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
📘 View easy version →
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 + 4d\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.

1STEP 1

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

1511\frac{15}{11}, 1915\frac{19}{15}, 1713\frac{17}{13} = 11+411\frac{11+4}{11}, 15+415\frac{15+4}{15}, 13+413\frac{13+4}{13}
2STEP 2

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}.

1511\frac{15}{11} = 1 + 411\frac{4}{11}, 1915\frac{19}{15} = 1 + 415\frac{4}{15}, 1713\frac{17}{13} = 1 + 413\frac{4}{13}
3STEP 3

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

415\frac{4}{15} ＜ 413\frac{4}{13} ＜ 411\frac{4}{11}
4STEP 4

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

1 + 415\frac{4}{15} ＜ 1 + 413\frac{4}{13} ＜ 1 + 411\frac{4}{11} ⟺ 1915\frac{19}{15} ＜ 1713\frac{17}{13} ＜ 1511\frac{15}{11}
5STEP 5

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

1915\frac{19}{15} ＜ 1713\frac{17}{13} ＜ 1511\frac{15}{11} → (E)
Answer
1915\frac{19}{15} ＜ 1713\frac{17}{13} ＜ 1511\frac{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!

More like this

Same archetype — closest grade level first.