You are in the middle of a problem and suddenly need to know whether \tfrac{7}{9} or \tfrac{11}{15} is bigger. No calculator, no time to find an LCM and rewrite both over 45. You need an answer in five seconds. Happily, there are three mental tricks that settle almost every fraction comparison without paper. Pick the right one for the situation and the answer drops out.

Trick 1: cross-multiply and compare products

This is the fastest universal move. For two fractions \tfrac{a}{b} and \tfrac{c}{d} with positive b, d, they are ordered the same way as the products ad and bc:

\frac{a}{b} \;<\; \frac{c}{d} \quad \iff \quad a \cdot d \;<\; b \cdot c

Multiply the top of the first fraction by the bottom of the second, and the top of the second by the bottom of the first. Compare the two products. Whichever side is bigger, the fraction on that side is bigger.

For \tfrac{7}{9} vs \tfrac{11}{15}:

7 \times 15 = 105, \qquad 9 \times 11 = 99

The 7 \cdot 15 product belongs to \tfrac{7}{9} (its numerator times the other denominator). 105 > 99, so \tfrac{7}{9} > \tfrac{11}{15}. That's the whole computation — two small multiplications that most students can do in their head, and a single "which is bigger?" comparison at the end.

Why cross-multiplication works: if you multiply both sides of \tfrac{a}{b} < \tfrac{c}{d} by the positive number bd, you get ad < bc. The inequality is preserved because bd is positive. So "which cross-product is bigger" is just "which fraction is bigger, in disguise." Note the rule needs both denominators positive — for negative denominators the inequality flips, which is why you want to keep fractions written with positive bottoms.

The direction rule

Students sometimes forget which cross-product goes with which fraction. The rule: the product that shares a numerator with fraction X "votes for" fraction X. In \tfrac{7}{9} vs \tfrac{11}{15}:

Whichever product wins, its fraction wins.

A visual trick: draw an arrow diagonally from top-left to bottom-right (7 \to 15, giving 7 \times 15) — that product belongs to the left fraction. The other diagonal (9 \nearrow 11) gives the right fraction's product. Writing both fractions side by side makes the diagonals obvious.

Cross-multiplication diagram for seven ninths versus eleven fifteenthsTwo fractions written side by side, seven over nine on the left and eleven over fifteen on the right. A diagonal arrow goes from the seven in the numerator of the left fraction down to the fifteen in the denominator of the right fraction, labelled seven times fifteen equals one hundred and five. A second diagonal arrow goes from the nine in the denominator of the left fraction up to the eleven in the numerator of the right fraction, labelled nine times eleven equals ninety nine. A caption below says one hundred and five is bigger than ninety nine, so seven ninths is bigger. 7 9 vs 11 15 7 × 15 = 105 9 × 11 = 99 105 > 99 ⟹ 7/9 > 11/15
Cross-multiplication drawn on the page. Each diagonal gives the product that belongs to its starting fraction. The bigger product wins — and here, $105 > 99$, so $\tfrac{7}{9}$ beats $\tfrac{11}{15}$.

Trick 2: compare distances from one

When both fractions are close to 1 — like \tfrac{7}{9} and \tfrac{11}{15} — a different trick is often even faster. Compute how far each is from 1:

1 - \frac{7}{9} = \frac{2}{9}, \qquad 1 - \frac{11}{15} = \frac{4}{15}

Whichever distance is smaller, that fraction is closer to 1 and therefore larger.

Now compare \tfrac{2}{9} and \tfrac{4}{15} (which are smaller, easier numbers). Cross-multiply again: 2 \times 15 = 30 and 9 \times 4 = 36. So \tfrac{2}{9} < \tfrac{4}{15}. That means \tfrac{7}{9} is closer to 1, so \tfrac{7}{9} is the larger fraction. Same answer as Trick 1, arrived at with slightly simpler mental arithmetic because the distance fractions had smaller numerators.

Why "distance from 1" is often easier: for \tfrac{a}{b}, the distance from 1 is \tfrac{b-a}{b} — a fraction whose numerator is the gap between numerator and denominator. For \tfrac{7}{9}, the gap is 2; for \tfrac{11}{15}, the gap is 4. When the gaps are small compared to the denominators, the distance fractions are small and easy to compare. Small numbers multiply faster than big ones, which is why the trick buys you speed.

This trick is spectacular when the fractions are all close to 1:

\frac{99}{100} \text{ vs } \frac{999}{1000}

Distances from 1: \tfrac{1}{100} = 0.01 and \tfrac{1}{1000} = 0.001. The second is smaller, so \tfrac{999}{1000} is closer to 1, so \tfrac{999}{1000} > \tfrac{99}{100}. No paper needed.

Trick 3: sniff out the decimal

If the fractions are close to nice decimals you already know, approximate and compare.

\tfrac{7}{9} \approx 0.78 (since \tfrac{7}{9} = 0.\overline{7} = 0.777\dots). \tfrac{11}{15} \approx 0.73 (since 11 \div 15 \approx 0.733\dots — roughly "eleven-fifteenths is a bit under three-quarters"). 0.78 > 0.73, so \tfrac{7}{9} > \tfrac{11}{15}.

This trick is fast when you have memorised a few standard decimal-fraction pairs. A short list worth knowing by heart:

\frac{1}{2} = 0.5 \quad \frac{1}{3} \approx 0.333 \quad \frac{2}{3} \approx 0.667 \quad \frac{1}{4} = 0.25 \quad \frac{3}{4} = 0.75
\frac{1}{5} = 0.2 \quad \frac{1}{8} = 0.125 \quad \frac{1}{9} \approx 0.111 \quad \frac{1}{7} \approx 0.143

With those in your head, most exam-scale fractions can be decimal-sniffed in under five seconds — you just anchor to the nearest memorised value and nudge.

When to use which trick

Try all three on a friend-versus-friend fraction race — you'll naturally learn which one fits which shape of problem.

Compare $\tfrac{13}{20}$ and $\tfrac{5}{8}$ in three different ways.

Trick 1 — cross-multiply. 13 \times 8 = 104, 20 \times 5 = 100. 104 > 100, so \tfrac{13}{20} > \tfrac{5}{8}.

Trick 2 — distance from \tfrac{1}{2}. \tfrac{13}{20} - \tfrac{1}{2} = \tfrac{3}{20} = 0.15. \tfrac{5}{8} - \tfrac{1}{2} = \tfrac{1}{8} = 0.125. \tfrac{13}{20} is further above \tfrac{1}{2}, so it is bigger.

Trick 3 — decimal sniff. \tfrac{13}{20} = \tfrac{65}{100} = 0.65. \tfrac{5}{8} = 0.625. 0.65 > 0.625, so \tfrac{13}{20} > \tfrac{5}{8}.

Three tricks, three routes to the same answer. Each takes about ten seconds. The decimal sniff is probably fastest here because both fractions have clean decimal expansions.

The takeaway

Fraction comparison is almost never worth the full LCM-and-rewrite ritual in your head. Cross-multiplication reduces it to one subtraction, distance-from-one often halves the numerator sizes, and decimal sniffing turns it into something your brain has been doing since class 4.

For a maths paper you have time to do it properly. For a mental-arithmetic moment — checking a shopping bill, reading a cricket score, guessing a probability — pick whichever trick fits the shape of the problem, and the answer arrives in seconds rather than minutes.

For the record, \tfrac{7}{9} > \tfrac{11}{15}, and now there are three different ways you can see why at a glance.

Related: Fractions and Decimals · Drag 2/7, 0.28 and 28% onto a Number Line — Which One Wins? · Is 0.45 Bigger Than 0.8 Because It Has More Digits? · Can I Add 1/2 + 1/3 by Adding Tops and Bottoms to Get 2/5?