Cross-multiplication (rule of three)
Cross-multiplication (the rule of three) finds a missing number when two amounts change in proportion. If 3 lb cost $87, 5 lb cost 87 × 5 ÷ 3 = $145.
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4 people finish the inventory in 6 hours. How long do 8 people take?
How to do it: put what you know on one row and what you're after underneath, each amount under its match. You cross-multiply and divide by the one left over. It's also called solving a proportion.
It's direct when both amounts rise together: more pounds, more money. It's inverse when one rises and the other falls: more people, fewer hours. For inverse, multiply across the row and divide: if 3 people take 8 hours, 6 take 3 × 8 ÷ 6 = 4.
It only works if the relationship holds all the way. A fixed cost, a bulk discount or a capacity limit breaks it. A percentage is a proportion too: the part is to the total as x is to 100.
Practice it in How do you calculate a rate per 100,000 people?.
How it's used
Step by step, in Excel
If 3 lb cost $87, what do 5 lb cost?
- 3 lb → $87What you know, on one row.
- 5 lb → xWhat you're after, under its match: pounds under pounds.
- marked: x = 87 × 5 ÷ 3 = $145Cross-multiply, then divide by the one left over.
- In Excel: =B2*A3/A2A2 and B2 are what you know; A3 is the new amount.
When to look closer
When cross-multiplication breaks
Courier: $99 shipping plus $29 a pound
- 2 lb cost $157 ($99 + 2 × $29).
- Cross-multiplied for 6 lb: 157 × 6 ÷ 2 = $471
If 2 lb cost $157 with shipping included, does cross-multiplying get the price of 6 lb right?
Ask your Maestro about Cross-multiplication
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You: What does "Cross-multiplication" mean, in plain words?
Your Maestro: I'll explain it with a case of yours: it sticks better than a definition. What do you figure out at work with cross-multiplication without calling it that?
Asking about: Answers from the page "Why won't pandas add up my amounts? Numbers stored as text"
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Related terms
- Percentage
A percentage says how many parts out of every 100 something is. You get it by dividing the part by the total and multiplying by 100: 45 out of 180 customers is 45 ÷ 180 × 100 = 25%.
- Percent change
Percent change says how much a number moved relative to where it started: (new − old) ÷ old × 100. If sales go from $80K to $100K, they're up 25%.
- Weighted average
A weighted average gives each value a weight for how much it counts, instead of treating them all the same: multiply each value by its weight, add them up and divide by the sum of the weights.