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Comparing
Rational Numbers

Master the art of comparing fractions, decimals, and percents by converting them into a common form and placing them on the number line.

ISEE Upper 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 fractions, terminating decimals, repeating decimals, percents, and integers.

\(\frac{3}{4}\)
Fraction form: numerator over denominator
\(0.75\)
Decimal form: terminates after two digits
\(75\%\)
Percent form: parts per hundred

Rational Numbers on the Number Line

Every rational number has a unique position on the number line. Numbers to the right are greater; numbers to the left are lesser.

-1 0 1 2
\(-\frac{3}{4}\)
\(\frac{1}{3}\)
 0.75
\(\frac{7}{4}\)
 Lesser Greater ← →

Why Comparing Matters

The ISEE often gives you rational numbers in different forms and asks which is largest, smallest, or how they relate. You need a systematic method to compare them quickly and accurately.

Quantitative Comparison
Determine which of two quantities is greater, or if they are equal
Ordering Problems
Arrange fractions, decimals, and percents from least to greatest

The Comparison Rule

To compare rational numbers in different forms, convert them all to the same form. You have three options:

1
Common denominator: Convert all values to fractions with the same denominator, then compare numerators.
2
Decimal form: Convert all values to decimals by dividing numerator by denominator.
3
Cross-multiplication: For comparing two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), compare \(a \times d\) with \(c \times b\).

Cross-Multiplication in Action

\(\frac{3}{5}\)
 vs
\(\frac{5}{8}\)
\(3 \times 8 = 24\)
\(<\)
\(5 \times 5 = 25\)
So \(\frac{3}{5} < \frac{5}{8}\)

Positive vs Negative Rationals

When negative numbers are involved, the rules change. A number closer to zero is greater among negatives, and every positive number is greater than every negative number.

Scenario Rule Example
Both positive Larger value wins \(\frac{3}{4} > \frac{2}{3}\)
Both negative Closer to zero wins \(-\frac{1}{3} > -\frac{1}{2}\)
Mixed signs Positive always greater \(0.1 > -5\)
Comparing to zero Positive > 0 > Negative \(\frac{1}{100} > 0 > -\frac{1}{100}\)

Step-by-Step Method

1
Identify the forms: Determine if each value is a fraction, decimal, percent, or integer.
2
Choose a common form: Convert everything to decimals (usually fastest) or find a common denominator.
3
Handle signs first: Separate positives and negatives. Any positive \(>\) any negative.
4
Compare: Use the converted values (or cross-multiply) to determine the order.
5
Double-check negatives: For negative values, remember that \(-0.3 > -0.5\) (closer to zero is greater).

Conversion Quick Reference

Fraction → Decimal
Divide numerator by denominator: \(\frac{3}{8} = 3 \div 8 = 0.375\)
Percent → Decimal
Divide by 100: \(65\% = 0.65\)
Decimal → Fraction
Place over power of 10: \(0.45 = \frac{45}{100} = \frac{9}{20}\)

Example 1: Decimal Method

Compare: \(\frac{7}{12}\) and \(0.58\). Which is greater?

0.55 0.56 0.57 0.58 0.59 0.58
\(\frac{7}{12} \approx 0.583\)
1
Convert: \(\frac{7}{12} = 7 \div 12 = 0.58\overline{3}\)
2
Compare digits: \(0.583\ldots\) vs \(0.580\): the thousandths digit \(3 > 0\).
3
Conclude: \(\frac{7}{12} > 0.58\) ✓

Example 2: Cross-Multiply

Compare: \(\frac{5}{9}\) and \(\frac{4}{7}\). Which is greater?

1
Cross-multiply: Compute \(5 \times 7 = 35\) and \(4 \times 9 = 36\).
2
Compare products: \(35 < 36\).
3
Apply rule: Since \(35 < 36\), we get \(\frac{5}{9} < \frac{4}{7}\). ✓

Example 3: Mixed Forms

Order from least to greatest: \(\frac{2}{3}\), \(0.7\), \(65\%\), \(\frac{5}{8}\).

1
Convert all to decimals: \(\frac{2}{3} \approx 0.667\), \(0.7 = 0.700\), \(65\% = 0.650\), \(\frac{5}{8} = 0.625\).
2
Sort the decimals: \(0.625 < 0.650 < 0.667 < 0.700\).
3
Write in original forms: \(\frac{5}{8} < 65\% < \frac{2}{3} < 0.7\). ✓

Example 4: Negative Rationals

Compare: \(-\frac{3}{5}\) and \(-\frac{2}{3}\). Which is greater?

-1 -0.5 0
\(-\frac{2}{3} \approx -0.667\)
\(-\frac{3}{5} = -0.6\)
1
Convert: \(-\frac{3}{5} = -0.6\) and \(-\frac{2}{3} \approx -0.667\).
2
Compare: \(-0.6\) is closer to zero than \(-0.667\).
3
Conclude: \(-\frac{3}{5} > -\frac{2}{3}\) because \(-0.6\) is to the right of \(-0.667\). ✓

Common Mistakes to Avoid

✕ Comparing Numerators Only
Claiming \(\frac{3}{8} > \frac{2}{5}\) because \(3 > 2\). You must use a common denominator or cross-multiply: \(3 \times 5 = 15 < 2 \times 8 = 16\), so \(\frac{3}{8} < \frac{2}{5}\).
✕ Reversing Negatives
Thinking \(-\frac{1}{2} < -\frac{1}{3}\) because \(\frac{1}{2} > \frac{1}{3}\). With negatives, the one with the smaller absolute value is greater: \(-\frac{1}{3} > -\frac{1}{2}\).
✕ Forgetting Percent Conversion
Comparing \(0.4\) to \(35\%\) without converting. \(35\% = 0.35\), so \(0.4 > 35\%\). Always convert percents before comparing.

Strategy Summary

Method Best For Speed Accuracy
Convert to decimals 3+ mixed-form values Fast Watch repeating decimals
Cross-multiply Comparing exactly 2 fractions Fastest Exact
Common denominator Many fractions to order Slower Exact
Benchmark values Quick estimation Fastest Approximate

Question 1 of 5

Which of the following is the greatest?

Question 2 of 5

Which inequality is correct?

Question 3 of 5

Which of the following is greater: \(-\frac{5}{6}\) or \(-\frac{7}{9}\)?

Question 4 of 5

Arrange from least to greatest: \(\frac{3}{4}\), \(70\%\), \(0.72\), \(\frac{5}{7}\).

Question 5 of 5

If \(\frac{a}{b} < \frac{c}{d}\) and both \(b > 0\) and \(d > 0\), which of the following must be true?