How this instrument works
An equivalent ratio is a second pair of numbers that carries the exact same proportion as the first, just scaled up or down. Two ratios a:b and newA:newB are equivalent precisely when a × newB equals b × newA — the cross-multiplication rule behind every 'solve for the missing term' problem taught under the name proportion.
That cross-multiplication test is not an arbitrary shortcut; it falls straight out of rearranging one equation. Start from the idea that both ratios share a single scale factor k, so newA = a×k and newB = b×k. Divide the first equation by a to isolate k, substitute it into the second, and k cancels to leave newB = b × newA ⁄ a — the exact formula this sheet evaluates. Comparing ratios by testing whether their cross products match goes back to Book V of Euclid's Elements, where Eudoxus's theory of proportion handled ratios of any lengths, including pairs with no common numeric unit.
One case breaks the formula outright: if a is 0, there is no scale factor to recover, since dividing by zero is undefined — the sheet's only guard rail. A gentler edge case is newA = 0, which is perfectly legal and returns newB = 0, the degenerate ratio 0:0 in scaled form. What tends to surprise people is how many different-looking number pairs describe one ratio: 2:3, 4:6, and 200:300 are the same relationship at different sizes, and this calculator simply finds one more member of that family.
- Enter the two numbers of your known ratio into Original ratio: a and Original ratio: b — say a 2:3 paint-mixing ratio or a 16:9 screen ratio.
- Enter the value you already have for the new, scaled-up first term into New value for a.
- Read Equivalent value for b — the sheet computes newB = b × newA ⁄ a and hands back the matching second term.
- Check the arithmetic by hand if you like: cross-multiply a × newB and b × newA — the two products must come out equal.
- Keep Original ratio: a away from 0 — the formula needs it to recover the scale factor, and division by zero has no answer.
Worked example — scaling a 2:3 mix to a first term of 10
A workshop mixes resin and hardener in a 2:3 ratio — 2 parts resin (a = 2) to 3 parts hardener (b = 3). A bigger batch calls for exactly 10 units of resin, so New value for a is 10. The sheet computes newB = b × newA ⁄ a = 3 × 10 ⁄ 2 = 15, meaning the batch needs 15 units of hardener to keep the mix identical.
The scale factor makes the answer easy to check by hand: 10 is 2 multiplied by 5, so the second term must grow by that same factor of 5, giving 3 × 5 = 15. Cross-multiplying confirms it independently — 2 × 15 = 30 and 3 × 10 = 30, the two products match exactly, which is the definition of equivalent ratios doing its job.
Questions
What makes two ratios equivalent?
Two ratios a:b and newA:newB are equivalent exactly when a × newB equals b × newA — their cross products match. Equivalently, one ratio is the other multiplied through by a single scale factor: newA = a×k and newB = b×k for the same k, whatever k happens to be.
How is an equivalent ratio different from an equivalent fraction?
A ratio compares two separate quantities, like 2 parts resin to 3 parts hardener, and stays meaningful as a pair. A fraction packages that same comparison into one number, 2⁄3, that you can add, subtract, or convert to a decimal. The scaling math is identical, but a ratio keeps both terms visible because the individual quantities usually matter, not just their quotient.
Why does Original ratio: a have to be nonzero?
The formula newB = b × newA ⁄ a divides by a, and division by zero has no defined answer. Geometrically, a = 0 means the first term of the ratio has vanished, so there is no scale factor linking it to any new first term — the proportion has nothing left to scale from.
Where does the cross-multiplication rule come from?
Set both ratios equal to a shared scale factor k: newA = a×k and newB = b×k. Dividing the first equation by a isolates k as newA ⁄ a, and substituting that into the second gives newB = b × newA ⁄ a. Multiplying that result through by a produces the familiar a × newB = b × newA.
What real situations use equivalent ratios?
Anywhere a proportion has to hold while one quantity changes: scaling a recipe, reading a map at 1:50,000, mixing concrete at 1:2:4, keeping a 16:9 screen ratio at a new width, or converting a gear or dosage rate. Knowing one new value and needing its untouched partner is this calculation, every time.
What's the most common mistake when solving for a missing ratio term?
Multiplying the wrong pair of numbers together. The correct products are diagonal: a with newB, and b with newA, not a with newA. Writing the two ratios stacked as a fraction, a⁄b = newA⁄newB, makes the diagonal pairing easier to see before you cross-multiply.