AMC 8 · 2019 · #3
Easy mode Grade 4Problem
Look at these three fractions: 1115, 1519, and 1317.
Put them in order from smallest to largest. Which choice below shows the correct order?
Pick an answer.
AMC 8 2019 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: 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
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
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
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."
💡 Noticing the same gap of $4$ across three different fractions is exactly the "find the rule in a number pattern" skill from Grade 4.
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.
💡 Rewriting an improper fraction as a whole part plus a proper fraction is the Grade 4 "fraction as a sum of unit fractions" idea.
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 $\frac{4}{15} < \frac{4}{13} < \frac{4}{11}$.
💡 Comparing same-numerator fractions by their denominators is a Grade 4 fraction-comparison rule — bigger denominator means smaller piece.
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.
💡 Adding the same $1$ to every part of an inequality keeps the order the same — still Grade 4 fraction reasoning.
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.
💡 Reading the order off the answer choices and crossing out the ones that disagree is the standard multiple-choice elimination move.
4.OA.C.5 Spot the pattern. Look at each fraction's numerator minus its denominator: $15-1 4.NF.B.3 Reorganize each fraction as $1$ plus a unit-style fractional part. Splitting $\f 4.NF.A.2 Since every fraction now starts with the same $1$, the order of the whole number 4.NF.A.2 Add the $1$ back to each side of the inequality and substitute the original frac 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.5Generate 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.3Understand 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.2Compare 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!
⭐ 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!
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