Cross Multiplication Calculator
Welcome to CalcuGo's cross multiplication calculator, where we solve for an unknown variable x in proportions. The expressions we tackle here are often called proportions, and there's an easy algorithmic way to deal with them: cross multiply the fractions. Whichever of the four values is unknown, once we cross multiply and divide, we're sure to get our result.
Solving for x with Fractions
We use cross multiplication when we have one-variable equations involving fractions. The typical form is:
a / b = c / d
where exactly one of the four values (a, b, c, or d) is unknown. For example:
- 2 / x = 5 / 7 — find x in the denominator of the left fraction
- x / 3 = 8 / 12 — find x in the numerator
- 7 / 4 = 21 / x — find x in the denominator of the right fraction
- -1 / 3.4 = 9x / 10 — negative numbers and decimals are fully supported
How to Cross Multiply Fractions
The name says it all: we multiply in a cross pattern. Given the proportion:
a / b = c / d
Cross multiplying gives us:
a × d = b × c
This single equation has no fractions, so it's easy to solve for whichever variable is unknown:
- Find a: a = b × c ÷ d
- Find b: b = a × d ÷ c
- Find c: c = a × d ÷ b
- Find d: d = b × c ÷ a
Step-by-Step Example
Problem: Solve 2/3 = 4/x
- Write the cross-multiplication equation: 2 × x = 3 × 4
- Simplify: 2x = 12
- Divide both sides by 2: x = 6
- Verification: 2/3 = 4/6 = 0.667 ✓ and 2 × 6 = 3 × 4 = 12 ✓
Try it in the calculator above — enter a = 2, b = 3, c = 4, select "Find d", and click Calculate.
Using US Customary & Metric Units
Cross multiplication is particularly useful for unit conversions. For example:
- Miles to kilometers: 1 mile / 1.60934 km = x miles / y km
- Pounds to kilograms: 1 lb / 0.453592 kg = x lb / y kg
- Gallons to liters: 1 gal / 3.78541 L = x gal / y L
- Feet to meters: 1 ft / 0.3048 m = x ft / y m
- Ounces to grams: 1 oz / 28.3495 g = x oz / y g
Currency Exchange with Cross Multiplication
Cross multiplication is also useful for currency conversions at known exchange rates:
- USD to RUB (₽): 1 USD / 90 RUB = x USD / y RUB — find how many rubles for any dollar amount
- EUR to USD ($): 1 EUR / 1.08 USD = x EUR / y USD
- GBP to USD ($): 1 GBP / 1.27 USD = x GBP / y USD
- USD to JPY (¥): 1 USD / 150 JPY = x USD / y JPY
- USD to CNY (¥): 1 USD / 7.2 CNY = x USD / y CNY
- USD to INR (₹): 1 USD / 83 INR = x USD / y INR
- EUR to RUB (₽): 1 EUR / 97 RUB = x EUR / y RUB
- USD to BRL (R$): 1 USD / 5.0 BRL = x USD / y BRL
Simply enter the known exchange rate and the amount you want to convert, select which value to find, and the calculator does the rest.
When Does Cross Multiplication Fail?
Cross multiplication requires that no denominator equals zero, and the known value you divide by in the formula must not be zero. Specifically:
- Finding a: d must not be 0
- Finding b: c must not be 0
- Finding c: b must not be 0
- Finding d: a must not be 0
The calculator will show an error if a division by zero would occur.
Frequently Asked Questions (FAQ)
What is cross multiplication?
Cross multiplication is a technique for solving a proportion (an equation with two equivalent fractions) by multiplying each numerator by the opposite denominator: a/b = c/d becomes a × d = b × c. This eliminates the fractions and produces a simple linear equation.
How do I cross multiply to find x?
Set up your proportion with x in one of the four positions. Cross multiply to get an equation without fractions, then isolate x by dividing. For example, x/5 = 6/15 → 15x = 30 → x = 2.
Can I use negative numbers?
Yes. Cross multiplication works for any real numbers — positive, negative, or decimal. The formula a × d = b × c holds regardless of sign.
Can the denominator be negative?
Yes. For example, 3/(−2) = (−6)/4 is a valid proportion. Cross multiplication still applies: 3 × 4 = (−2) × (−6) → 12 = 12 ✓.
What is a proportion?
A proportion states that two ratios (fractions) are equal: a/b = c/d. Cross multiplication is the standard way to solve for an unknown in a proportion.
How is this different from a ratio calculator?
A ratio calculator typically simplifies a given ratio (e.g., 4:6 → 2:3). A cross multiplication (proportion) calculator solves for an unknown in an equivalence of two ratios.