Cross Multiplication Calculator
Solve proportions with one unknown variable
Understanding Cross Multiplication
Cross multiplication is a powerful technique for solving proportions where two ratios are equal. When you have an equation in the form a/b = c/d, cross multiplication states that a × d = b × c.
How Cross Multiplication Works
The principle behind cross multiplication is based on multiplying both sides of the equation by the denominators to eliminate fractions:
- Start with the proportion: a/b = c/d
- Multiply both sides by b: a = (c × b)/d
- Multiply both sides by d: a × d = c × b
- This gives us the cross multiplication formula: a × d = b × c
Real-World Applications
Example: If 3 apples cost $2, how much do 12 apples cost?
Set up proportion: 3/2 = 12/x
Cross multiply: 3x = 24
Solve: x = $8
Common Use Cases
- Scale drawings: Converting measurements on maps or blueprints
- Unit conversions: Converting between different units of measurement
- Recipe scaling: Adjusting ingredient amounts for different serving sizes
- Rate problems: Calculating speed, price per unit, or efficiency
- Similar triangles: Finding missing sides in geometry
- Mixture problems: Determining proportions in solutions
Tips for Success
- Always set up your proportion with corresponding values in the same position
- Label your variables clearly to avoid confusion
- Check your answer by substituting it back into the original proportion
- Make sure units match on each side of the proportion
- Simplify fractions before cross multiplying when possible
Frequently Asked Questions
What is cross multiplication?
Cross multiplication is a method for solving proportions by multiplying the numerator of each fraction by the denominator of the other fraction. For a/b = c/d, it gives us a×d = b×c. This eliminates the fractions and creates a simple equation to solve.
When should I use cross multiplication?
Use cross multiplication when you have two equal ratios with one unknown value. It's particularly useful for word problems involving rates, proportions, scale factors, and similar figures. It's the fastest method for solving simple proportion equations.
Can I use cross multiplication with more than one unknown?
No, cross multiplication only works when you have exactly one unknown variable in a proportion. If you have multiple unknowns, you would need additional equations or information to solve the problem. Each proportion can only solve for one variable.
How do I check if my answer is correct?
Substitute your answer back into the original proportion and verify that both sides are equal. Convert each fraction to a decimal and compare them - they should match. For example, if you found x=9 for 3/4 = x/12, check that 3/4 = 0.75 and 9/12 = 0.75.
What's the difference between a ratio and a proportion?
A ratio is a comparison of two quantities (like 3:4 or 3/4), while a proportion is an equation stating that two ratios are equal (like 3/4 = 6/8). Cross multiplication is specifically used to solve proportions, not individual ratios.
Can I cross multiply with negative numbers?
Yes, cross multiplication works with negative numbers. Follow the same process, but be careful with signs when multiplying. Remember that a negative times a positive gives a negative result, and a negative times a negative gives a positive result.