Comparing Fractions by Cross Multiplication
Learn the cross multiplication method to quickly compare any two fractions, even with different denominators.
Definition
- - If , then
- - If , then
- - If , then
Try it now
Worked Examples
Which is greater: or ?
Set up the cross multiplication
vs → Ready to cross multiply
Multiply first numerator by second denominator
→ Left product:
Multiply second numerator by first denominator
→ Right product:
Compare the products
, so → is greater
Answer:
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.
Interactive Visual
Click on the circle to change the fraction
Fraction Number Line
Select two fractions to compare them.
Interactive Sandbox
Expression Calculator
Try these:
History
No calculations yet
Practice Problems
16 problemsWhich fraction is greater: or ?
Why It Matters
- 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
- 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 and (items per euro). Cross multiply: vs . Since , you get more items per euro with the 5-pack.
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 and :
Recipe Scaling
When comparing recipes, cross multiplication helps determine which uses more of an ingredient proportionally.
Example:
Recipe A uses cup of sugar for 12 cookies. Recipe B uses cup for 12 cookies. Which is sweeter? Cross multiply: vs . Since , Recipe A uses more sugar.
Recipe A uses cup of flour. Recipe B uses cup of flour.
Which recipe uses more flour?
Step 1: Write the mathematical expression
Cross multiply to compare:
Key Takeaways
- 1Cross multiplication compares fractions by multiplying each numerator by the opposite denominator
- 2For vs : compare with
- 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
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 and