Comparing Fractions Without Dividing: 4 CAT DI Shortcuts

DILR

A large share of the time lost in DI sets is lost on a single repeated chore: deciding which of two fractions is bigger. Which year had the higher profit margin? Which state had the better ratio of students to teachers? Each comparison looks small, and long division makes it slow, so a set with ten such comparisons quietly eats five or six minutes.

You rarely need the actual value. You only need the order. This guide gives you four shortcuts that settle the order without dividing, each with a worked example you can check, and a simple rule for choosing which shortcut to try first.

Why Division Is the Wrong Default in DI

Dividing two numbers to compare them feels safe, because it gives a definite answer. But it is slow, error-prone under time pressure, and usually more precise than the question needs. If the options differ by five percent or more, a rough comparison is enough, and the exact decimals are wasted effort.

Many DI questions are really about order. Which is larger, which grew faster, which is closer to a target? Each of these can be answered by logic about numerators and denominators, without a single division. The shortcuts below are four such pieces of logic, arranged from the fastest to the slowest.

The aim is not to avoid arithmetic altogether. It is to do the smallest amount that settles the question. If your first check settles it, stop. If it does not, move to the next, and only reach for full division when the numbers are so close that nothing else works. Estimation tricks like these also appear in our guide on when estimation beats exact calculation in CAT Quant.

Dominance First: Four Shortcuts in Order

The shortcuts are most effective when you try them in a fixed order, from the quickest check to the most laborious. Each one either settles the comparison or tells you to move on.

The first two are mental checks that take seconds. The third is a small calculation. The fourth is a way of thinking about change over time, and it is the shortcut that pays off most in sets with year-on-year data. Let us take each in turn.

Shortcuts One and Two: Dominance and Benchmarks

Dominance is the simplest. Compare 62/150 with 58/160. The first has a bigger numerator and a smaller denominator, so it is clearly the bigger fraction. No arithmetic is needed. You can confirm that 62/150 is about 0.413 and 58/160 is about 0.363, and the first is larger. The check works whenever both parts point the same way.

When dominance fails, because one fraction has the bigger numerator and also the bigger denominator, use a benchmark. Benchmarks are easy fractions like one half, one third, one quarter, or percentages like ten percent. Compare 37/120 with one third. Three times 37 is 111, which is less than 120, so 37/120 is below one third.

The benchmark is most useful when you need to compare two fractions that are each close to a benchmark on opposite sides. If one is below one third and the other is above it, you have your order without further work. Even when both sit on the same side, noticing how far each is from the benchmark often settles the question.

Memorise a handful of benchmark equivalents. One half is 50 percent, one third is about 33.3 percent, one quarter is 25 percent, one fifth is 20 percent, one eighth is 12.5 percent, and so on. With these in your head, a percentage comparison can be turned into a benchmark check in seconds. For example, 18/240 is 7.5 percent, and 21/300 is 7 percent, so the first is slightly larger, found by scaling both to a common base of one hundred.

Shortcut	When it works	Example
Dominance	Numerator up and denominator down	62/150 is larger than 58/160
Benchmark	Fractions near an easy reference	37/120 is below one third, since 3 times 37 is 111
Cross-multiply	Numbers small or easy to round	47/83 vs 5/9, since 423 is more than 415
Growth rate	Both parts change over time	Revenue up 15 percent, cost up 10 percent

Shortcut Three: Cross-Multiplying Without Pain

Cross-multiplication compares a/b with c/d by comparing a times d with c times b. If a times d is larger, a/b is the larger fraction, provided the denominators are positive. It is exact, and it replaces a long division with two multiplications.

Take 47/83 and 5/9. Multiply 47 by 9 to get 423, and 5 by 83 to get 415. Since 423 is larger, 47/83 is larger. You can check by division: 47/83 is about 0.566, and 5/9 is about 0.556. The fractions are close, which is exactly the situation where cross-multiplication earns its keep.

When the numbers are large, round them first. If you are comparing 4,730/8,320 with 5/9, you can treat the first as about 47/83 and proceed. Rounding both numerator and denominator by the same factor leaves the fraction almost unchanged, and the check remains reliable when the difference is not tiny.

Be careful with rounding when the fractions are very close. If the two products differ by only one or two percent, rounding errors can flip the answer, and you should compute more carefully or use another method. In most DI options, the values are separated enough for a rough check to work, but it is worth noticing when they are not.

Shortcut Four: Compare the Growth Rates

Many DI questions ask whether a ratio rose or fell between two years. The ratio could be profit to revenue, cost per unit, or students per teacher. You do not need to compute the ratio in each year. You only need to compare how the numerator and the denominator changed.

The rule is simple. If the numerator grew by a larger percentage than the denominator, the ratio rose. If it grew by a smaller percentage, the ratio fell. If they grew by the same percentage, the ratio is unchanged. This is true for decreases too, with the roles reversed: the part that fell faster has the larger effect.

Suppose revenue rose from 240 to 276 and cost rose from 180 to 198. Revenue grew by 36 on 240, which is 15 percent. Cost grew by 18 on 180, which is 10 percent. Revenue grew faster, so the ratio of revenue to cost rose. You can verify it: 240/180 is about 1.333, and 276/198 is about 1.394, which is higher.

This shortcut is very powerful in sets with several years of data, because it lets you answer ratio questions from the growth percentages alone, which are often given directly or are easy to read from a chart. Our guide on percentage change and CAGR traps in CAT DI shows the pitfalls to avoid when working with growth rates.

Choosing Between the Shortcuts Under Pressure

In the exam you will not stop to name the shortcut. The order matters because it matches speed: dominance and benchmarks are nearly instant, cross-multiplication takes a few seconds, and growth rate is a mode of thought for time-based data. If your first check settles the comparison, you are done.

A good habit is to look at the table or chart and ask what kind of comparison each question wants. Does it compare two entities in one year? Dominance, benchmark and cross-multiplication are the likely tools. Does it compare one entity across years? Growth rate is the natural choice. Matching the tool to the question type saves time.

Be aware of the limits. These shortcuts give the order, not the size of the gap. If a question asks for the exact percentage difference, you will have to calculate. And if the fractions are extremely close, you may need to compute, since estimates can mislead when the gap is smaller than your rounding error.

These shortcuts also help when you are filling in missing data, since you can check that a computed value is plausible. Our piece on filling the blanks in tables with missing data shows how to combine totals with comparisons in that situation.

Practising the Shortcuts So They Stick

Shortcuts only help if they come to mind when you need them. The best way to build that reflex is to practise comparisons in isolation. Take a table from a DI set, pick twenty pairs of fractions, and for each pair decide the order using a named shortcut before you check with a calculator or long division.

Keep a count of which shortcut settled each pair. Most aspirants find that dominance and benchmarks cover more than half of the comparisons, cross-multiplication covers a good share of the rest, and growth rate dominates in time-series sets. Seeing this makes the order feel natural.

Keep a short list of the four shortcuts in your CAT formula sheets and cheatsheets and glance at it before practice. And if you want the practice spread evenly across the weeks, a CAT study planner can schedule short DI sessions without crowding out other sections.