ALGEBRAArithmeticMathematics Calculator
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Cross Multiplication

Solve proportions a/b = c/d by cross multiplying: aร—d = bร—c. Find any missing value, verify proportions, and compute scale factors. Essential for unit conversion, recipe scaling, and similar figures.

Concept Fundamentals
aร—d = bร—c
Rule
d = (bร—c)/a
Solve for d

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Unit conversion: 1 km = 0.621 miles โ†’ 5 km = ? Recipe scaling: double ingredients proportionally. Similar triangles: corresponding sides are proportional.

Key quantities
aร—d = bร—c
Rule
Key relation
d = (bร—c)/a
Solve for d
Key relation

Ready to run the numbers?

Why: Proportions express equivalent ratios. Cross multiplication converts the proportion into a simple equation.

How: Multiply diagonally: aร—d and bร—c. If equal, the proportion holds. To solve for a variable, isolate it algebraically.

Unit conversion: 1 km = 0.621 miles โ†’ 5 km = ?Recipe scaling: double ingredients proportionally.
Sources:NCTM StandardsRule of Three

Run the calculator when you are ready.

Start CalculatingEnter three values and solve for the fourth.

Enter Values

cross_multiply.sh
CALCULATED
$ cross_multiply --a 2 --b 3 --c 4 --d --solve=d
Result
D = 6
Verified
Yes โœ“
Scale Factor
โ€”
Proportion
a/b = c/d
Cross Multiplication Calculator
D = 6
Verified โœ“
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Proportion Values

Verification

๐Ÿ“ Step-by-Step Breakdown

SETUP
Proportion
a/b = c/d โ†’ aร—d = bร—c
METHOD
Solve for d
d = (bร—c)/a = (3ร—4)/2 = 6
VERIFY
Verify
aร—d = 12, bร—c = 12 โ†’ Equal โœ“
RESULT
Result
D = 6

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

๐Ÿ“œ

Cross multiplication is the "rule of three" from medieval arithmetic.

โ€” History

๐Ÿ”„

Used in unit conversion: 1 km = 0.621 miles โ†’ 5 km = ?

โ€” Conversion

๐Ÿ“‹ Key Takeaways

  • โ€ข For a/b = c/d, cross multiply: aร—d = bร—c
  • โ€ข Solve for any variable: a = (bร—c)/d, d = (bร—c)/a, etc.
  • โ€ข Proportions are equivalent ratios; scale factors relate them

๐Ÿ’ก Did You Know?

๐Ÿ“œCross multiplication is the "rule of three" from medieval arithmetic.Source: History
๐Ÿ”„Used in unit conversion: 1 km = 0.621 miles โ†’ 5 km = ?Source: Conversion
๐ŸณRecipe scaling: double ingredients by scaling both numerator and denominator.Source: Cooking
๐Ÿ“Similar triangles: corresponding sides are proportional.Source: Geometry
๐Ÿ“ŠPercent problems: part/whole = percent/100Source: Percentages
๐Ÿ—บ๏ธMap scales: 1 inch = 10 miles uses the same proportion logic.Source: Maps

๐Ÿ“– How It Works

If a/b = c/d, then multiplying both sides by bร—d gives aร—d = bร—c. To solve for a missing value, isolate it algebraically. Example: 2/3 = 4/d โ†’ 2d = 12 โ†’ d = 6. Verify: 2ร—6 = 3ร—4 = 12.

๐Ÿ“ Worked Example: 2/3 = 4/d

Step 1: Cross multiply: 2ร—d = 3ร—4

Step 2: 2d = 12

Step 3: d = 12/2 = 6

Verify: 2ร—6 = 12, 3ร—4 = 12 โœ“

๐Ÿš€ Real-World Applications

๐Ÿ”„ Unit Conversion

Convert km to miles, cups to ml, etc.

๐Ÿณ Recipe Scaling

Double or halve recipes proportionally.

๐Ÿ“ Similar Figures

Find missing sides in similar triangles.

๐Ÿ“Š Percent Problems

part/whole = percent/100

๐Ÿ—บ๏ธ Map Scales

Convert map distance to real distance.

๐Ÿ’ฐ Financial Ratios

Compare proportions in finance.

โš ๏ธ Common Mistakes to Avoid

  • Division by zero: Denominators cannot be zero. The proportion has no valid solution.
  • Wrong cross: aร—d and bร—c โ€” not aร—c or bร—d.
  • Not verifying: Always plug the solution back and check aร—d = bร—c.

๐ŸŽฏ Expert Tips

๐Ÿ’ก Always verify

Plug the solution back and check aร—d = bร—c

๐Ÿ’ก Denominators โ‰  0

Avoid division by zero

๐Ÿ’ก Scale factor

When proportion holds, scale factor = c/a = d/b

๐Ÿ’ก Reduce first

Reduce fractions first to simplify: 4/6 = 2/3

๐Ÿ“Š Reference Table

Solve forFormula
aa = (bร—c)/d
bb = (aร—d)/c
cc = (aร—d)/b
dd = (bร—c)/a

โ“ FAQ

What is cross multiplication?

For a/b = c/d, cross multiply to get aร—d = bร—c. Used to solve proportions.

When can I use it?

Whenever you have two equal ratios (proportions): fractions, unit conversion, scaling.

Why does aร—d = bร—c?

Multiply both sides of a/b = c/d by bร—d. Left: aร—d. Right: bร—c.

What if a denominator is zero?

Division by zero is undefined. The proportion has no valid solution.

How to verify a proportion?

Check that aร—d equals bร—c. If equal, the proportion holds.

What is the scale factor?

When a/b = c/d, the scale factor is c/a = d/b (how much one ratio is scaled).

๐Ÿ“Œ Summary

Cross multiplication solves proportions a/b = c/d by using aร—d = bร—c. Solve for any variable algebraically. Always verify by plugging back. Essential for unit conversion, recipe scaling, and similar figures.

๐Ÿ”— Next Steps

Try the Ratio Calculator for comparing quantities, or the Unit Rate Calculator for per-unit comparisons.

โš ๏ธ Disclaimer: Results are for educational purposes. Verify critical calculations independently.

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โฌ…๏ธJump in and explore the concept!
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