Compare 4/7, 5/9, 7/13 — Don't Eyeball, Convert to Decimals or Use a Common Denominator

An MCQ lands on your screen:

Which is the greatest among \tfrac{4}{7}, \tfrac{5}{9}, \tfrac{7}{13}?

The fractions are all close to each other. The numerators are similar in size. The denominators are similar in size. You cannot eyeball this. And if you try to, you will get it wrong more than half the time. The reliable thinking move is to switch to a single comparison format — either decimals or a common denominator — and let the comparison happen in that cleaner space.

Why "eyeballing" fails for close fractions

For fractions where one is obviously bigger (say \tfrac{1}{2} vs \tfrac{7}{8}), your intuition is fine. For fractions where numerators and denominators are similar, intuition is useless. Watch:

\tfrac{4}{7} \approx 0.5714
\tfrac{5}{9} \approx 0.5556
\tfrac{7}{13} \approx 0.5385

All three sit between 0.5 and 0.6, within a range of 0.033. No amount of visual inspection will reliably separate them. You need a numerical comparison.

The general recognition: if the fractions differ by less than about 10\% of their size, do not trust eyeballing. Commit to a method.

Method 1: convert all to decimals

Do the division on each fraction (mentally, with long division, or with a table). Write the decimals to 3-4 places. Read off the order.

For the trio above:

\tfrac{4}{7} = 0.5714\ldots, \quad \tfrac{5}{9} = 0.5555\ldots, \quad \tfrac{7}{13} = 0.5384\ldots

Order (biggest to smallest): \tfrac{4}{7} > \tfrac{5}{9} > \tfrac{7}{13}.

This is the fastest method if you can do the divisions quickly. For common denominators like 2, 3, 4, 5, 6, 8, 9, 10, 12, you should know the decimal forms cold. For unfamiliar ones (7, 11, 13, 17), you do long division — two or three places of precision is usually enough.

Once the three fractions are converted to decimals, they become three points on a number line, and the ordering is immediate. Before conversion, the fractions all "look similar." After, $4/7$ is clearly the biggest.

Method 2: common denominator

The classical move. Find the lcm of the denominators, rewrite all fractions with that denominator, and compare numerators.

For \tfrac{4}{7}, \tfrac{5}{9}, \tfrac{7}{13}: \text{lcm}(7, 9, 13) = 819.

\frac{4}{7} = \frac{4 \cdot 117}{819} = \frac{468}{819}, \quad \frac{5}{9} = \frac{5 \cdot 91}{819} = \frac{455}{819}, \quad \frac{7}{13} = \frac{7 \cdot 63}{819} = \frac{441}{819}.

Ordering: 468 > 455 > 441, so \tfrac{4}{7} > \tfrac{5}{9} > \tfrac{7}{13}.

This is slower than decimals for unrelated denominators (because the lcm can get large) but it is exact — no rounding, no guessing about which decimal is really larger. For fractions whose decimals are close to one another (0.333, 0.332, 0.331), common denominator is safer.

Method 3: pairwise cross-multiplication

If you only need to compare two fractions, cross-multiply. For \tfrac{a}{b} vs \tfrac{c}{d} with positive b, d: compare ad and bc.

\tfrac{4}{7} vs \tfrac{5}{9}: 4 \cdot 9 = 36 vs 7 \cdot 5 = 35. Since 36 > 35, \tfrac{4}{7} > \tfrac{5}{9}. \tfrac{5}{9} vs \tfrac{7}{13}: 5 \cdot 13 = 65 vs 9 \cdot 7 = 63. Since 65 > 63, \tfrac{5}{9} > \tfrac{7}{13}.

Transitivity gives the full order. For three or four fractions, cross-multiplication at the sorting stage is actually competitive with decimals. For five or more, one of the single-pass methods (decimals or common denominator) is faster.

When each method is best

Two fractions, one comparison. Cross-multiply. One multiplication each side, done.
Three to four fractions, moderate denominators. Decimal conversion is usually fastest — if you know your decimal-equivalents table, this takes seconds.
Three to four fractions, ugly denominators, exact answer required. Common denominator. Slower but exact.
Fractions very close to each other (< 1\% apart). Common denominator. Decimals introduce rounding at the level of the difference and may give wrong answers.

Match the method to the fractions.

The trap: "\tfrac{4}{7} is bigger because the numerator is bigger"

A common student trap: "\tfrac{4}{7} has numerator 4, \tfrac{5}{9} has numerator 5, \tfrac{7}{13} has numerator 7 — so the biggest numerator wins." This would be true only if the denominators were equal. They are not.

An even more stubborn trap: "\tfrac{7}{13} is bigger because it has the bigger numerator and the bigger denominator." No. Bigger denominator means each piece is smaller, and whether "seven small pieces" beats "four bigger pieces" depends on the exact sizes. You cannot know without computing.

Why eyeballing gives wrong answers: fractions measure both "how many pieces" (numerator) and "how big is each piece" (reciprocal of denominator). Without fixing one, you cannot compare. Fixing the denominator (Method 2) makes the comparison about numerators only; converting to decimals (Method 1) converts both pieces of information into a single number per fraction, after which comparison is trivial.

A worked JEE-style question

Arrange in ascending order: \tfrac{11}{16}, \tfrac{13}{20}, \tfrac{17}{25}, \tfrac{9}{14}.

Method 1 (decimals): 11/16 = 0.6875. 13/20 = 0.65. 17/25 = 0.68. 9/14 \approx 0.6429. Ordering: 9/14 < 13/20 < 17/25 < 11/16, i.e. 0.643 < 0.65 < 0.68 < 0.6875.

Fast because 16, 20, 25 are "nice" denominators. Only 9/14 needs actual long division, and 9 \div 14 is 0.64\ldots — a quick mental check is enough.

Method 2 (common denominator): \text{lcm}(16, 20, 25, 14) = 2800. Multiplying out gives 11/16 = 1925/2800, 13/20 = 1820/2800, 17/25 = 1904/2800, 9/14 = 1800/2800. Same ordering, but the arithmetic is much heavier.

For this problem, Method 1 wins. For problems where the denominators happen to be small and coprime, Method 2 is competitive.

Arrange in descending order: $\tfrac{3}{5}$, $\tfrac{7}{11}$, $\tfrac{11}{17}$

Decimals: 3/5 = 0.6. 7/11 = 0.6363\ldots 11/17 = 0.6470\ldots

Descending: \tfrac{11}{17} > \tfrac{7}{11} > \tfrac{3}{5}, i.e. 0.647 > 0.636 > 0.6.

Sanity check with a pairwise cross-multiply. \tfrac{11}{17} vs \tfrac{7}{11}: 11 \cdot 11 = 121 vs 17 \cdot 7 = 119. 121 > 119, so \tfrac{11}{17} > \tfrac{7}{11}. Matches.

Why cross-checking with a second method is worth a few seconds: rounding errors in decimals can flip ties. If two fractions look like they have the same two-decimal approximation, you need more precision or a direct cross-multiply to break the tie.

What to remember

If fractions are close, do not trust eyeballing — convert to a common format first.
Decimals are usually fastest: divide each, compare digits.
Common denominator (lcm) is slower but exact — use it when decimals are too close to be safe.
For two fractions, cross-multiply. It is a one-line comparison.
Match the method to the problem: familiar denominators → decimals; coprime denominators of moderate size → common denominator; any pairwise case → cross-multiply.

Comparing fractions is a mechanical skill. The moment the exam hands you a comparison, pick a method and commit — do not wait for a flash of intuition that is not going to come. Intuition is accurate for obvious cases and unreliable for close ones, and exams test the close ones.