AMC 8 · 2019 · #3

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition

fraction-arithmeticpattern-recognition pattern-recognitionidentify-subproblems ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 2 insights

Problem

Which of the following is the correct order of the fractions $\frac{15}{11},\frac{19}{15},$ and $\frac{17}{13},$ from least to greatest?

$\textbf{(A) }\frac{15}{11}< \frac{17}{13}< \frac{19}{15} \qquad\textbf{(B) }\frac{15}{11}< \frac{19}{15}<\frac{17}{13} \qquad\textbf{(C) }\frac{17}{13}<\frac{19}{15}<\frac{15}{11} \qquad\textbf{(D) } \frac{19}{15}<\frac{15}{11}<\frac{17}{13} \qquad\textbf{(E) } \frac{19}{15}<\frac{17}{13}<\frac{15}{11}$

Pick an answer.

(A)

$\frac{15}{11} < \frac{17}{13} < \frac{19}{15}$

(B)

$\frac{15}{11} < \frac{19}{15} < \frac{17}{13}$

(C)

$\frac{17}{13} < \frac{19}{15} < \frac{15}{11}$

(D)

$\frac{19}{15} < \frac{15}{11} < \frac{17}{13}$

(E)

$\frac{19}{15} < \frac{17}{13} < \frac{15}{11}$

View mode:

Toolkit + CCSS Solution

Understand

Restated: Three fractions are given: $\frac{15}{11}$, $\frac{19}{15}$, and $\frac{17}{13}$. Each is slightly greater than $1$. We must put them in order from least to greatest and pick the matching answer choice (A)-(E).

Givens: Three fractions: $\frac{15}{11}$, $\frac{19}{15}$, $\frac{17}{13}$; All three are improper fractions (numerator $>$ denominator), so each is greater than $1$; In every fraction, the numerator is exactly $4$ more than the denominator: $15 - 11 = 19 - 15 = 17 - 13 = 4$; Answer choices (A)-(E) list every possible left-to-right ordering

Unknowns: The correct least-to-greatest ordering of the three fractions, matching one of choices (A)-(E)

Understand

Plan

Primary tool: #15 Organize Information in More Ways

Secondary: #5 Look for a Pattern, #3 Eliminate Possibilities

Tool #5 (Pattern) is the first thing to notice: in each fraction the numerator is $4$ bigger than the denominator. That is the secret of the problem — without that observation, three random-looking fractions would force ugly cross-multiplication. Tool #15 (Organize Information in More Ways) then rewrites every fraction as $1 + \frac{4}{d}$, turning three two-number comparisons into one same-numerator comparison. Once we have the fractional parts in order, Tool #3 (Eliminate) walks us straight to the unique choice that matches.

Execute — Answer: E

#5 Look for a Pattern 4.OA.C.5 Step 1

Spot the pattern.
Look at each fraction's numerator minus its denominator: $15-11=4$, $19-15=4$, $17-13=4$.
Every fraction is "its denominator plus $4$, over its denominator."

$$\frac{15}{11},\ \frac{19}{15},\ \frac{17}{13} \;=\; \frac{11+4}{11},\ \frac{15+4}{15},\ \frac{13+4}{13}$$

💡 Noticing the same gap of $4$ across three different fractions is exactly the "find the rule in a number pattern" skill from Grade 4.

#15 Organize Information in More Ways 4.NF.B.3 Step 2

Reorganize each fraction as $1$ plus a unit-style fractional part.
Splitting $\frac{d+4}{d}$ into $\frac{d}{d} + \frac{4}{d}$ converts every fraction into $1 + \frac{4}{d}$, which is much easier to compare.

$$\frac{15}{11} = 1 + \frac{4}{11},\quad \frac{19}{15} = 1 + \frac{4}{15},\quad \frac{17}{13} = 1 + \frac{4}{13}$$

💡 Rewriting an improper fraction as a whole part plus a proper fraction is the Grade 4 "fraction as a sum of unit fractions" idea.

#15 Organize Information in More Ways 4.NF.A.2 Step 3

Since every fraction now starts with the same $1$, the order of the whole numbers is decided entirely by the order of the fractional parts $\frac{4}{11}$, $\frac{4}{15}$, $\frac{4}{13}$.
These three fractions all share the numerator $4$, so the one with the largest denominator is the smallest, and the one with the smallest denominator is the largest.
Ordering denominators: $11 < 13 < 15$, so $\frac{4}{15} < \frac{4}{13} < \frac{4}{11}$.

$$\frac{4}{15} < \frac{4}{13} < \frac{4}{11}$$

💡 Comparing same-numerator fractions by their denominators is a Grade 4 fraction-comparison rule — bigger denominator means smaller piece.

#15 Organize Information in More Ways 4.NF.A.2 Step 4

Add the $1$ back to each side of the inequality and substitute the original fractions to get the final order.

$$1 + \frac{4}{15} < 1 + \frac{4}{13} < 1 + \frac{4}{11} \;\;\Longleftrightarrow\;\; \frac{19}{15} < \frac{17}{13} < \frac{15}{11}$$

💡 Adding the same $1$ to every part of an inequality keeps the order the same — still Grade 4 fraction reasoning.

#3 Eliminate Possibilities 4.NF.A.2 Step 5

Match the ordering to the answer choices.
The order $\frac{19}{15} < \frac{17}{13} < \frac{15}{11}$ is exactly choice (E).
Choices (A)-(D) put at least one fraction out of place and are eliminated.

$$\frac{19}{15} < \frac{17}{13} < \frac{15}{11} \;\Rightarrow\; \textbf{(E)}$$

💡 Reading the order off the answer choices and crossing out the ones that disagree is the standard multiple-choice elimination move.

[1] #5 4.OA.C.5 Spot the pattern. Look at each fraction's numerator minus its denominator: $15-1

[2] #15 4.NF.B.3 Reorganize each fraction as $1$ plus a unit-style fractional part. Splitting $\f

[3] #15 4.NF.A.2 Since every fraction now starts with the same $1$, the order of the whole number

[4] #15 4.NF.A.2 Add the $1$ back to each side of the inequality and substitute the original frac

[5] #3 4.NF.A.2 Match the ordering to the answer choices. The order $\frac{19}{15} < \frac{17}{1

Review

Reasonableness: Sanity-check with rough decimals: $\frac{15}{11} \approx 1.364$, $\frac{17}{13} \approx 1.308$, $\frac{19}{15} \approx 1.267$. Sorted: $1.267 < 1.308 < 1.364$, which is $\frac{19}{15} < \frac{17}{13} < \frac{15}{11}$ — matches choice (E). Also, the denominator pattern ($11 < 13 < 15$) and the value pattern (largest first, then middle, then smallest) line up the way the same-numerator rule predicts.

Alternative: Tool #6 (Guess and Check) by cross-multiplying pairs: to compare $\frac{15}{11}$ and $\frac{17}{13}$, compare $15 \times 13 = 195$ with $17 \times 11 = 187$; since $195 > 187$, $\frac{15}{11} > \frac{17}{13}$. Do the same for the other two pairs and assemble the full order. This works, but takes three separate cross-multiplications instead of one structural insight.

CCSS standards used (min grade 4)

4.OA.C.5 Generate a number or shape pattern following a given rule (Spotting the common pattern across the three fractions — numerator is always denominator $+ 4$.)
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions (Decomposing each improper fraction $\frac{d+4}{d}$ into $1 + \frac{4}{d}$.)
4.NF.A.2 Compare two fractions with different numerators and different denominators (Ordering $\frac{4}{11}, \frac{4}{13}, \frac{4}{15}$ via the same-numerator rule (bigger denominator $\Rightarrow$ smaller fraction) and translating that order back to the original fractions.)

⭐ This AMC 8 problem only needs the Grade 4 fraction-comparison rule you already know — when the top numbers match, the fraction with the smaller bottom number is bigger!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 4)

More like this