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\(a\)
\(b\)
\(<\)

Comparing
Rational Numbers

Master the skill of comparing fractions, decimals, and mixed numbers in any form: find common ground and determine which value is greater, less, or equal.

ISEE Middle Level

What Are Rational Numbers?

A rational number is any number that can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). This includes whole numbers, fractions, decimals that terminate or repeat, and negative values.

Fractions
\(\frac{3}{4}\), \(-\frac{2}{5}\), \(\frac{7}{1}\)
Decimals
\(0.75\), \(-0.4\), \(0.\overline{3}\)
Mixed Numbers
\(2\frac{1}{3}\), \(-1\frac{3}{8}\)

The Number Line Test

Every rational number has an exact position on the number line. Numbers to the right are greater, and numbers to the left are smaller. This is the fundamental way to think about comparison.

-1 -0.5 0 0.5 1
\(-\frac{3}{4}\)
\(\frac{1}{4}\)
\(\frac{3}{4}\)
 Greater → ← Lesser

Why Comparing Matters

The ISEE Quantitative Reasoning section often asks you to determine which of two quantities is larger, or to order a set of values. You will see fractions, decimals, and mixed numbers presented side by side in different forms.

Quantitative Comparison
Is Column A greater, less, or equal to Column B?
Ordering Problems
Arrange from least to greatest or greatest to least.

Method 1: Common Denominators

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

\(\frac{2}{3}\)
 original →
\(\frac{\color{#818cf8}8}{12}\)
\(\times 4\)
 vs
\(\frac{\color{#4ade80}9}{12}\)
\(\times 3\)

Method 2: Convert to Decimals

Divide the numerator by the denominator to get a decimal. Then compare the decimals digit by digit from left to right.

1
Divide: \(\frac{5}{8} = 5 \div 8 = 0.625\)
2
Divide: \(\frac{7}{11} = 7 \div 11 = 0.\overline{63}\)
3
Compare digit by digit: \(0.625\) vs \(0.6363...\) The tens and tenths digits match. At the hundredths place: \(2 < 3\).
4
Conclude: \(\frac{5}{8} < \frac{7}{11}\)

Method 3: Cross-Multiplication

The fastest shortcut for comparing two fractions: multiply diagonally and compare the products. No need to find an LCD.

\(\frac{3}{7}\)
\(\frac{2}{5}\)
\(3 \times 5 = 15\)
\(2 \times 7 = 14\)
 ?

Positive vs Negative Rationals

Comparing negative rational numbers follows opposite intuition. The number closer to zero is greater. A number with a larger absolute value is actually smaller when negative.

Positive Negative
Larger absolute value means Greater Smaller
Closer to zero means Smaller Greater
Example \(\frac{5}{6} > \frac{1}{3}\) \(-\frac{1}{3} > -\frac{5}{6}\)
Think of it as Farther right \(=\) bigger Farther right \(=\) bigger

How to Compare Any Two Rationals

1
Check signs first. Any positive number is greater than any negative number. If signs differ, you are done.
2
Convert to the same form. Choose the method that fits: common denominators, decimals, or cross-multiplication.
3
Compare the converted values. For fractions with the same denominator, compare numerators. For decimals, compare digit by digit.
4
If both are negative, flip. Compare absolute values, then reverse the inequality sign.
5
Write the answer. Use \(<\), \(>\), or \(=\) to state the relationship.

Example 1: Fraction vs Decimal

Compare \(\frac{5}{6}\) and \(0.8\).

1
Both positive: Signs are the same, so we proceed to convert.
2
Convert fraction to decimal: \(\frac{5}{6} = 5 \div 6 = 0.8\overline{3}\)
3
Compare: \(0.8\overline{3}\) vs \(0.800\). At the hundredths place: \(3 > 0\).
4
Conclusion: \(\frac{5}{6} > 0.8\)

Example 2: Mixed Numbers

Compare \(2\frac{3}{8}\) and \(2\frac{1}{3}\).

1
Whole parts are equal (both are \(2\)), so compare only the fractional parts: \(\frac{3}{8}\) vs \(\frac{1}{3}\).
2
Cross-multiply: \(3 \times 3 = 9\) and \(1 \times 8 = 8\).
3
Compare products: \(9\) > \(8\), so \(\frac{3}{8} > \frac{1}{3}\).
4
Conclusion: \(2\frac{3}{8} > 2\frac{1}{3}\)

Example 3: Negative Rationals

Compare \(-\frac{4}{9}\) and \(-0.5\).

-1 -0.5 0 -0.5
\(-\frac{4}{9}\)
1
Convert: \(-\frac{4}{9} = -0.\overline{4} \approx -0.444\)
2
Compare absolute values: \(0.444 < 0.5\), so \(-\frac{4}{9}\) is closer to zero.
3
Conclusion: \(-\frac{4}{9} > -0.5\) because \(-0.444\) is to the right of \(-0.5\) on the number line.

Benchmark Fractions Strategy

Use common benchmarks like \(0\), \(\frac{1}{4}\), \(\frac{1}{2}\), \(\frac{3}{4}\), and \(1\) to quickly estimate and compare without converting.

0
\(\frac{1}{4}\)
\(\frac{1}{2}\)
\(\frac{3}{4}\)
 1
\(\frac{3}{8}\)
\(\frac{5}{8}\)

Ordering Multiple Values

Order from least to greatest: \(\frac{2}{5}\), \(0.45\), \(\frac{3}{7}\).

1
Convert all to decimals: \(\frac{2}{5} = 0.400\), already have \(0.450\), \(\frac{3}{7} \approx 0.4286\)
2
Line them up: \(0.400\), \(0.4286\), \(0.450\)
3
Order: \(\frac{2}{5} < \frac{3}{7} < 0.45\)

Common Mistakes

✕ Comparing Numerators Only
Thinking \(\frac{3}{8} > \frac{2}{5}\) because \(3 > 2\). You must account for different denominators.
✕ Negative Number Reversal
Forgetting to flip the inequality for negatives. \(-3 < -2\), not greater, even though \(3 > 2\).
✕ Decimal Place Errors
Thinking \(0.5 < 0.45\) because \(45\) looks bigger than \(5\). Pad zeros: \(0.50 > 0.45\).

Method Comparison

Method Best For Speed Accuracy
Common Denominators Simple fractions with small denominators Medium High
Decimal Conversion Mixed forms (fraction vs decimal) Fast High
Cross-Multiplication Two fractions, any denominators Fastest High
Benchmarks Quick estimation, eliminating answers Fastest Medium

Question 1 of 5

Which symbol correctly completes the statement?
\(\frac{7}{12}\) ___ \(\frac{5}{8}\)

Question 2 of 5

Which of the following is the greatest?

Question 3 of 5

Which symbol correctly completes the statement?
\(-\frac{2}{3}\) ___ \(-\frac{3}{4}\)

Question 4 of 5

Which list is in order from least to greatest?

Question 5 of 5

Column A: \(3\frac{2}{5}\)
Column B: \(3.45\)

Which column has the greater value?