AMC 10 · 2009 · #2
Grade 6 arithmeticWhich of the following is equal to 21−3131−41?
Pick an answer.
AMC 10 2009 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: Find the single value of the compound fraction $\dfrac{\frac{1}{3}-\frac{1}{4}}{\frac{1}{2}-\frac{1}{3}}$, where the top and the bottom are each a difference of two fractions.
Givens: The numerator is $\frac{1}{3}-\frac{1}{4}$; The denominator is $\frac{1}{2}-\frac{1}{3}$; The big fraction bar means: divide the numerator by the denominator; Answer choices: (A) $\frac{1}{4}$, (B) $\frac{1}{3}$, (C) $\frac{1}{2}$, (D) $\frac{2}{3}$, (E) $\frac{3}{4}$
Unknowns: The single number that the whole compound fraction equals
Understand
Restated: Find the single value of the compound fraction $\dfrac{\frac{1}{3}-\frac{1}{4}}{\frac{1}{2}-\frac{1}{3}}$, where the top and the bottom are each a difference of two fractions.
Givens: The numerator is $\frac{1}{3}-\frac{1}{4}$; The denominator is $\frac{1}{2}-\frac{1}{3}$; The big fraction bar means: divide the numerator by the denominator; Answer choices: (A) $\frac{1}{4}$, (B) $\frac{1}{3}$, (C) $\frac{1}{2}$, (D) $\frac{2}{3}$, (E) $\frac{3}{4}$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #3 Eliminate Possibilities
The expression is really three tasks stacked together: simplify the top, simplify the bottom, then divide one by the other. Tool #7 (Identify Subproblems) handles them one at a time instead of all at once, so no common-denominator step gets skipped. Tool #3 (Eliminate Possibilities) is a quick sanity guard: the top ($\frac{1}{12}$) is much smaller than the bottom ($\frac{1}{6}$) but not tiny, so the ratio should be a modest fraction near one-half — which lets us reject the extreme-looking choices at a glance.
Execute — Answer: C
5.NF.A.1 Step 1 Simplify the top
- Work out the numerator $\frac{1}{3}-\frac{1}{4}$ by itself.
- The denominators $3$ and $4$ are different, so rewrite both over their least common denominator $12$: $\frac{1}{3}=\frac{4}{12}$ and $\frac{1}{4}=\frac{3}{12}$.
- Subtracting gives $\frac{4}{12}-\frac{3}{12}=\frac{1}{12}$.
💡 You can only subtract fractions once the pieces are the same size, so give them a common denominator first.
5.NF.A.1 Step 2 Simplify the bottom
- Now the denominator $\frac{1}{2}-\frac{1}{3}$.
- The least common denominator of $2$ and $3$ is $6$, so $\frac{1}{2}=\frac{3}{6}$ and $\frac{1}{3}=\frac{2}{6}$.
- Subtracting gives $\frac{3}{6}-\frac{2}{6}=\frac{1}{6}$.
💡 Same idea again: match the denominators, then the tops just subtract.
6.NS.A.1 Step 3 Divide top by bottom
- The big fraction bar now means $\frac{1}{12}$ divided by $\frac{1}{6}$.
- Dividing by a fraction is multiplying by its flip, so $\frac{1}{12}\div\frac{1}{6}=\frac{1}{12}\times\frac{6}{1}=\frac{6}{12}$.
- Both parts share a factor of $6$, so $\frac{6}{12}=\frac{1}{2}$.
- That is choice (C).
- It sits right at one-half, which rules out the smaller (A), (B) and the larger (D), (E).
💡 Dividing by $\frac{1}{6}$ asks how many sixths fit in the top, and flipping-and-multiplying answers that in one move.
5.NF.A.1 Work out the numerator $\frac{1}{3}-\frac{1}{4}$ by itself. The denominators $3$ 5.NF.A.1 Now the denominator $\frac{1}{2}-\frac{1}{3}$. The least common denominator of $ 6.NS.A.1 The big fraction bar now means $\frac{1}{12}$ divided by $\frac{1}{6}$. Dividing Review
Reasonableness: A quick size check confirms it: the top $\frac{1}{12}$ is exactly half of the bottom $\frac{1}{6}$ (since $\frac{1}{6}=\frac{2}{12}$), so their ratio has to be $\frac{1}{2}$. Because top and bottom are close in size and the top is the smaller one, the answer had to be a fraction a bit under $1$, not something tiny like $\frac{1}{4}$ nor near $\frac{3}{4}$ — and $\frac{1}{2}$ fits.
Alternative: Skip the division by clearing all the little fractions at once: multiply the top and the bottom of the compound fraction by $12$. The top becomes $12(\frac{1}{3}-\frac{1}{4})=4-3=1$ and the bottom becomes $12(\frac{1}{2}-\frac{1}{3})=6-4=2$, giving $\frac{1}{2}$ directly — the same choice (C).
CCSS standards used (min grade 6)
5.NF.A.1Add and subtract fractions with unlike denominators (Rewriting $\frac{1}{3}-\frac{1}{4}$ over $12$ and $\frac{1}{2}-\frac{1}{3}$ over $6$ to get $\frac{1}{12}$ and $\frac{1}{6}$.)6.NS.A.1Interpret and compute quotients of fractions and solve word problems (Dividing the top by the bottom: $\frac{1}{12}\div\frac{1}{6}=\frac{1}{12}\times\frac{6}{1}=\frac{1}{2}$.)
⭐ Simplify the top and the bottom into single fractions first, then a big fraction bar just means divide — and dividing by a fraction is flip-and-multiply.
⭐ Simplify the top and the bottom into single fractions first, then a big fraction bar just means divide — and dividing by a fraction is flip-and-multiply.
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