Equivalent proportions
Test or solve an equality between two ratios by cross multiplication.
Equivalent proportions is one of 1 ratios formula in the arithmetic section of this library, and it is used at middle school level.
Why equivalent proportions works
If two fractions are equal, multiplying both sides by both bottom numbers must keep them equal. The b cancels on the left and the d cancels on the right, and what survives is ad = bc. Cross multiplying is not a trick, it is that clearing of denominators done in one step.
What each symbol means
$a,b,c,d$ are quantities arranged in matching order.
Equivalent proportions: when it holds
$b\ne0$ and $d\ne0$; units or categories in corresponding positions must match.
When it stops applying
It only handles relationships that scale up together. One painter taking 6 hours does not mean 6 painters take 36 hours; that is an inverse relationship where the product stays fixed instead of the ratio.
Equivalent proportions: a worked example
$3/5=x/20$ gives $5x=60$, so $x=12$.
The mistake to avoid
What people do: Students build the two ratios with the units in different orders, such as miles over hours on one side and hours over miles on the other.
Why it goes wrong: Flipping one side turns the equation upside down and the answer comes out as the reciprocal of what it should be.
Do this instead: Set up both sides in the same order and write the units in. With 3 books for 5 dollars and x books for 20 dollars, books stay on top on both sides: 3/5 = x/20 gives 5x = 60, so x = 12.
Equivalent proportions: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,b,c,d$ are quantities arranged in matching order.
- Check the conditions before substituting. $b\ne0$ and $d\ne0$; units or categories in corresponding positions must match.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most arithmetic slips.
Where this formula fits
- Subject
- Arithmetic formulas — 13 entries in this library
- Topic
- Ratios
- Level
- Middle school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where equivalent proportions comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Ratios and Proportions — the lesson behind this formula: compare quantities and scale equivalent relationships.
- Arithmetic Calculator — check your substitution and the value it produces.
- Study arithmetic — the subject guide that explains the ideas these formulas compress.
- Arithmetic Practice — questions that make you retrieve the formula instead of recognising it.
- All 13 arithmetic formulas — the full grouped reference, or the complete formula library.
Questions about equivalent proportions
Which numbers get multiplied together?
The ones diagonally across from each other: top left with bottom right, and top right with bottom left. Drawing the X over the two fractions makes the pairing hard to get wrong.
Can I use this to check whether two fractions are equal?
Yes, and it is faster than finding a common denominator. Since 7 times 12 and 21 times 4 both give 84, you know 4/7 and 12/21 are the same number.
Is cross multiplication safe with inequalities?
Only when both bottom numbers are positive. Multiplying an inequality by a negative value flips the direction of the sign, so check the signs before you cross multiply.
How do I set up a word problem as a proportion?
Pick a label for the top and a label for the bottom, then keep them in that order on both sides. Writing the units next to each number makes a mismatched setup obvious before you solve.