denominator is $12$.
💡 Stack the additions the same way, one pair at a time.
6.NS.B.2 Step 4 Divide top by bottom
- Now the fraction is $\dfrac{48}{12}$.
- Dividing, $48\div12=4$ because $12\cdot4=48$.
- So $D(2,4,6)=4$, which is choice (C).
- Notice $24$ in (E) is just the top $2\cdot4\cdot6$ with the division skipped — keeping the denominator in the picture avoids that trap.
💡 Dividing asks how many denominators fit in the numerator: four $12$s make $48$.
6.EE.A.2 The letters $a$, $b$, $c$ are just placeholders. Replace them with the given num 3.OA.C.7 The top is the product $abc=2\cdot4\cdot6$. Multiply in steps: $2\cdot4=8$, then 2.NBT.B.5 The bottom is the sum $a+b+c=2+4+6$. Add in steps: $2+4=6$, then $6+6=12$. So th 6.NS.B.2 Now the fraction is $\dfrac{48}{12}$. Dividing, $48\div12=4$ because $12\cdot4=4 Review
Reasonableness: A quick sanity check: the numerator $48$ is a few times bigger than the denominator $12$, so the answer should be a small whole number, not a huge one — $4$ fits, while $24$ (choice E) would need the denominator to be $2$, which it is not. Also $48\div12=4$ reverses cleanly to $12\cdot4=48$, confirming the division. The three tempting wrong answers each drop a piece of the work: $24$ skips dividing, $6$ is just the sum-adjacent trap, and $1$ or $2$ come from mis-adding or mis-multiplying.
Alternative: Cancel before multiplying. Write $\dfrac{2\cdot4\cdot6}{2+4+6}=\dfrac{2\cdot4\cdot6}{12}$ and note $12=2\cdot6$, so the $2$ and the $6$ on top cancel the $12$ on the bottom, leaving just the $4$: $\dfrac{2\cdot4\cdot6}{2\cdot6}=4$. This reaches (C) with no large multiplication at all.
CCSS standards used (min grade 6)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Reading the definition $D(a,b,c)=\dfrac{abc}{a+b+c}$ and substituting $a=2$, $b=4$, $c=6$ to evaluate it.)3.OA.C.7Fluently multiply and divide within 100 (Computing the numerator product $2\cdot4\cdot6=48$.)2.NBT.B.5Fluently add and subtract within 100 (Computing the denominator sum $2+4+6=12$.)6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm (Dividing the numerator by the denominator, $48\div12=4$.)
⭐ To use a formula, drop each number into its letter's spot, then finish the top and the bottom separately before you divide.
⭐ To use a formula, drop each number into its letter's spot, then finish the top and the bottom separately before you divide.
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