Cross Multiply Fractions: How To & Examples

Cross multiplying fractions is a shortcut for comparing two fractions or solving for a missing numerator or denominator in a proportion. You multiply the numerator of one fraction by the denominator of the other, and do the same with the remaining pair. This article explains exactly how it works, when to use it, and where people commonly go wrong.

Formula

For two fractions a/b and c/d, cross multiplication produces two products:

• a × d (numerator of the first × denominator of the second)
• c × b (numerator of the second × denominator of the first)

If a × d > c × b, then a/b > c/d. If the products are equal, the fractions are equal. When solving for a missing value, set the two products equal and solve the resulting equation.

Worked Example

Using the calculator defaults: compare 3/4 and 5/6.

1. First product: 3 × 6 = 18
2. Second product: 5 × 4 = 20
3. Since 18 < 20, 3/4 < 5/6

The calculator shows the first product as 18, the second as 20, and the comparison result as “Second”. So the larger fraction is 5/6.

Why Cross Multiplication Works

Cross multiplication is simply a way to give both fractions the same denominator without writing it out. To compare a/b and c/d, you could rewrite them with the common denominator b×d:

• a/b becomes (a×d)/(b×d)
• c/d becomes (c×b)/(b×d)

Since the denominators are identical, comparing the numerators (a×d vs. c×b) is enough. This is why the shortcut works for any two fractions, regardless of size.

When to Use Cross Multiplication

Cross multiplication is appropriate in two main situations:

• Comparing two fractions to see which is larger or if they are equal.
• Solving a proportion where one numerator or denominator is unknown, such as 3/4 = x/12.

It is not for adding, subtracting, multiplying, or dividing fractions. Those operations have their own rules. Cross multiplication only applies to equality or inequality comparisons.

Common Mistakes and Pitfalls

Even experienced students make these errors:

• Cross multiplying when adding or subtracting. You cannot cross multiply to add fractions; you need a common denominator.
• Misaligning numerators and denominators. Always multiply the first numerator by the second denominator, and the second numerator by the first denominator. Swapping them gives wrong results.
• Forgetting to multiply both sides when solving. In an equation like 3/4 = x/12, you set 3×12 = 4×x, then solve 36 = 4x, so x = 9. Missing one product breaks the equation.
• Ignoring signs. If a fraction is negative, the sign carries through the product. For example, comparing -3/4 and -5/6: -3×6 = -18, -5×4 = -20. Since -18 > -20, -3/4 > -5/6.

Cross Multiplication vs. Other Methods

Cross multiplication is not the only way to compare fractions. Other approaches include:

• Common denominator: Rewrite both fractions with the same denominator, then compare numerators. This is essentially what cross multiplication does in one step.
• Decimal conversion: Divide numerator by denominator for each fraction and compare the decimals. This is often easier with a calculator but can be tedious by hand.
• Benchmarking: Compare each fraction to 1/2 or 1. For example, 3/4 is clearly larger than 1/2, while 2/5 is smaller. This works only for rough comparisons.

Cross multiplication is the most efficient for exact comparisons and for solving proportions, which is why it is taught early in algebra.

Real-World Applications

Proportions appear in everyday contexts:

• Cooking: Scaling a recipe from 2 servings to 6 servings uses a proportion like 2/6 = 1/x.
• Maps: If a map scale is 1 inch = 10 miles, then 3 inches = 30 miles is a direct proportion.
• Unit conversion: Converting kilometers to miles uses a fixed ratio, and cross multiplication solves for the unknown distance.
• Finance: Calculating interest or discounts often involves proportions, though these typically require additional formulas.

Limits of Cross Multiplication

Cross multiplication works only for simple fractions (a/b = c/d). It cannot handle:

• More than two fractions in a single comparison.
• Variables in both the numerator and denominator on the same side without additional algebra.
• Nonlinear relationships like x² or square roots.

For those cases, you need more advanced algebraic techniques. Also, cross multiplication does not tell you how much larger one fraction is; it only tells you which is larger or if they are equal.

Frequently Asked Questions

Can you cross multiply when adding fractions?

No. Cross multiplication is only for comparing or solving proportions. To add fractions, you need a common denominator. For example, 1/2 + 1/3 requires converting to 3/6 + 2/6 = 5/6. Cross multiplying would give 1×3 and 1×2, which are not related to the sum.

What if the fractions have different signs?

Treat the signs as part of the numerator. For example, comparing -2/3 and 1/4: cross multiply gives -2×4 = -8 and 1×3 = 3. Since -8 < 3, -2/3 < 1/4. The sign of the product matters, so do not drop it.

Is cross multiplication the same as the butterfly method?

Yes. The “butterfly method” is a visual way to cross multiply: draw lines connecting the numerator of one fraction to the denominator of the other, forming a butterfly shape. It is the same arithmetic, just presented differently for visual learners.

How do you cross multiply with three fractions?

You cannot directly cross multiply three fractions. For three fractions a/b, c/d, e/f, you would first compare two of them, then compare the winner with the third. Or, convert all to a common denominator and compare numerators.

Does cross multiplication work for mixed numbers?

Only after converting mixed numbers to improper fractions. For example, 1 1/2 becomes 3/2. Then you can cross multiply with another fraction. Trying to cross multiply mixed numbers directly gives incorrect results.

Why do we cross multiply when solving proportions?

Because a proportion states that two ratios are equal. Cross multiplication is a direct consequence of multiplying both sides by the product of the denominators, which eliminates the fractions and leaves a simple linear equation. It is the fastest reliable method.

Related

• Compression Ratio Calculation Explained
• Cell Density Equation: Formula & How to Calculate
• Buffer Calculation Formula & How to Use It

in a | Numerator of first fraction | - | 3 | 0
in b | Denominator of first fraction | - | 4 | 0
in c | Numerator of second fraction | - | 5 | 0
in d | Denominator of second fraction | - | 6 | 0
out cross1 | a × d | - | a * d | 0
out cross2 | c × b | - | c * b | 0
out compare | Which fraction is larger | - | (a * d > c * b) ? "First" : "Second" | 0