AMC 8 · 2012 · #20

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

fraction-arithmeticfraction-decimal-conversionratio-proportion identify-subproblemseasier-related-problem ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 3 insights

Problem

What is the correct ordering of the three numbers $\frac{5}{19}$ , $\frac{7}{21}$ , and $\frac{9}{23}$ , in increasing order?

Pick an answer.

(A)

$\frac{9}{23}<\frac{7}{21}<\frac{5}{19}$

(B)

$\frac{5}{19}<\frac{7}{21}<\frac{9}{23}$

(C)

$\frac{9}{23}<\frac{5}{19}<\frac{7}{21}$

(D)

$\frac{5}{19}<\frac{9}{23}<\frac{7}{21}$

(E)

$\frac{7}{21}<\frac{5}{19}<\frac{9}{23}$

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Toolkit + CCSS Solution

Understand

Restated: Order the three fractions $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$ from least to greatest, and pick the matching answer choice.

Givens: Three fractions: $\frac{5}{19}$, $\frac{7}{21}$, $\frac{9}{23}$; Answer choices give the five possible orderings of these three fractions; $\frac{7}{21}$ simplifies (numerator and denominator share a factor of $7$)

Unknowns: The increasing-order arrangement of the three fractions, expressed as one of choices (A)-(E)

Understand

Restated: Order the three fractions $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$ from least to greatest, and pick the matching answer choice.

Plan

Primary tool: #5 Look for a Pattern

Secondary: #9 Solve an Easier Related Problem

Before doing any arithmetic, scan the three fractions for structure. The numerators are $5, 7, 9$ and the denominators are $19, 21, 23$ — each denominator is exactly $14$ more than its numerator. That turns the question into one easy-to-reason-about family: $\frac{n}{n+14}$. Tool #5 (Look for a Pattern) lets us answer the whole ordering with one observation about that family, without grinding through three pairwise cross-multiplications. Tool #9 (Easier Related Problem) supports it: $\frac{7}{21}$ simplifies to the familiar landmark $\frac{1}{3}$, which we can use as a sanity check against the other two.

Execute — Answer: B

#5 Look for a Pattern 4.NF.A.1 Step 1

Spot the shared structure.
Write each denominator as numerator $+ 14$: $\frac{5}{19} = \frac{5}{5+14}$, $\frac{7}{21} = \frac{7}{7+14}$, $\frac{9}{23} = \frac{9}{9+14}$.
Every fraction has the form $\frac{n}{n+14}$.

$$\frac{5}{19},\;\frac{7}{21},\;\frac{9}{23} \;\longleftrightarrow\; \frac{n}{n+14} \text{ for } n = 5, 7, 9$$

💡 Naming the common form is the Tool #5 move — once the pattern is named, one rule will sort all three.

#5 Look for a Pattern 5.NF.B.3 Step 2

Rewrite the family to make the comparison obvious.
Use the identity $\frac{n}{n+14} = 1 - \frac{14}{n+14}$.
The $14$ on top is fixed, so the size of $\frac{14}{n+14}$ depends only on its denominator $n+14$.

$$\frac{n}{n+14} = \frac{(n+14) - 14}{n+14} = 1 - \frac{14}{n+14}$$

💡 Splitting a fraction into "$1$ minus a leftover piece" turns an ordering problem into a much simpler ordering of the leftover pieces.

#5 Look for a Pattern 4.NF.A.2 Step 3

Order the leftover pieces.
As $n$ grows from $5$ to $7$ to $9$, the denominator $n+14$ grows from $19$ to $21$ to $23$, so $\frac{14}{n+14}$ shrinks.
A smaller piece subtracted from $1$ leaves a larger result.

$$\frac{14}{19} > \frac{14}{21} > \frac{14}{23} \;\Longrightarrow\; 1 - \frac{14}{19} < 1 - \frac{14}{21} < 1 - \frac{14}{23}$$

💡 Same numerator, bigger denominator means smaller fraction — a Grade 4 fraction-sense fact, just used at scale.

#5 Look for a Pattern 6.NS.C.7 Step 4

Translate the leftover-piece order back to the original fractions.
Bigger $n$ gives a smaller leftover, so bigger $n$ gives a larger value of $\frac{n}{n+14}$.

$$\frac{5}{19} < \frac{7}{21} < \frac{9}{23}$$

💡 Ordering a list of numbers by ordering a single varying quantity is Grade 6 rational-number reasoning.

#9 Solve an Easier Related Problem 4.NF.A.2 Step 5

Sanity check the middle fraction against a familiar landmark with Tool #9.
$\frac{7}{21} = \frac{1}{3}$.
Cross-check: $\frac{5}{19}$ vs $\frac{1}{3}$: $5 \times 3 = 15 < 19 \times 1 = 19$, so $\frac{5}{19} < \frac{1}{3}$.
And $\frac{9}{23}$ vs $\frac{1}{3}$: $9 \times 3 = 27 > 23 \times 1 = 23$, so $\frac{9}{23} > \frac{1}{3}$.
Both checks agree with the pattern result.

$$\frac{5}{19} < \frac{1}{3} = \frac{7}{21} < \frac{9}{23} \;\Rightarrow\; \textbf{(B)}$$

💡 Reducing the middle fraction to the easier $\frac{1}{3}$ is Tool #9 — solve the easier related comparison first, then confirm the harder claim.

[1] #5 4.NF.A.1 Spot the shared structure. Write each denominator as numerator $+ 14$: $\frac{5}

[2] #5 5.NF.B.3 Rewrite the family to make the comparison obvious. Use the identity $\frac{n}{n+

[3] #5 4.NF.A.2 Order the leftover pieces. As $n$ grows from $5$ to $7$ to $9$, the denominator

[4] #5 6.NS.C.7 Translate the leftover-piece order back to the original fractions. Bigger $n$ gi

[5] #9 4.NF.A.2 Sanity check the middle fraction against a familiar landmark with Tool #9. $\fra

Review

Reasonableness: Convert to decimals as a fast double-check: $\frac{5}{19} \approx 0.263$, $\frac{7}{21} = \frac{1}{3} \approx 0.333$, $\frac{9}{23} \approx 0.391$. These decimals are clearly in increasing order, matching $\frac{5}{19} < \frac{7}{21} < \frac{9}{23}$ and confirming choice (B). The values also fit the pattern: all are less than $1$ and grow toward $1$ as $n$ grows, exactly as $\frac{n}{n+14}$ predicts.

Alternative: Tool #6 (Guess and Check) via pairwise cross-multiplication on the original three fractions. Compare $\frac{5}{19}$ and $\frac{7}{21}$: $5 \times 21 = 105$ vs $7 \times 19 = 133$, so $\frac{5}{19} < \frac{7}{21}$. Compare $\frac{7}{21}$ and $\frac{9}{23}$: $7 \times 23 = 161$ vs $9 \times 21 = 189$, so $\frac{7}{21} < \frac{9}{23}$. Chain: $\frac{5}{19} < \frac{7}{21} < \frac{9}{23}$, the same answer (B). This is more arithmetic but uses no pattern recognition.

CCSS standards used (min grade 6)

4.NF.A.1 Explain equivalent fractions and recognize fraction structure (Rewriting each fraction in the common form $\frac{n}{n+14}$ to expose the shared structure across the three fractions.)
4.NF.A.2 Compare two fractions with different numerators and different denominators (Using "same numerator, bigger denominator means smaller fraction" on $\frac{14}{19}, \frac{14}{21}, \frac{14}{23}$, and cross-multiplying against the landmark $\frac{1}{3}$ for the sanity check.)
5.NF.B.3 Interpret a fraction as division and rewrite using equivalent expressions (Rewriting $\frac{n}{n+14}$ as $1 - \frac{14}{n+14}$, which turns the comparison into ordering the simpler leftover pieces.)
6.NS.C.7 Understand ordering and absolute value of rational numbers (Translating the order of the leftover pieces $\frac{14}{n+14}$ back into the order of the original three rational numbers and writing the final chain $\frac{5}{19} < \frac{7}{21} < \frac{9}{23}$.)

⭐ Look for hidden structure first: once you see all three fractions are $\frac{n}{n+14}$, ordering them is a one-line Grade 6 observation — no big arithmetic needed.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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