Cross multiplication means multiplying each fraction's numerator by the other fraction's denominator. For a/b and c/d, you compare a × d against b × c. Whichever side is bigger tells you which fraction is bigger. It's the fastest way to compare fractions, solve proportions, or check if two fractions are equal, and you don't need a common denominator to do it.
Quick Summary
Cross multiplication = multiply each numerator by the other fraction's denominator, then compare the two products.
Use it for three things only: comparing fractions, solving proportions, and checking equivalence.
Don't use it for adding, subtracting, multiplying, or dividing fractions.
Always convert mixed numbers to improper fractions before cross multiplying.
The bigger cross product tells you which fraction is bigger. Equal products mean the fractions are equal.
What Is Cross Multiplication?
Here's the idea. You've got two fractions, a/b and c/d, and you want to know how they relate to each other. Instead of finding a common denominator, you multiply diagonally: a × d and b × c. That's it. Compare the two products, and you've got your answer. The name "cross multiplication" comes from the X-shaped pattern those diagonal multiplications make.
So if a × d comes out bigger than b × c, then a/b is the bigger fraction. If both products match, the fractions are equal. Why does this work? You're really just multiplying both fractions by both denominators at the same time, which wipes out the fractions and leaves you with whole numbers you can actually compare. If you want the full proof, Wikipedia's cross-multiplication article walks through it, and Khan Academy's proportions unit covers it with video examples.
One thing to remember: cross multiplication only tells you which fraction is bigger (or if they're equal). It doesn't give you a new fraction as a result.
The quick-reference rule: For fractions a/b and c/d, if a × d > b × c, then a/b > c/d. If a × d = b × c, the fractions are equal. If a × d < b × c, then a/b < c/d. That one rule covers comparing, equivalence checking, and proportions. Most students first see cross multiplication in 6th or 7th grade during ratio and proportion units, right around Common Core standard 7.RP.
How Do You Cross Multiply to Compare Fractions?
This is what most people use cross multiplication for. You've got two fractions and you want to know which one's bigger. Instead of messing with common denominators, you just multiply diagonally and let the numbers do the talking.
Step-by-Step: Cross Multiply to Compare Two Fractions
1 Write your two fractions next to each other
Let's say you want to compare 3/7 and 5/9.
2 Multiply diagonally
Take each numerator and multiply it by the other fraction's denominator.
3 × 9 = 27 7 × 5 = 35
3 The bigger product wins
The product 27 goes with the first fraction (3/7), and 35 goes with the second (5/9). Since 27 < 35, the first fraction is smaller: 3/7 < 5/9.
Cross Multiplication Examples for Comparing Fractions
Example 1: Which is larger, 2/5 or 3/8?
Multiply diagonally:
2 × 8 = 16 5 × 3 = 15
16 beats 15, so 2/5 > 3/8. Not by much, but 2/5 wins.
Example 2: Which is larger, 4/11 or 7/19?
4 × 19 = 76 11 × 7 = 77
This one's close. 76 vs. 77. Since 76 < 77, we get 4/11 < 7/19. Good luck spotting that without cross multiplication.
Example 3: Are 3/9 and 5/15 the same?
3 × 15 = 45 9 × 5 = 45
Both products are 45, so yes: 3/9 = 5/15. Makes sense if you simplify them. Both reduce to 1/3.
Cross multiplication isn't the only way to compare fractions. You can also use common denominators or convert to decimals. Our full comparing fractions guide covers all three methods, and the comparing fractions calculator lets you check your answers instantly.
How Do You Solve a Proportion by Cross Multiplying?
A proportion says two fractions are equal: a/b = c/d. If one of those four values is missing, cross multiplication turns the whole thing into a basic algebra problem you can solve in seconds.
Step-by-Step: Solve a Proportion Using Cross Multiplication
1 Write the proportion with the unknown
Use x for whatever value you're trying to find.
3/4 = x/20
2 Cross multiply and set the products equal
Same diagonal multiplication, but this time you set both sides equal instead of comparing them.
3 × 20 = 4 × x → 60 = 4x
3 Solve for x
Divide both sides by whatever's attached to x.
x = 60 ÷ 4 = 15
So 3/4 = 15/20. Quick check: 15/20 simplifies to 3/4. It works.
Cross Multiplication Examples for Proportions
Example 4: Solve x/6 = 10/15
x × 15 = 6 × 10 → 15x = 60
Divide by 15:
x = 60 ÷ 15 = 4
Sanity check: 4/6 simplifies to 2/3. So does 10/15. We're good.
Example 5: A recipe needs 2 cups of flour for every 3 cups of milk. How much flour for 12 cups of milk?
Set up the proportion and cross multiply:
2/3 = x/12
2 × 12 = 3 × x → 24 = 3x → x = 8
You need 8 cups of flour. This is where cross multiplication really shines: real-world scaling problems.
When to Use Cross Multiplication (and When Not To)
This is where people get tripped up. Cross multiplication works great for some things and completely fails for others. Here's the breakdown.
Situation
Cross Multiply?
Comparing two fractions (which is larger?)
✓ Yes
Checking if two fractions are equivalent
✓ Yes
Solving a proportion (a/b = c/d)
✓ Yes
Adding fractions
✗ No, use a common denominator
Subtracting fractions
✗ No, use a common denominator
Multiplying fractions
✗ No, multiply straight across
Dividing fractions
✗ No, multiply by the reciprocal
What Are the Most Common Cross Multiplication Mistakes?
Four mistakes come up over and over again. If you can avoid these, you're in good shape.
Don't use it for adding or subtracting. This is the #1 mistake. Cross multiplication is only for comparing, checking equivalence, and solving proportions. If you try to add 1/3 + 1/4 by cross multiplying, you'll get garbage. You need a common denominator for that, and the answer is 7/12.
Keep your products on the right side. When comparing a/b and c/d, the product a × d goes with the first fraction and b × c goes with the second. Flip them, and you'll get the comparison backwards. It sounds obvious, but it's easy to mix up during a test.
Convert mixed numbers first. Got 1 2/3 and 2 1/4? You can't cross multiply them as-is. Turn them into improper fractions (5/3 and 9/4) and then cross multiply. Skipping this step is a guaranteed wrong answer.
Watch out for negatives. If one or both fractions are negative, the sign of each cross product matters. Comparing -2/3 and 1/4 gives (-2 × 4) = -8 and (3 × 1) = 3, so -2/3 < 1/4. Don't lose track of the minus sign halfway through.
Practice Problems
Try these five. Click "Show Answer" when you're ready to check.
1. Which is larger: 5/8 or 7/11?
5 × 11 = 55 and 8 × 7 = 56. 55 is less than 56, so 5/8 < 7/11. Really close, but 7/11 edges it out.
2. Are 6/14 and 9/21 equivalent?
6 × 21 = 126 and 14 × 9 = 126. Same product on both sides, so yes, 6/14 = 9/21. Both simplify to 3/7.
3. Solve for x: x/8 = 15/24
Set up: x × 24 = 8 × 15, which gives 24x = 120, so x = 5. Quick check: 5/8 and 15/24 both simplify to 5/8.
4. Which is larger: 11/13 or 9/10?
11 × 10 = 110, 13 × 9 = 117. Since 110 < 117, 11/13 < 9/10. The fraction 9/10 is bigger.
5. Solve for x: 7/x = 21/30
7 × 30 = x × 21 gives you 210 = 21x. Divide both sides: x = 10. Verify: 7/10 and 21/30 both equal 7/10.
Frequently Asked Questions
It's a shortcut for comparing fractions without finding a common denominator. You take two fractions, a/b and c/d, and multiply diagonally: a × d and b × c. Whichever product is bigger, that's the bigger fraction. You can also use it to solve proportions and check if two fractions are equal.
Three situations: comparing two fractions to see which is bigger, solving a proportion where one value is unknown, or checking whether two fractions are equivalent. That's it. Don't try to use it for adding, subtracting, multiplying, or dividing fractions. Those all have their own methods.
No. This doesn't work for addition or subtraction. If you need to add fractions, find a common denominator first, then add the numerators. Cross multiplication is strictly for comparing, solving proportions, and checking equivalence.
Yes, but you have to convert them to improper fractions first. For example, turn 2 1/3 into 7/3 before you cross multiply. If you skip that step and try to cross multiply the mixed number directly, you'll get the wrong answer.
It's basically a shortcut for multiplying both sides of a comparison by both denominators. When you compare a/b and c/d, multiplying everything by b × d cancels out the fractions and gives you a × d vs. b × c. The relationship stays the same; you just don't have fractions anymore.
Both can tell you which fraction is bigger, but they're good at different things. Cross multiplication is faster when you just need to compare or solve a proportion. Common denominators are what you need when you're actually adding or subtracting fractions, because you need the fractions rewritten with the same bottom number.
For two fractions a/b and c/d: compare a × d with b × c. If a × d is bigger, then a/b is the bigger fraction. If they're equal, the fractions are equivalent. That's the whole formula. It works for comparing, equivalence checks, and proportions.
Write your two fractions side by side. Multiply the first numerator by the second denominator. Then multiply the first denominator by the second numerator. Now compare: the bigger product tells you which fraction is bigger. If both products match, the fractions are equal.
Not directly. Cross multiplication only works with two fractions at a time. If you need to compare three or more, you can either cross multiply each pair separately or convert everything to a common denominator. To rank 1/3, 2/5, and 3/8, for instance, you'd need three pairwise comparisons.
Totally different operations. Cross multiplication goes diagonally (each numerator times the opposite denominator) and tells you which fraction is bigger. Multiplying fractions goes straight across (numerator × numerator, denominator × denominator) and gives you a new fraction. They have different purposes and different results.