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Ordering
Rational Numbers

Master the art of comparing and ordering integers, fractions, and decimals: a core skill for the SSAT Quantitative section.

SSAT Upper Level

What Are Rational Numbers?

A rational number is any number that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). This includes integers, terminating decimals, and repeating decimals.

Integers
\(-3,\; 0,\; 7\) are rational because \(7 = \frac{7}{1}\)
Fractions
\(\frac{2}{5},\; -\frac{3}{4}\) are already in \(\frac{a}{b}\) form
Decimals
\(0.75 = \frac{3}{4}\) and \(0.\overline{3} = \frac{1}{3}\)

Visualizing Order on a Number Line

On a number line, numbers increase from left to right. A number further left is less than a number further right. This is the foundation for comparing all rational numbers.

-3 -2 -1 0 1 2 3 -1.5 0.5 2.5 ← smaller · · · larger →

Why Ordering Matters on the SSAT

The SSAT Quantitative section frequently asks you to identify the greatest or least value in a set, place numbers in order, or compare fractions and decimals under time pressure. Speed and accuracy come from a solid method.

Compare
Determine if \(a > b\), \(a < b\), or \(a = b\)
Order
Arrange a set from least to greatest (or greatest to least)

The Common Denominator Rule

To compare two fractions, rewrite them with the same denominator. Then the fraction with the larger numerator is larger.

\(\frac{2}{3}\)
\(=\frac{8}{12}\)
 vs
\(\frac{3}{4}\)
\(=\frac{9}{12}\)
\(8 < 9\)
so \(\frac{2}{3} < \frac{3}{4}\)

Cross Multiplication Shortcut

A faster way to compare two fractions: cross multiply. Compare \(\frac{a}{b}\) and \(\frac{c}{d}\) by computing \(a \times d\) and \(c \times b\).

\(a\)
\(b\)
 vs
\(c\)
\(d\)
\(a \times d\)
?
\(c \times b\)
If \(a \times d > c \times b\), then \(\frac{a}{b} > \frac{c}{d}\)

Positive vs. Negative Rationals

Negative numbers reverse intuition. A number with a larger absolute value that is negative is actually smaller. Compare carefully near zero.

Positive side
Bigger absolute value = bigger number. \(5 > 3\) because \(|5| > |3|\).
Negative side
Bigger absolute value = smaller number. \(-5 < -3\) because \(|-5| > |-3|\).
-5 -3 0 3 5
\(-5 < -3\)
\(3 < 5\)

Step-by-Step Method

1
Identify the forms. Are all numbers fractions, decimals, or a mix?
2
Convert to a common form. Either convert all to decimals, or find a common denominator for fractions.
3
Handle negatives separately. Sort positives and negatives in distinct groups, then merge.
4
Compare and order. Line up the converted values and arrange from least to greatest (or as directed).
5
Verify with estimation. Quick mental check: does the order make sense on a number line?

Example 1: Comparing Two Fractions

Which is greater: \(\frac{5}{8}\) or \(\frac{7}{12}\)?

Example 2: Mixed Set Ordering

Order from least to greatest: \(\frac{3}{5},\; 0.4,\; \frac{1}{2},\; 0.65\)

0 0.4 0.5 0.6 0.65 1

Example 3: Negative Rationals

Order from least to greatest: \(-\frac{3}{4},\; -0.5,\; \frac{1}{3},\; -\frac{2}{3}\)

-1 -0.75 -0.667 -0.5 0 0.333 1

Example 4: Tricky Fractions

Which is greater: \(\frac{7}{11}\) or \(\frac{5}{8}\)?

Common Mistakes

✕ Comparing Numerators Only
Thinking \(\frac{3}{7} > \frac{2}{5}\) just because \(3 > 2\). You must use a common denominator or cross multiply.
✕ Ignoring Negative Signs
Treating \(-\frac{3}{4}\) as \(\frac{3}{4}\) when ordering. Remember: the negative flips the position on the number line.
✕ Wrong Cross Multiply Side
Mixing up which product corresponds to which fraction. Each product stays with the numerator on its own side.

Conversion Strategies at a Glance

Strategy Best For Speed
Common Denominator 2-3 fractions with small denominators Fast
Convert to Decimals Mixed sets (fractions + decimals) Fast
Cross Multiplication Comparing exactly 2 fractions Very Fast
Benchmark Fractions Quick estimation (\(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{3}{4}\)) Fastest

Using Benchmark Fractions

Benchmark fractions like \(\frac{1}{4}\), \(\frac{1}{3}\), \(\frac{1}{2}\), \(\frac{2}{3}\), and \(\frac{3}{4}\) help you quickly estimate where other fractions fall.

0
\(\frac{1}{4}\)
 0.25
\(\frac{1}{3}\)
\(\frac{1}{2}\)
 0.5
\(\frac{2}{3}\)
\(\frac{3}{4}\)
 0.75 1

Question 1 of 5

Which of the following is the correct order from least to greatest?

Question 2 of 5

Which fraction is greater: \(\frac{4}{7}\) or \(\frac{5}{9}\)?

Question 3 of 5

Which value is the smallest?

Question 4 of 5

Arrange from greatest to least:
\(0.8,\; \frac{7}{9},\; \frac{4}{5},\; 0.78\)

Question 5 of 5

If \(x = -\frac{5}{6}\) and \(y = -\frac{7}{9}\), which statement is true?