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Teacher Guide: Comparing Fractions by Cross Multiplication

Learn the cross multiplication method to quickly compare any two fractions, even with different denominators.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Comparing & Ordering. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Apply cross multiplication to compare two fractions
  • Determine which of two fractions is greater, lesser, or if they are equal
  • Explain why cross multiplication works mathematically
  • Use cross multiplication in real-world comparison scenarios
Prerequisites
  • • Understanding of fractions as parts of a whole
  • • Basic multiplication of whole numbers
  • • Familiarity with inequality symbols (<, >, =)
Discussion Starters
  • 1. Why might cross multiplication be faster than finding a common denominator?
  • 2. Can you think of a time when you needed to compare two quantities that were not in the same units?
  • 3. If two fractions cross multiply to give equal products, what does that tell you?
  • 4. How could you use cross multiplication when shopping?
Common Misconceptions

The cross products are the values of the fractions

Remediation: Clarify that cross products only tell us which fraction is larger. Calculate the decimal values to show they differ from the cross products.

Cross multiplication only works with proper fractions

Remediation: Demonstrate with improper fractions and mixed numbers (convert to improper first). The method works universally.

Differentiation Ideas

For Struggling Students:

  • • Use visual arrow diagrams showing the cross pattern
  • • Start with fractions where one denominator divides evenly into the other
  • • Provide multiplication tables for reference

For On-Level Students:

  • • Compare fractions with two-digit numbers in numerators or denominators
  • • Apply to word problems involving rates or unit prices
  • • Verify answers by converting to decimals

For Advanced Students:

  • • Compare three or more fractions by pairwise cross multiplication
  • • Explore algebraic proofs of why cross multiplication works
  • • Create real-world problems for classmates to solve
Standards Alignment
  • 5.NF.A.1 (CCSS.MATH.CONTENT.5.NF.A.1)

    Add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions

  • 5.NF.A.2 (CCSS.MATH.CONTENT.5.NF.A.2)

    Solve word problems involving addition and subtraction of fractions referring to the same whole

  • 6.NS.C.7 (CCSS.MATH.CONTENT.6.NS.C.7)

    Understand ordering and absolute value of rational numbers

Lesson Resources
  • visualCross Multiplication Diagram

    Interactive arrows showing the cross pattern

  • activityFraction Face-Off

    Students compete to quickly compare fractions using cross multiplication

  • worksheetCross Multiply Practice

    20 fraction comparison problems with increasing difficulty

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Cross multiplication is a quick method to compare two fractions without finding a common denominator.
To compare ab\frac{a}{b} and cd\frac{c}{d}:
  1. 1.Multiply the numerator of the first fraction by the denominator of the second: a×da \times d
  2. 2.Multiply the numerator of the second fraction by the denominator of the first: c×bc \times b
  3. 3.Compare the two products:
  • If a×d>c×ba \times d > c \times b, then ab>cd\frac{a}{b} > \frac{c}{d}
  • If a×d<c×ba \times d < c \times b, then ab<cd\frac{a}{b} < \frac{c}{d}
  • If a×d=c×ba \times d = c \times b, then ab=cd\frac{a}{b} = \frac{c}{d}
abvscd  ⟹  a×dvsc×b\frac{a}{b} \quad \text{vs} \quad \frac{c}{d} \implies a \times d \quad \text{vs} \quad c \times b

Worked Examples

Which is greater: 34\frac{3}{4} or 57\frac{5}{7}?
1

Set up the cross multiplication

34\frac{3}{4} vs 57\frac{5}{7} → Ready to cross multiply
2

Multiply first numerator by second denominator

3×7=213 \times 7 = 21 → Left product: 2121
3

Multiply second numerator by first denominator

5×4=205 \times 4 = 20 → Right product: 2020
4

Compare the products

21>2021 > 20, so 34>57\frac{3}{4} > \frac{5}{7} → 34\frac{3}{4} is greater

Common Mistakes

Multiplying numerator by numerator instead of cross multiplying

Why it's wrong: Students sometimes confuse cross multiplication with regular multiplication. Cross multiplication means diagonally: numerator of first with denominator of second.

Correct: Always multiply across: numerator of one fraction times denominator of the other.

Mixing up which product corresponds to which fraction

Why it's wrong: After calculating both products, students forget which product belongs to which fraction.

Correct: The product using the first fraction's numerator tells you about the first fraction. Write the products directly below or beside each fraction.

Thinking cross multiplication gives the actual fraction value

Why it's wrong: Students may think the products are the fractions' values. They are not; they are only useful for comparison.

Correct: Cross multiplication tells you which fraction is larger, not what the fractions equal as decimals or in lowest terms.

Why It Matters

Cross multiplication is a powerful shortcut because:
  • Speed: No need to find the least common denominator
  • Accuracy: Works for any two fractions, even with large or unfriendly denominators
  • Foundation: The same technique is used later to solve proportions and equations
Real-life applications:
  • Comparing prices per unit (which deal is better?)
  • Determining which recipe uses more of an ingredient proportionally
  • Checking if two ratios are equivalent

Real World Applications

Shopping Comparisons

Stores often display different package sizes at different prices. Cross multiplication helps find the better deal.

Example:

If a 3-pack costs 8 euros and a 5-pack costs 12 euros, compare 38\frac{3}{8} and 512\frac{5}{12} (items per euro). Cross multiply: 3×12=363 \times 12 = 36 vs 5×8=405 \times 8 = 40. Since 36<4036 < 40, you get more items per euro with the 5-pack.

1Try It Yourself

A bakery sells 4 muffins for 6 euros or 7 muffins for 10 euros.

Which deal gives more muffins per euro?

Step 1: Write the mathematical expression

Compare the cross products 4×104 \times 10 and 7×67 \times 6 with < or >:

Recipe Scaling

When comparing recipes, cross multiplication helps determine which uses more of an ingredient proportionally.

Example:

Recipe A uses 23\frac{2}{3} cup of sugar for 12 cookies. Recipe B uses 35\frac{3}{5} cup for 12 cookies. Which is sweeter? Cross multiply: 2×5=102 \times 5 = 10 vs 3×3=93 \times 3 = 9. Since 10>910 > 9, Recipe A uses more sugar.

2Try It Yourself

Recipe A uses 34\frac{3}{4} cup of flour. Recipe B uses 56\frac{5}{6} cup of flour.

Which recipe uses more flour?

Step 1: Write the mathematical expression

Compare the cross products 3×63 \times 6 and 5×45 \times 4 with < or >:

Key Takeaways

  • 1Cross multiplication compares fractions by multiplying each numerator by the opposite denominator
  • 2For ab\frac{a}{b} vs cd\frac{c}{d}: compare a×da \times d with c×bc \times b
  • 3The fraction whose numerator gives the larger product is the greater fraction
  • 4If both products are equal, the fractions are equivalent
  • 5This method works for any fractions, regardless of their denominators

Frequently Asked Questions

Why does cross multiplication work?

Cross multiplication is actually finding a common denominator behind the scenes. When you compare ab\frac{a}{b} and cd\frac{c}{d}, you could rewrite them as a×db×d\frac{a \times d}{b \times d} and c×bd×b\frac{c \times b}{d \times b}. Since both denominators become b×db \times d, you only need to compare the numerators: a×da \times d vs c×bc \times b.

Can I use cross multiplication with more than two fractions?

Cross multiplication compares exactly two fractions at a time. For three or more fractions, compare them in pairs, or convert all fractions to the same denominator or to decimals.

Does the order matter?

The order determines which product belongs to which fraction. Be consistent: the product using the first fraction's numerator corresponds to the first fraction. If you switch the fractions, switch your interpretation too.

Glossary

Cross multiplication
A method to compare fractions by multiplying each numerator by the opposite fraction's denominator
Numerator
The top number in a fraction, representing parts taken
Denominator
The bottom number in a fraction, representing total equal parts
Equivalent fractions
Fractions that represent the same value, such as 24\frac{2}{4} and 12\frac{1}{2}

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