Cross multiplication is a shortcut for solving a proportion: if ab = cd, then ad = bc. Multiply each numerator by the denominator on the other side and set the two products equal. To solve x6 = 1015, write 15x = 60, so x = 4. It works only when one fraction equals one fraction.
In Spanish
multiplicación cruzada (productos cruzados)
Also called
cross multiply, cross products, means and extremes, butterfly method
Each numerator is multiplied by the opposite denominator, along the two diagonals of the X. The two products are equal, which gives a simple equation.
Formal definition and notation
For real numbers with b ≠ 0 and d ≠ 0, ab = cd if and only if ad = bc. This follows from multiplying both sides of the proportion by bd: ab × bd = cd × bd gives ad = cb. The products ad and bc are called the cross products; checking that they are equal also tests whether two ratios are equivalent.
Symbol
Read it as
Meaning
ad = bc
a times d equals b times c
the cross products of ab = cd
a : b = c : d
a is to b as c is to d
the same proportion in ratio form; a and d are the extremes, b and c the means
Key facts
It is the same as multiplying both sides by both denominators; it is a shortcut, not a new rule.
Use it only on one fraction = one fraction. With extra terms, such as x2 + 1 = 34, use a common denominator instead.
Equal cross products mean two ratios are equivalent: 68 and 912 give 72 = 72.
Keep the units in the same places on both sides: cups over cookies equals cups over cookies.
When a side is a binomial, use parentheses: x + 25 = 93 gives 3(x + 2) = 45.
Cross-multiplication rule
Cross-multiplication rule
ab = cd ⇔ ad = bc
a, d
first numerator and second denominator (one diagonal)
b, c
first denominator and second numerator (the other diagonal)
b, d ≠ 0
neither denominator may be zero
After cross multiplying, divide by the number next to the variable. Always check the answer in the original proportion.
Try it: equivalent ratios
Interactive
Every row of this table is equivalent to 3 : 4. Pick any two rows: their cross products are equal, such as 3 × 20 = 4 × 15.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
A recipe uses 3 cups of flour for 24 cookies. How many cups of flour are needed for 60 cookies?
Keep the same units in the same places: 3 cups24 cookies = c cups60 cookies.
Cross multiply: 24c = 3 × 60 = 180.
Divide: c = 180 ÷ 24 = 7.5.
Answer: You need 7.5 cups of flour.
Common mistakes
✗ Mistakex2 + 1 = 34: cross multiply to get 4x + 1 = 6.
✓ InsteadSubtract 1 first: x2 = −14, then 4x = −2, so x = −12.
Cross multiplication needs exactly one fraction on each side. Isolate the fraction before using it.
✗ MistakeFor 34 = x20: 3x = 4 × 20, so x = 803.
✓ InsteadMultiply along the diagonals: 4x = 3 × 20, so x = 15.
Multiplying the two numerators and the two denominators is the rule for multiplying fractions, not for solving proportions.
✗ Mistake3 cups24 cookies = 60 cookiesc cups, giving 3c = 1440, so 480 cups.
✓ Instead324 = c60 (cups over cookies on both sides), so c = 7.5.
If the units do not match top to top and bottom to bottom, the proportion is wrong before you multiply. 480 cups should fail a sense check.
Not to be confused with
Cross Multiplicationvs.Multiplying fractions
To multiply fractions you go straight across: 23 × 45 = 815. Cross multiplication works only on an equation with one fraction on each side.
Cross Multiplicationvs.Cross-cancelling
Cross-cancelling simplifies before multiplying fractions: in 49 × 38, divide 4 and 8 by 4 and 3 and 9 by 3. It is a simplifying step, not a way to solve equations.
A proportion is the equation ab = cd. Cross multiplication is one method for solving it.
Quick check: 3 questions
Question 1 of 3
Solve 5x = 1521.
Show answer and explanation
Answer: C. Cross multiply: 15x = 5 × 21 = 105, so x = 7. 63 multiplies 3 by 21, 9 adds 4 to 5, and 3 divides 15 by 5.
Question 2 of 3
Which pair of ratios forms a proportion?
Show answer and explanation
Answer: D.6 × 15 = 90 and 9 × 10 = 90: equal cross products. A gives 32 and 36, B gives 9 and 25, C gives 16 and 21.
Question 3 of 3
A car travels 150 miles on 6 gallons. At the same rate, how many gallons does it need for 425 miles?
Show answer and explanation
Answer: A.6150 = g425, so 150g = 2550 and g = 17. 70.8 divides 425 by 6 (miles per gallon set up upside down), and 10,625 multiplies 425 by 25.
Cross Multiplication: frequently asked questions
Why does cross multiplication work?
It is a shortcut for multiplying both sides of ab = cd by both denominators, bd. On the left, the b cancels and leaves ad; on the right, the d cancels and leaves bc. Since you did the same thing to both sides, the equation stays true: ad = bc.
When can you use cross multiplication?
Only when the equation is a single fraction equal to a single fraction, like x4 = 912. If either side has another term added or subtracted, move it first or clear denominators another way. It also works with variables in the denominator, as long as no denominator ends up being 0.
Is cross multiplication the same as solving a proportion?
It is the most common way to solve one. You can also scale: in 34 = x20, notice that 4 × 5 = 20, so x = 3 × 5 = 15. Or use the unit rate. All methods give the same answer; cross multiplication is the one that always works for any numbers.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.RP.A.3
Cite this entry
Effortless Math Team. “Cross Multiplication.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/cross-multiplication/
Flashcard
TermCross Multiplicationab = cd → ?
DefinitionMultiply along the diagonals of a proportion and set the products equal: ad = bc. 34 = x20 gives 4x = 60, so x = 15.
Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2ProportionA proportion is an equation that says two ratios are equal, such as 12 = 36. It is read 1 is to 2 as 3 is to 6. In a true proportion the cross products are equal, which lets you find a missing value. Proportions are often used to solve scale, percent, and rate problems.Ratios and Proportions · Grade 6, Grade 7, Grade 8RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5Common DenominatorA common denominator is a number that can be used as the denominator of two or more fractions at once. It is any common multiple of the denominators. For 13 and 14, 12 works: 412 and 312. You need one to add, subtract or compare fractions with different denominators.Number System · Grade 4, Grade 5, Grade 6BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2Unit RateA unit rate is a rate that compares a quantity to one unit of another quantity, such as 50 miles per hour or 4 dollars per notebook. To find it, divide the first quantity by the second: 150 miles in 3 hours is 150 ÷ 3 = 50 miles per hour. The word per means for each one.Ratios and Proportions · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2FractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5Equivalent FractionsEquivalent fractions are fractions that name the same value, like 12, 24 and 36. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.Number System · Grade 3, Grade 4, Grade 5Scale DrawingA scale drawing is a drawing of a real object in which every length is multiplied by the same number, so it has the same shape but a different size. Its scale, such as 1 cm = 2 m, links a length on the drawing to the actual length. Maps, floor plans and blueprints are scale drawings.Ratios and Proportions · Grade 7Similar FiguresSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Geometry · Grade 7, Grade 8, Algebra 1PercentA percent is a number out of 100. Percent means per hundred, so 25% means 25 out of 100, or 25100 = 0.25. To find a percent of a quantity, change the percent to a decimal and multiply: 20% of 80 is 0.20 × 80 = 16. The symbol for percent is %.Number System · Grade 5, Grade 6, Grade 7Proportional RelationshipA proportional relationship is a relationship between two quantities in which one is always the same multiple of the other, so y = kx for a fixed number k. The ratio yx is the same for every pair, and the graph is a straight line through the origin. Example: at 3 dollars per pound, cost = 3 × pounds.Ratios and Proportions · Grade 7, Grade 8, Algebra 1Directly ProportionalDirectly proportional describes two quantities whose ratio never changes: y = kx for a fixed nonzero number k, the constant of proportionality. If x doubles, y doubles; if x is cut in half, so is y. The graph is a straight line through the origin, and yx = k for every pair.Ratios and Proportions · Grade 7, Grade 8, Algebra 1