Comparing Fractions Worksheets Grade 7 - Step-by-Step Solutions

Master advanced comparing fractions: rational numbers, negative fractions, complex comparisons, and advanced mathematical reasoning through these 5 detailed exercises.

Detailed Summary: Advanced Comparing Fractions
Key Definitions:

Rational Number: Any number that can be expressed as \(\frac{p}{q}\) where p and q are integers and q ≠ 0

Proper Fraction: A fraction where the absolute value of the numerator is less than the denominator (\(|a| < |b|\))

Improper Fraction: A fraction where the absolute value of the numerator is greater than or equal to the denominator

Mixed Number: A number combining a whole number and a proper fraction

Equivalent Fractions: Fractions that represent the same value but use different numerators and denominators

Like Fractions: Fractions with the same denominator

Unlike Fractions: Fractions with different denominators

Unit Fraction: A fraction with numerator ±1

Core Rules and Principles:

Rule 1: When denominators are the same, compare numerators directly

Rule 2: When numerators are the same, compare denominators inversely

Rule 3: For unlike fractions, find common denominator or use cross multiplication

Rule 4: Any fraction greater than 1 is improper; less than 1 is proper

Rule 5: Equivalent fractions have equal cross products

Rule 6: To compare mixed numbers, compare whole numbers first, then fractional parts

Rule 7: Negative fractions: compare absolute values, then apply sign rules

Rule 8: When comparing negative fractions, reverse the inequality: if \(\frac{a}{b} > \frac{c}{d}\), then \(-\frac{a}{b} < -\frac{c}{d}\)

Step-by-Step Methods:

Method 1 - Same Denominator: Compare numerators directly

Method 2 - Same Numerator: Compare denominators (inverse relationship)

Method 3 - Cross Multiplication: Multiply diagonally and compare products

Method 4 - Common Denominator: Convert to same denominator, then compare numerators

Method 5 - Decimal Conversion: Convert fractions to decimals and compare

Method 6 - Benchmark Comparison: Compare to known values like \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{3}{4}\)

Method 7 - Distance from 1: For fractions close to 1, compare how far each is from 1

Method 8 - Absolute Value: For negative fractions, compare absolute values first

Examples (Simple to Advanced):

Simple: \(\frac{3}{5}\) vs \(\frac{2}{5}\) → Same denominator, 3 > 2, so \(\frac{3}{5} > \frac{2}{5}\)

Moderate: \(\frac{2}{3}\) vs \(\frac{2}{5}\) → Same numerator, 3 < 5, so \(\frac{2}{3} > \frac{2}{5}\)

Advanced: \(\frac{3}{4}\) vs \(\frac{5}{6}\) → Cross multiply: \(3 \times 6 = 18\), \(4 \times 5 = 20\), so \(\frac{5}{6} > \frac{3}{4}\)

Complex: \(\frac{7}{12}\) vs \(\frac{5}{8}\) → LCD = 24: \(\frac{14}{24}\) vs \(\frac{15}{24}\), so \(\frac{5}{8} > \frac{7}{12}\)

Very Complex: \(2\frac{3}{4}\) vs \(2\frac{5}{6}\) → Compare fractional parts: \(\frac{3}{4} = \frac{9}{12}\), \(\frac{5}{6} = \frac{10}{12}\), so \(2\frac{5}{6} > 2\frac{3}{4}\)

Grade 7 Advanced: \(-\frac{5}{7}\) vs \(-\frac{3}{4}\) → Compare absolute values: \(\frac{5}{7}\) vs \(\frac{3}{4}\) → \(\frac{5}{7} < \frac{3}{4}\), so \(-\frac{5}{7} > -\frac{3}{4}\)

Tips and Common Pitfalls:

Tip 1: Always identify whether denominators or numerators are the same first

Tip 2: Remember: larger denominator = smaller fraction (when numerators are equal)

Tip 3: Use benchmarks like \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{3}{4}\) to estimate

Tip 4: For mixed numbers, convert to improper fractions if easier to compare

Tip 5: Use prime factorization to efficiently find LCD for complex denominators

Tip 6: For negative fractions, compare absolute values first, then reverse the inequality

Pitfall 1: Don't just compare numerators without considering denominators

Pitfall 2: Forgetting that \(\frac{4}{4} = 1\) and \(\frac{5}{4} > 1\)

Pitfall 3: Not simplifying fractions before comparing

Pitfall 4: Confusing negative fraction comparisons

Pitfall 5: Forgetting to reverse inequality for negative fractions

Key Notes for Memorization:

Memory Aid: "More pieces, smaller pieces" - when denominator increases, each piece gets smaller

Formula: For \(\frac{a}{b}\) vs \(\frac{c}{d}\): if \(a \times d > b \times c\), then \(\frac{a}{b} > \frac{c}{d}\)

Pattern: Fractions increase as they approach 1 (from below) or decrease as they move away from 1

Verification: Always check your answer by converting to decimals or drawing models

Shortcut: If one fraction is clearly greater than 1 and the other is less than 1, the comparison is obvious

Advanced Shortcut: For fractions close to 1, compare their distances from 1

Negative Rule: The larger the absolute value of a negative number, the smaller the number itself

Solution: Exercises 1 to 3
1 Negative Fraction Comparison
Exercise 1
Compare: \(-\frac{5}{7}\) and \(-\frac{3}{4}\)
Rule:

Negative Fractions: Compare absolute values first, then reverse the inequality. The larger absolute value corresponds to the smaller negative number.

Visual Model:

vs

\(-\frac{5}{7}\) vs \(-\frac{3}{4}\)

Step-by-Step Method:
  1. Compare absolute values: \(\frac{5}{7}\) vs \(\frac{3}{4}\)
  2. Use cross multiplication: \(5 \times 4 = 20\) vs \(7 \times 3 = 21\)
  3. Since \(20 < 21\), then \(\frac{5}{7} < \frac{3}{4}\)
  4. Reverse the inequality for negatives: \(-\frac{5}{7} > -\frac{3}{4}\)
Step 1: Compare absolute values

Compare \(\left|-\frac{5}{7}\right| = \frac{5}{7}\) vs \(\left|-\frac{3}{4}\right| = \frac{3}{4}\)

Step 2: Cross multiply

\(5 \times 4 = 20\) and \(7 \times 3 = 21\)

Step 3: Compare products

20 vs 21 → 20 is less than 21

Step 4: Apply the absolute value comparison

Since \(20 < 21\), then \(\frac{5}{7} < \frac{3}{4}\)

Step 5: Apply the negative fraction rule

For negative numbers, if \(\frac{5}{7} < \frac{3}{4}\), then \(-\frac{5}{7} > -\frac{3}{4}\)

Decimal Verification: \(-\frac{5}{7} ≈ -0.714\) and \(-\frac{3}{4} = -0.75\), confirming \(-0.714 > -0.75\)

\(-\frac{5}{7} > -\frac{3}{4}\)
Final answer:

\(-\frac{5}{7}\) is greater than \(-\frac{3}{4}\)

Applied rules:

• Negative Fraction Rule: Compare absolute values first, then reverse inequality

• Cross Multiplication: Compare absolute values using standard method

• Verification: Convert to decimals to confirm result

2 Mixed Positive and Negative Fractions
Exercise 2
Compare: \(\frac{7}{9}\) and \(-\frac{4}{5}\)
Rule:

Positive vs Negative: Any positive number is greater than any negative number. This is a fundamental principle of the number line.

Number Line Model:

0
\(\frac{7}{9}\)
\(-\frac{4}{5}\)
Step-by-Step Method:
  1. Identify signs: \(\frac{7}{9}\) is positive, \(-\frac{4}{5}\) is negative
  2. Apply fundamental rule: positive numbers are always greater than negative numbers
  3. Therefore: \(\frac{7}{9} > -\frac{4}{5}\)
Step 1: Identify signs

\(\frac{7}{9}\) is positive (numerator and denominator both positive)

\(-\frac{4}{5}\) is negative (preceded by minus sign)

Step 2: Apply the fundamental rule

On the number line, all positive numbers lie to the right of zero

All negative numbers lie to the left of zero

Therefore, any positive number is greater than any negative number

Step 3: State the conclusion

\(\frac{7}{9} > -\frac{4}{5}\)

Step 4: Verification

Convert to decimals: \(\frac{7}{9} ≈ 0.778\) and \(-\frac{4}{5} = -0.8\)

Since \(0.778 > -0.8\), our conclusion is verified

Decimal Verification: \(\frac{7}{9} ≈ 0.778\) and \(-\frac{4}{5} = -0.8\), confirming \(0.778 > -0.8\)

\(\frac{7}{9} > -\frac{4}{5}\)
Final answer:

\(\frac{7}{9}\) is greater than \(-\frac{4}{5}\)

Applied rules:

• Sign Comparison: Positive numbers are always greater than negative numbers

• Number Line Principle: Numbers to the right are greater than numbers to the left

• Verification: Convert to decimals to confirm result

3 Complex Rational Number Comparison
Exercise 3
Compare: \(\frac{-13}{18}\) and \(\frac{-7}{12}\)
Rule:

Negative Rational Numbers: Compare absolute values and reverse the inequality. The fraction with the smaller absolute value is greater when both are negative.

Visual Model:

vs

\(\frac{-13}{18}\) vs \(\frac{-7}{12}\)

Step-by-Step Method:
  1. Compare absolute values: \(\frac{13}{18}\) vs \(\frac{7}{12}\)
  2. Find LCD of 18 and 12: \(18 = 2 \times 3^2\), \(12 = 2^2 \times 3\), LCD = \(2^2 \times 3^2 = 36\)
  3. Convert: \(\frac{13}{18} = \frac{26}{36}\), \(\frac{7}{12} = \frac{21}{36}\)
  4. Compare numerators: 26 vs 21 → 26 > 21
  5. Since \(\frac{13}{18} > \frac{7}{12}\), then \(\frac{-13}{18} < \frac{-7}{12}\)
Step 1: Compare absolute values

Compare \(\left|\frac{-13}{18}\right| = \frac{13}{18}\) vs \(\left|\frac{-7}{12}\right| = \frac{7}{12}\)

Step 2: Find LCD using prime factorization

\(18 = 2 \times 3^2\), \(12 = 2^2 \times 3\)

LCD = highest power of each prime = \(2^2 \times 3^2 = 4 \times 9 = 36\)

Step 3: Convert fractions

\(\frac{13}{18} = \frac{13 \times 2}{18 \times 2} = \frac{26}{36}\)

\(\frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36}\)

Step 4: Compare numerators

26 vs 21 → 26 is greater than 21

Step 5: Apply the negative fraction rule

Since \(\frac{13}{18} > \frac{7}{12}\), then \(\frac{-13}{18} < \frac{-7}{12}\)

Step 6: Verification

Cross multiply: \(13 \times 12 = 156\) vs \(18 \times 7 = 126\)

Since \(156 > 126\), then \(\frac{13}{18} > \frac{7}{12}\), confirming our result

Decimal Verification: \(\frac{-13}{18} ≈ -0.722\) and \(\frac{-7}{12} ≈ -0.583\), confirming \(-0.722 < -0.583\)

\(\frac{-13}{18} < \frac{-7}{12}\)
Final answer:

\(\frac{-13}{18}\) is less than \(\frac{-7}{12}\)

Applied rules:

• Absolute Value Comparison: Compare absolute values first for negative fractions

• Prime Factorization: Efficient method to find LCD for complex denominators

• Negative Inequality Rule: Reverse the inequality when dealing with negative numbers

Solution: Exercises 4 to 5
4 Complex Equivalent Rational Numbers
Exercise 4
Are \(\frac{-21}{35}\) and \(\frac{-15}{25}\) equivalent? Show your work using simplification.
Rule:

Equivalent Rational Numbers: Two rational numbers are equivalent if their simplified forms are identical, or if their cross products are equal (including sign).

Visual Model:

=

\(\frac{-21}{35}\) vs \(\frac{-15}{25}\) (both equal \(\frac{-3}{5}\))

Step-by-Step Method:
  1. Method 1: Cross multiply: \((-21) \times 25 = -525\) vs \(35 \times (-15) = -525\)
  2. Method 2: Simplify both fractions to lowest terms
  3. \(\frac{-21}{35} = \frac{-3}{5}\) and \(\frac{-15}{25} = \frac{-3}{5}\)
Step 1: Cross multiply

\((-21) \times 25 = -525\) and \(35 \times (-15) = -525\)

Step 2: Compare products

-525 vs -525 → Products are equal

Step 3: Simplify \(\frac{-21}{35}\)

Find GCD of 21 and 35: Factors of 21: 1, 3, 7, 21; Factors of 35: 1, 5, 7, 35

GCD = 7: \(\frac{-21}{35} = \frac{-21 ÷ 7}{35 ÷ 7} = \frac{-3}{5}\)

Step 4: Simplify \(\frac{-15}{25}\)

Find GCD of 15 and 25: Factors of 15: 1, 3, 5, 15; Factors of 25: 1, 5, 25

GCD = 5: \(\frac{-15}{25} = \frac{-15 ÷ 5}{25 ÷ 5} = \frac{-3}{5}\)

Step 5: Conclusion

Both fractions simplify to \(\frac{-3}{5}\), so they are equivalent

Decimal Verification: \(\frac{-21}{35} = -0.6\) and \(\frac{-15}{25} = -0.6\), confirming they are equivalent

\(\frac{-21}{35} = \frac{-15}{25}\)
Final answer:

Yes, \(\frac{-21}{35}\) and \(\frac{-15}{25}\) are equivalent fractions

Applied rules:

• Equivalence Test: Cross products must be equal (including signs)

• Simplification: Reduce to lowest terms using GCD

• Verification: Both methods confirm equivalence

5 Advanced Ordering Including Negative Fractions
Exercise 5
Order from least to greatest: \(\frac{2}{3}, -\frac{3}{4}, \frac{5}{6}, -\frac{1}{2}, \frac{7}{8}\)
Rule:

Ordering Mixed Signs: Separate positive and negative numbers, order each group individually, then combine. Negative numbers are always less than positive numbers.

Number Line:

\(-\frac{3}{4}\)
0
\(\frac{2}{3}\)
\(\frac{5}{6}\)
\(\frac{7}{8}\)
Step-by-Step Method:
  1. Separate positive and negative fractions: Negatives: \(-\frac{3}{4}, -\frac{1}{2}\); Positives: \(\frac{2}{3}, \frac{5}{6}, \frac{7}{8}\)
  2. Order negative fractions (remember to reverse inequalities): \(-\frac{3}{4}\) vs \(-\frac{1}{2}\)
  3. Order positive fractions: \(\frac{2}{3}\) vs \(\frac{5}{6}\) vs \(\frac{7}{8}\)
  4. Combine results with negatives first, then positives
Step 1: Separate by sign

Negative fractions: \(-\frac{3}{4}, -\frac{1}{2}\)

Positive fractions: \(\frac{2}{3}, \frac{5}{6}, \frac{7}{8}\)

Step 2: Order negative fractions

Compare absolute values: \(\frac{3}{4}\) vs \(\frac{1}{2}\)

\(\frac{3}{4} = \frac{3}{4}\) and \(\frac{1}{2} = \frac{2}{4}\)

Since \(\frac{3}{4} > \frac{2}{4}\), then \(-\frac{3}{4} < -\frac{1}{2}\)

Step 3: Order positive fractions

Find LCD of 3, 6, 8: \(3=3\), \(6=2 \times 3\), \(8=2^3\), LCD = \(2^3 \times 3 = 24\)

\(\frac{2}{3} = \frac{16}{24}\), \(\frac{5}{6} = \frac{20}{24}\), \(\frac{7}{8} = \frac{21}{24}\)

Order by numerators: \(\frac{16}{24} < \frac{20}{24} < \frac{21}{24}\)

So: \(\frac{2}{3} < \frac{5}{6} < \frac{7}{8}\)

Step 4: Combine results

Negatives come first: \(-\frac{3}{4} < -\frac{1}{2}\)

Positives come next: \(\frac{2}{3} < \frac{5}{6} < \frac{7}{8}\)

Final order: \(-\frac{3}{4} < -\frac{1}{2} < \frac{2}{3} < \frac{5}{6} < \frac{7}{8}\)

Decimal Verification: -0.75, -0.5, 0.667, 0.833, 0.875, confirming the order

\(-\frac{3}{4} < -\frac{1}{2} < \frac{2}{3} < \frac{5}{6} < \frac{7}{8}\)
Final answer:

From least to greatest: \(-\frac{3}{4}, -\frac{1}{2}, \frac{2}{3}, \frac{5}{6}, \frac{7}{8}\)

Applied rules:

• Sign Separation: Order positive and negative numbers separately

• Negative Inequality: Reverse the inequality when comparing negative fractions

• Verification: Convert to decimals to confirm order

Questions & Answers

Question: I'm confused about comparing \(-\frac{7}{10}\) and \(-\frac{5}{8}\). Why is the one with the larger absolute value actually smaller?

Answer: This is a fundamental concept about negative numbers! Think of it this way:

  • On the number line, numbers to the left are smaller than numbers to the right
  • Both \(-\frac{7}{10}\) and \(-\frac{5}{8}\) are to the left of zero
  • First, compare absolute values: \(\frac{7}{10}\) vs \(\frac{5}{8}\)
  • Cross multiply: \(7 \times 8 = 56\) vs \(10 \times 5 = 50\)
  • Since \(56 > 50\), then \(\frac{7}{10} > \frac{5}{8}\)
  • For negative numbers, the one with the larger absolute value is actually smaller
  • Therefore: \(-\frac{7}{10} < -\frac{5}{8}\)

Think of debt: owing $70 (represented as -$70) is worse than owing $50 (represented as -$50). So -$70 < -$50, even though 70 > 50.

Question: My child is working with problems like ordering \(\frac{3}{4}, -\frac{2}{3}, \frac{5}{6}, -\frac{1}{2}\). How can they approach this systematically?

Answer: Use a systematic approach by separating positive and negative numbers:

  1. Separate by sign: Positives: \(\frac{3}{4}, \frac{5}{6}\); Negatives: \(-\frac{2}{3}, -\frac{1}{2}\)
  2. Order negatives (reverse inequality): Compare \(\frac{2}{3}\) vs \(\frac{1}{2}\) → \(\frac{2}{3} > \frac{1}{2}\), so \(-\frac{2}{3} < -\frac{1}{2}\)
  3. Order positives normally: \(\frac{3}{4}\) vs \(\frac{5}{6}\) → Cross multiply: \(3 \times 6 = 18\), \(4 \times 5 = 20\) → \(\frac{5}{6} > \frac{3}{4}\)
  4. Combine results: Negatives first, then positives: \(-\frac{2}{3} < -\frac{1}{2} < \frac{3}{4} < \frac{5}{6}\)

Memory trick: "Separate, order within groups, then combine with negatives first!"

Verification: Convert to decimals: -0.667, -0.5, 0.75, 0.833, confirming the order.

Question: When comparing fractions like \(\frac{-12}{15}\) and \(\frac{-8}{10}\), should I simplify first or use cross multiplication?

Answer: Both methods work, but simplification can make calculations easier:

Method 1 (Simplification):

  • \(\frac{-12}{15} = \frac{-4}{5}\) (divide by GCD = 3)
  • \(\frac{-8}{10} = \frac{-4}{5}\) (divide by GCD = 2)
  • They are equal!

Method 2 (Cross Multiplication):

  • \((-12) \times 10 = -120\) vs \(15 \times (-8) = -120\)
  • Products are equal, so the fractions are equivalent

Simplification is often faster when GCDs are easy to identify. Cross multiplication is more systematic and works for all cases. For teaching purposes, showing both methods helps students understand the underlying concepts.