AMC 10 · 2007 · #2
Grade 7 arithmeticDefine a@b=ab−b2 and a#b=a+b−ab2. What is 6#26@2?
Pick an answer.
AMC 10 2007 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: Two made-up operations are defined by rules: $a@b = ab - b^{2}$ and $a\#b = a + b - ab^{2}$. Using these rules, find the value of the fraction $\dfrac{6@2}{6\#2}$.
Givens: The rule $a@b = ab - b^{2}$; The rule $a\#b = a + b - ab^{2}$; The target is the fraction $\dfrac{6@2}{6\#2}$; Answer choices: (A) $-\frac{1}{2}$, (B) $-\frac{1}{4}$, (C) $\frac{1}{8}$, (D) $\frac{1}{4}$, (E) $\frac{1}{2}$
Unknowns: The numerical value of $\dfrac{6@2}{6\#2}$
Understand
Restated: Two made-up operations are defined by rules: $a@b = ab - b^{2}$ and $a\#b = a + b - ab^{2}$. Using these rules, find the value of the fraction $\dfrac{6@2}{6\#2}$.
Givens: The rule $a@b = ab - b^{2}$; The rule $a\#b = a + b - ab^{2}$; The target is the fraction $\dfrac{6@2}{6\#2}$; Answer choices: (A) $-\frac{1}{2}$, (B) $-\frac{1}{4}$, (C) $\frac{1}{8}$, (D) $\frac{1}{4}$, (E) $\frac{1}{2}$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #3 Eliminate Possibilities
A fraction is just a top divided by a bottom, so Tool #7 (Identify Subproblems) splits the job into three clean pieces: compute the numerator $6@2$, compute the denominator $6\#2$, then divide. Each custom operation is only a substitution into its rule. Tool #3 (Eliminate Possibilities) gives a quick sanity check: once the top comes out positive and the bottom negative, the answer must be negative, which already rules out three of the five choices.
Execute — Answer: A
6.EE.A.2 Step 1 Read the two rules
- Each new symbol is a recipe.
- In $a@b = ab - b^{2}$, the letter $a$ marks the first number and $b$ the second.
- In $a\#b = a + b - ab^{2}$ the same holds.
- For both $6@2$ and $6\#2$ we substitute $a = 6$ and $b = 2$.
- Handle the top and the bottom of the fraction as two separate problems.
💡 A made-up symbol is just an instruction sheet — copy the numbers into the slots the letters mark.
6.EE.A.1 Step 2 Compute the numerator
- Apply the $@$ rule: $6@2 = (6)(2) - 2^{2}$.
- First the power: $2^{2} = 4$.
- Then the product: $6 \times 2 = 12$.
- Subtract: $12 - 4 = 8$.
- So the numerator equals $8$.
💡 Do the exponent first, then the multiply, then the subtract — order of operations keeps the recipe honest.
7.NS.A.1 Step 3 Compute the denominator
- Apply the $\#$ rule: $6\#2 = 6 + 2 - 6\cdot 2^{2}$.
- Again the power first: $2^{2} = 4$, so $6\cdot 2^{2} = 6 \times 4 = 24$.
- The front part is $6 + 2 = 8$.
- Now $8 - 24 = -16$, a negative number.
- So the denominator equals $-16$.
💡 Subtracting a bigger number from a smaller one drops you below zero — the result is negative.
7.NS.A.2 Step 4 Divide top by bottom
- The fraction is $\dfrac{6@2}{6\#2} = \dfrac{8}{-16}$.
- A positive divided by a negative is negative, and $\dfrac{8}{16} = \dfrac{1}{2}$, so the value is $-\dfrac{1}{2}$.
- That matches choice (A).
💡 Same digits, opposite signs on top and bottom means the fraction reduces to the simple size with a minus in front.
6.EE.A.2 Each new symbol is a recipe. In $a@b = ab - b^{2}$, the letter $a$ marks the fir 6.EE.A.1 Apply the $@$ rule: $6@2 = (6)(2) - 2^{2}$. First the power: $2^{2} = 4$. Then t 7.NS.A.1 Apply the $\#$ rule: $6\#2 = 6 + 2 - 6\cdot 2^{2}$. Again the power first: $2^{2 7.NS.A.2 The fraction is $\dfrac{6@2}{6\#2} = \dfrac{8}{-16}$. A positive divided by a ne Review
Reasonableness: Recheck each piece: $6@2 = 12 - 4 = 8$ and $6\#2 = 8 - 24 = -16$, so the ratio is $8/(-16) = -1/2$. The numerator is positive and the denominator is negative, so the answer has to be negative — that alone leaves only (A) $-\frac{1}{2}$ and (B) $-\frac{1}{4}$, and the exact division picks (A). The size is also sensible: $8$ is half of $16$, so the magnitude $\frac{1}{2}$ is expected.
Alternative: Use Eliminate Possibilities before finishing the arithmetic. Notice the top $6@2 = 12 - 4$ is clearly positive and the bottom $6\#2 = 8 - 24$ is clearly negative, so the fraction is negative — cross out (C), (D), (E) immediately. Then a single division $8 \div (-16)$ separates the two survivors and lands on (A).
CCSS standards used (min grade 7)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Reading the custom rules $a@b$ and $a\#b$ and substituting $a = 6$, $b = 2$ into them.)6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Evaluating the squares $2^{2} = 4$ inside both operations and following order of operations.)7.NS.A.1Apply and extend understanding of addition and subtraction to rational numbers (Computing $8 - 24 = -16$ for the denominator, where the subtraction crosses below zero.)7.NS.A.2Apply and extend understanding of multiplication and division of rational numbers (Dividing $8$ by $-16$ and reducing the signed fraction to $-\frac{1}{2}$.)
⭐ A strange new symbol is just a recipe: put the numbers where the letters are, follow order of operations, and work the top and bottom of a fraction one at a time.
⭐ A strange new symbol is just a recipe: put the numbers where the letters are, follow order of operations, and work the top and bottom of a fraction one at a time.
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