Compare Two Fractions in Three Seconds — The Cross-Multiplication Shortcut

You are handed \tfrac{5}{8} and \tfrac{7}{11} and asked which is bigger. The textbook method is to find the LCM of 8 and 11, rewrite both fractions over 88, and compare the numerators. That works, but it is four steps and a multiplication table. There is a one-line shortcut that gives the same answer in three seconds, and every competitive-exam problem-solver has it on autopilot.

The trick

To compare two positive fractions \dfrac{a}{b} and \dfrac{c}{d}, cross-multiply:

Compute a \times d — the left numerator times the right denominator.
Compute b \times c — the left denominator times the right numerator.

The bigger of those two products tells you which fraction is bigger.

If ad > bc, then \dfrac{a}{b} > \dfrac{c}{d}.
If ad < bc, then \dfrac{a}{b} < \dfrac{c}{d}.
If ad = bc, the fractions are equal.

Apply it to \tfrac{5}{8} vs \tfrac{7}{11}:

5 \times 11 = 55
8 \times 7 = 56

Since 56 > 55, the right-hand fraction \tfrac{7}{11} is bigger. Done — one multiplication on each side, one comparison, one answer.

The two diagonal products are $5 \times 11 = 55$ and $8 \times 7 = 56$. The bigger product comes from the diagonal that includes the $7$ (on the right), so $\tfrac{7}{11}$ is the bigger fraction. The entire comparison is two multiplications and one inequality — no LCM, no rewriting.

Why this works

The fraction \dfrac{a}{b} is really \dfrac{a \times d}{b \times d} and the fraction \dfrac{c}{d} is really \dfrac{c \times b}{d \times b}. Both are now written over the same (positive) denominator bd, so comparing them reduces to comparing their numerators: ad versus bc.

Why: any two positive fractions can be given the common denominator bd in one move — multiply each top and bottom by the other fraction's denominator. Once they share a denominator, the fraction with the bigger numerator is the bigger number. The cross-products ad and bc are exactly those numerators over the shared denominator bd.

So cross-multiplication is not a trick — it is the standard "find a common denominator" method, with the rewriting step skipped because you only need the numerators to compare, not the fractions themselves.

One warning: positive denominators only

Cross-multiplication reverses if one of the denominators is negative, because multiplying an inequality by a negative number flips it. Every fraction you will meet in a school exam has a positive denominator (the convention is to put the minus sign in the numerator, e.g. \tfrac{-3}{5} rather than \tfrac{3}{-5}), so this does not come up in practice. But if you ever see \tfrac{a}{b} with b < 0, rewrite it as \tfrac{-a}{|b|} before cross-multiplying.

Try it — drag and compare

Drag the four points to change $a$, $b$, $c$, $d$. The two readouts show the cross-products $a \times d$ and $b \times c$. Whichever is larger tells you which of $\tfrac{a}{b}$ and $\tfrac{c}{d}$ is larger — no common denominator required. Try it on a few JEE-style pairs and notice how quickly the answer pops out.

Three exam-style uses

Comparing two fractions in an MCQ. Options list \tfrac{13}{17} and \tfrac{15}{19} — which is bigger? Cross-multiply: 13 \times 19 = 247, 17 \times 15 = 255. 255 wins, so \tfrac{15}{19} is bigger.

Comparing a fraction to an integer or to \tfrac{1}{2}. Is \tfrac{4}{9} > \tfrac{1}{2}? Cross-multiply: 4 \times 2 = 8, 9 \times 1 = 9. 9 > 8, so \tfrac{1}{2} is bigger. (In general, \tfrac{a}{b} > \tfrac{1}{2} exactly when 2a > b — the doubled numerator beats the denominator. Useful for a quick sniff test on probability answers.)

Ranking three fractions. Given \tfrac{3}{5}, \tfrac{4}{7}, \tfrac{5}{9}, rank them. Compare pairs. \tfrac{3}{5} vs \tfrac{4}{7}: 3 \times 7 = 21, 5 \times 4 = 20. So \tfrac{3}{5} > \tfrac{4}{7}. \tfrac{4}{7} vs \tfrac{5}{9}: 4 \times 9 = 36, 7 \times 5 = 35. So \tfrac{4}{7} > \tfrac{5}{9}. By transitivity, \tfrac{3}{5} > \tfrac{4}{7} > \tfrac{5}{9}.

Near-ties: when cross-products are close

If ad and bc come out very close — like 55 and 56 in the opening example — the fractions are also very close. That is worth knowing, because it tells you when a problem is asking you to distinguish small differences and you need to be careful. Specifically, the difference between the two fractions is

\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}

so if the cross-products differ by just 1 and the denominator product bd is large (say, 88), the fractions differ by only \tfrac{1}{88} \approx 0.011. Your intuition should register this as "these two are almost the same, and I should double-check the cross-multiplication before committing."

What to remember

Compare \dfrac{a}{b} and \dfrac{c}{d} by computing ad and bc. Bigger product → bigger fraction.
Works for any two positive fractions. No LCM, no rewriting.
It is secretly the common-denominator method, compressed to one line.
Good for MCQ comparisons, \tfrac{1}{2} sanity-checks, and ranking problems.
Repeat for multiple fractions by comparing pairs — the ranking follows.

One simple rule; dozens of exam questions reduced to two multiplications. This is the cheapest comparison tool in your mental toolkit, and it belongs on autopilot.