Competition · AMC preparation · step 4 of 4

AMC 8 · 2019 · #3

Grade 4 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation
fraction-arithmeticpattern-recognition pattern-recognitionidentify-subproblems ↑ Prerequisites: fraction-arithmetic
📏 Medium solution 💡 2 insights
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Problem
Three fractions are given: 1511\frac{15}{11}, 1915\frac{19}{15}, and 1713\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).

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

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Organize Information in More Ways

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 + 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.

1STEP 1

Spot the shared pattern

Spot the pattern: in every fraction the numerator is exactly 4 more than the denominator.

15/11, 19/15, 17/13 = (11+4)/11, (15+4)/15, (13+4)/13
2STEP 2

Split each fraction into two parts

Split each d+4d\frac{d+4}{d} into 1 + 4d\frac{4}{d}, so every fraction becomes 1 plus a small piece 4d\frac{4}{d}.

15/11 = 1 + 4/11, 19/15 = 1 + 4/15, 17/13 = 1 + 4/13
3STEP 3

Compare the fractional parts

Same numerator 4 means a bigger denominator gives a smaller part, so 415\frac{4}{15} ＜ 413\frac{4}{13} ＜ 411\frac{4}{11}.

4/15 ＜ 4/13 ＜ 4/11
4STEP 4

Add the 1 back

Add 1 back to every part; the order holds, giving 1915\frac{19}{15} ＜ 1713\frac{17}{13} ＜ 1511\frac{15}{11}.

1 + 4/15 ＜ 1 + 4/13 ＜ 1 + 4/11 ⟺ 19/15 ＜ 17/13 ＜ 15/11
5STEP 5

Match against the choices

This order matches choice (E); the other choices misplace a fraction and drop out.

19/15 ＜ 17/13 ＜ 15/11 → (E)
Answer
19/15 ＜ 17/13 ＜ 15/11
Sanity-check with rough decimals: 1511\frac{15}{11} ≈ 1.364, 1713\frac{17}{13} ≈ 1.308, 1915\frac{19}{15} ≈ 1.267. Sorted: 1.267 ＜ 1.308 ＜ 1.364, which is 1915\frac{19}{15} ＜ 1713\frac{17}{13} ＜ 1511\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.
💡Key takeaway

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!

  • Spot the shared pattern
  • Split each fraction into two parts
  • Compare the fractional parts
  • Add the 1 back
  • Match against the choices

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