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\(\frac{1}{3}\)
.33

Comparing
Rational Forms

Master the skill of comparing fractions, decimals, and percents by converting between forms and placing values on a number line.

SSAT Upper Level: Quantitative

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 whole numbers, fractions, terminating decimals, and repeating decimals.

Fractions
\(\frac{3}{4}\), \(\frac{-7}{2}\), \(\frac{11}{5}\)
Decimals
\(0.75\), \(-3.5\), \(2.2\)
Percents
\(75\%\), \(350\%\), \(12.5\%\)

One Value, Three Forms

Same Value
\(\frac{3}{4}\)
0.75 75% ÷ ÷ ×100

\(\frac{3}{4}\), \(0.75\), and \(75\%\) all represent the exact same quantity. Comparing rational numbers means recognizing when different forms are equal, and determining which is greater or lesser.

Why Does This Matter?

On the SSAT, you will see values in mixed forms within the same problem. You need to quickly determine which is larger, which is smaller, or whether two expressions are equal.

Quantitative Comparisons
Compare Column A to Column B where values appear as fractions, decimals, or percents
Ordering Problems
Arrange a set of rational numbers from least to greatest (or greatest to least)

The Conversion Rules

1
Fraction → Decimal: Divide the numerator by the denominator: \(\frac{3}{8} = 3 \div 8 = 0.375\)
2
Decimal → Percent: Multiply by \(100\): \(0.375 \times 100 = 37.5\%\)
3
Percent → Fraction: Place over \(100\) and simplify: \(37.5\% = \frac{37.5}{100} = \frac{3}{8}\)
4
Decimal → Fraction: Use place value: \(0.375 = \frac{375}{1000} = \frac{3}{8}\)

The Conversion Cycle

Fraction Decimal Percent ÷ denominator × 100 ÷ 100, simplify

Fractions vs. Decimals: When to Use Each

Use Fractions When
Denominators share common factors, making cross-multiplication or common denominators fast: \(\frac{5}{6}\) vs. \(\frac{7}{9}\)
Use Decimals When
Denominators are powers of 10 or divide evenly: \(\frac{3}{8} = 0.375\) is easier to compare with \(0.4\)

Step-by-Step Comparison Method

1
Identify the forms. Are you comparing a fraction to a decimal, a percent to a fraction, or a mix of all three?
2
Choose a target form. Pick the form that makes the comparison simplest (often decimals for speed).
3
Convert all values to the chosen form using the conversion rules.
4
Compare directly. Use \(<\), \(=\), or \(>\) to state the relationship.
5
Double-check with estimation or a second method if time allows.

Example 1: Fraction vs. Decimal

Which is greater: \(\frac{5}{8}\) or \(0.6\)?

0 0.5 1 0.6 0.625
1
Convert \(\frac{5}{8}\) to a decimal: \(5 \div 8 = \) \(0.625\)
2
Compare: \(0.625\) > \(0.600\)
3
Answer: \(\frac{5}{8}\) is greater than \(0.6\)

Example 2: Three-Way Comparison

Order from least to greatest: \(40\%\), \(\frac{3}{7}\), \(0.43\)

1
Convert \(40\%\): \(40\% = 0.40\)
2
Convert \(\frac{3}{7}\): \(3 \div 7 \approx 0.4286\)
3
Compare decimals: \(0.40\) < \(0.4286\) < \(0.43\)

Example 3: Cross-Multiplication

Which is greater: \(\frac{5}{9}\) or \(\frac{7}{12}\)?

\(\frac{5}{9}\)
\(\frac{7}{12}\)
\(5 \times 12 = 60\)
\(7 \times 9 = 63\)
vs

Handling Repeating Decimals

Some fractions produce repeating decimals. When comparing, carry the division out to enough decimal places to see the difference.

\(\frac{1}{3} = 0.\overline{3}\)
\(= 0.3333...\)
\(\frac{2}{9} = 0.\overline{2}\)
\(= 0.2222...\)
\(\frac{5}{11} = 0.\overline{45}\)
\(= 0.4545...\)

Benchmark Values to Memorize

Fraction Decimal Percent
\(\frac{1}{2}\)\(0.5\)\(50\%\)
\(\frac{1}{3}\)\(0.\overline{3}\)\(33.\overline{3}\%\)
\(\frac{1}{4}\)\(0.25\)\(25\%\)
\(\frac{1}{5}\)\(0.2\)\(20\%\)
\(\frac{1}{8}\)\(0.125\)\(12.5\%\)
\(\frac{2}{3}\)\(0.\overline{6}\)\(66.\overline{6}\%\)
\(\frac{3}{4}\)\(0.75\)\(75\%\)
\(\frac{3}{5}\)\(0.6\)\(60\%\)

Comparing Negative Rationals

-1 0 1 -0.75
\(-\frac{1}{2}\)
-25% ← smaller | larger →

For negative rational numbers, the rules reverse: a number closer to zero is greater. So \(-0.25 > -0.5 > -0.75\).

Common Mistakes to Avoid

✕ Comparing Digits, Not Values
Thinking \(\frac{3}{5} > \frac{2}{3}\) because \(3 > 2\) and \(5 > 3\). Actually, \(\frac{2}{3} \approx 0.667 > 0.6 = \frac{3}{5}\).
✕ Forgetting to Align Decimals
Comparing \(0.5\) with \(0.35\) by looking at digits: \(35 > 5\). Add trailing zeros: \(0.50 > 0.35\).
✕ Reversing Negatives
Saying \(-\frac{3}{4} > -\frac{1}{4}\) because \(3 > 1\). Actually, \(-\frac{1}{4}\) is closer to \(0\), so it is greater.

Strategy Summary

Situation Best Strategy Example
Fraction vs. Decimal Convert fraction to decimal \(\frac{5}{8}\) vs. \(0.6\)
Two fractions (easy denominators) Common denominator \(\frac{3}{4}\) vs. \(\frac{5}{6}\)
Two fractions (hard denominators) Cross-multiply \(\frac{5}{9}\) vs. \(\frac{7}{12}\)
Percent vs. Fraction Convert both to decimals \(65\%\) vs. \(\frac{2}{3}\)
Negative values Compare absolute values, flip sign \(-0.4\) vs. \(-\frac{3}{8}\)

Question 1 of 5

Which of the following is the greatest?

Question 2 of 5

Which symbol correctly completes the statement: \(\frac{7}{12}\) ___ \(0.58\)?

Question 3 of 5

Which list is ordered from least to greatest?

Question 4 of 5

Which of the following is closest in value to \(\frac{5}{7}\)?

Question 5 of 5

Which is greater: \(-\frac{2}{5}\) or \(-0.45\)?