AMC 10 · 2011 · #1

Easy mode Grade 5

Problem

Add the even numbers $2$ , $4$ , and $6$ to get one total. Add the odd numbers $1$ , $3$ , and $5$ to get another total.

Make a fraction with the even total on top and the odd total on the bottom. Then make a second fraction with those same two totals flipped over.

Subtract the second fraction from the first. What do you get?

Pick an answer.

(A)

-1

(B)

$\frac{5}{36}$

(C)

$\frac{7}{12}$

(D)

$\frac{147}{60}$

(E)

$\frac{43}{3}$

AMC 10 2011 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.

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Toolkit + CCSS Solution

Understand

Restated: Evaluate $\dfrac{2+4+6}{1+3+5} - \dfrac{1+3+5}{2+4+6}$, a single fraction built from two smaller fractions that are reciprocals of each other.

Givens: The first fraction has the even sum $2+4+6$ on top and the odd sum $1+3+5$ on the bottom; The second fraction is the same two sums flipped: $1+3+5$ over $2+4+6$; Answer choices: (A) $-1$, (B) $\dfrac{5}{36}$, (C) $\dfrac{7}{12}$, (D) $\dfrac{147}{60}$, (E) $\dfrac{43}{3}$

Unknowns: The single value the whole expression equals

Understand

Restated: Evaluate $\dfrac{2+4+6}{1+3+5} - \dfrac{1+3+5}{2+4+6}$, a single fraction built from two smaller fractions that are reciprocals of each other.

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities

The expression stacks several small jobs — two sums, two fractions, one subtraction — so Tool #7 (Identify Subproblems) does them one at a time: add the numbers, build and simplify each fraction, then subtract. Tool #3 (Eliminate Possibilities) gives a fast sanity check: the bigger fraction ($\tfrac43$) comes first and the smaller ($\tfrac34$) is subtracted, so the result is positive but small — that alone kills the negative (A) and the large answers (D) and (E).

Execute — Answer: C

#7 Identify Subproblems 2.NBT.B.5 Step 1

Add the evens and the odds

Handle the four little sums first.
The evens give $2+4+6=12$ and the odds give $1+3+5=9$.
These are the only two numbers the whole problem is really about, so replace every sum with its value.

$$2+4+6 = 12, \qquad 1+3+5 = 9$$

💡 Turn each pile of numbers into one number first, so the fractions become plain and easy to read.

#7 Identify Subproblems 4.NF.A.1 Step 2

Build and simplify the two fractions

Now the expression is $\dfrac{12}{9} - \dfrac{9}{12}$.
Simplify each fraction by dividing top and bottom by a shared factor: $\dfrac{12}{9}=\dfrac{4}{3}$ (divide by $3$) and $\dfrac{9}{12}=\dfrac{3}{4}$ (divide by $3$).
Notice the two fractions are reciprocals — one is the other flipped over.

$$\dfrac{12}{9} = \dfrac{4}{3}, \qquad \dfrac{9}{12} = \dfrac{3}{4}$$

💡 Dividing top and bottom by the same number keeps a fraction's value but makes the numbers small enough to work with.

#3 Eliminate Possibilities 5.NF.A.1 Step 3

Subtract using a common denominator

Subtract $\dfrac{4}{3} - \dfrac{3}{4}$.
The lowest common denominator of $3$ and $4$ is $12$, so rewrite $\dfrac{4}{3}=\dfrac{16}{12}$ and $\dfrac{3}{4}=\dfrac{9}{12}$, then subtract the tops: $16-9=7$.
The result is $\dfrac{7}{12}$, which is positive and less than $1$ just as expected, matching choice (C).

$$\dfrac{4}{3} - \dfrac{3}{4} = \dfrac{16}{12} - \dfrac{9}{12} = \dfrac{7}{12}\;\Rightarrow\;\textbf{(C)}$$

💡 Two fractions can only be subtracted once they share a denominator; then you just subtract the numerators.

[1] #7 2.NBT.B.5 Handle the four little sums first. The evens give $2+4+6=12$ and the odds give $

[2] #7 4.NF.A.1 Now the expression is $\dfrac{12}{9} - \dfrac{9}{12}$. Simplify each fraction by

[3] #3 5.NF.A.1 Subtract $\dfrac{4}{3} - \dfrac{3}{4}$. The lowest common denominator of $3$ and

Review

Reasonableness: The expression is $x - \tfrac1x$ with $x=\tfrac43$, which is a bit bigger than $1$. Subtracting a number from its slightly-smaller reciprocal gives a small positive result, so an answer between $0$ and $1$ is exactly right — $\tfrac{7}{12}\approx 0.58$ fits. This immediately rules out the negative (A), the greater-than-$2$ (D), and the huge (E); and $\tfrac{5}{36}\approx 0.14$ (B) is far too small.

Alternative: Check with decimals: $\tfrac{12}{9}=1.333\ldots$ and $\tfrac{9}{12}=0.75$, so $1.333\ldots-0.75 = 0.5833\ldots$. Since $\tfrac{7}{12}=0.5833\ldots$ matches exactly while none of the other choices are near $0.58$, the answer (C) is confirmed.

CCSS standards used (min grade 5)

2.NBT.B.5 Fluently add and subtract within 100 (Computing the two sums $2+4+6=12$ and $1+3+5=9$.)
4.NF.A.1 Explain why a fraction is equivalent to another fraction (Simplifying $\dfrac{12}{9}=\dfrac{4}{3}$ and $\dfrac{9}{12}=\dfrac{3}{4}$ by dividing top and bottom by $3$.)
5.NF.A.1 Add and subtract fractions with unlike denominators (Rewriting $\dfrac{4}{3}-\dfrac{3}{4}$ over the common denominator $12$ to get $\dfrac{7}{12}$.)

⭐ Add each little pile into one number first, simplify the fractions, then match denominators before subtracting.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Add the evens and the odds

Build and simplify the two fractions

Subtract using a common denominator

Review

CCSS standards used (min grade 5)

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