Proportion Calculator
Result
Missing term
- Cross product (A × D = B × C)
- 48.0000
- Scale factor (C ÷ A)
- 4.0000
A proportion calculator solves a statement that two ratios are equal when one of the four terms is missing. Write 3 : 4 = 12 : ? and the answer is 16, because 3 : 4 and 12 : 16 are equivalent ratios — the second is the first with every term multiplied by four. The form is four number slots and nothing else, and the rule is that you leave exactly one of them blank: the blank one is the term being solved for. There is no menu asking which term you want, because the blank already says so, and all four positions work the same way — you can solve for the first term just as easily as the last, which is the question this layout most often gets asked. Only three of the four terms are needed, and the arithmetic is always the same one: cross-multiplying the proportion gives a single equation, A × D = B × C, and solving for the missing term means multiplying the other pair and dividing by its diagonal partner. That is why the panel shows the product of the pair you filled in — it is the number you would arrive at by hand — and, beside it, the factor by which the second ratio scales the first. Ratios may be negative, so the terms may be; zero is the one value that cannot appear anywhere, because a ratio with a zero term either has no solution or has infinitely many.
Ten proportions, and the four ways to leave one slot blank
| First term (A) | Second term (B) | Third term (C) | Fourth term (D) | Missing term | Cross product | Scale factor |
|---|---|---|---|---|---|---|
| 3 | 4 | 12 | 16 | D | 48 | 4 |
| 3 | 4 | 12 | 16 | A | 48 | 4 |
| 3 | 4 | 12 | 16 | B | 48 | 4 |
| 3 | 4 | 12 | 16 | C | 48 | 4 |
| 12 | 5 | 2.4 | 1 | C | 12 | 0.2 |
| 5 | 3 | 20 | 12 | D | 60 | 4 |
| 2 | 3 | 4 | 6 | D | 12 | 2 |
| 1.5 | 4.5 | 2.5 | 7.5 | C | 11.25 | 1.6667 |
| -3 | 4 | -12 | 16 | D | -48 | 4 |
| 7 | 7 | 7 | 7 | D | 49 | 1 |
Each row is a solved proportion, and the columns are the four terms, which slot was left blank, the product of the pair that was filled in, and the factor scaling the first ratio into the second. The first four rows are the same proportion, 3 : 4 = 12 : 16, with the blank moving through D, A, B and C in turn, and every reading is identical across all four — 48 for the cross product and 4 for the scale factor. Reading those four rows down is the fastest answer to whether this page can solve for the first term: it can solve for any of them. They also explain the missing-term column, which prints the letter of the empty slot rather than a number, since the terms and the letters are the same in every language. Row five is 12 : 5 = ? : 1, solved as 2.4, which is the same arithmetic as a price per item and the only row whose answer is not a whole number. Row six is 5 : 3 = 20 : ?, solved as 12. Row seven is the smallest integer proportion, 2 : 3 = 4 : 6. Row eight has decimals in every slot, and its scale factor of 1.6667 is the one reading on the table that does not come out even. Row nine has negative terms, where the cross product is negative and the scale factor is positive. Row ten is 7 : 7 = 7 : 7, whose scale factor of 1 says the two ratios are identical — a proportion can be trivially true, and it still has to be solved like any other.
Formula
A ÷ B = C ÷ D cross-multiplied: A × D = B × C missing term = (the pair you filled in, multiplied) ÷ its diagonal partner
- First term (A)
- The first number of the left-hand ratio: 3 in 3 : 4 = 12 : 16. It may be left blank, and then it is the term this page solves for — the answer is printed on the same line whatever slot was empty. It may be negative, and it may not be zero.
- Second term (B)
- The second number of the left-hand ratio: 4 in the same proportion. As the second member of a ratio it is the one being compared against, so a zero here does not describe anything — a ratio of 3 : 0 is not a ratio. Negative values are fine.
- Third term (C)
- The first number of the right-hand ratio: 12 in the example. It may be blank, and it may be negative. Zero is refused here too, because C ÷ D would then be zero for every D, and no D makes zero equal to a non-zero ratio.
- Fourth term (D)
- The second number of the right-hand ratio: 16 in the example, and the slot that is usually blank because the form supplies the first three. It is the term most often searched for, and it is solved the same way as the other three: multiply the other pair, divide by the diagonal partner.
- Missing term
- Whichever term was left blank, solved for: 16 for 3 : 4 = 12 : ?. It is printed on one line regardless of which of the four slots was empty, because a line labelled after a particular position would be pointing at the wrong slot three times out of four. The answer is never zero: it is one of the two products divided by a non-zero partner.
- Cross product
- The product of the pair you filled in: 48 for 3 : 4 = 12 : ?, from 4 × 12. Cross-multiplying is the statement that the two products are equal, so this is the number on both sides of the equal sign and the one to check by hand. Which pair is shown depends on which slot was blank, and it is always the complete pair — the one that does not need the solved value, which is itself a rounded reading.
- Scale factor
- C divided by A: 4 for 3 : 4 = 12 : 16, because the second ratio is the first with every term multiplied by four. It is not the value of either ratio — those are equal by definition, which is what a proportion says — it is how many times the second ratio scales the first, and it is the quickest way to check the answer by eye.
The first use is the textbook one: three terms of a proportion are known and the fourth is wanted, whether that is a scale drawing, a recipe, a mixture or a currency conversion at a fixed rate. The second is checking a solved proportion — the cross product line is the multiplication you would do by hand, and the scale factor is the one-glance version of the same check. The third is a proportion whose first term is unknown, which some tools cannot express because they assume the answer is always in the last slot; here you simply leave the first slot blank instead. The fourth is a pair of ratios that involve decimals, such as 1.5 : 4.5 = 2.5 : 7.5, where the answer is a decimal and this page prints decimals rather than turning the result into a fraction. The fifth is the boundary with a unit rate: 12 : 5 = ? : 1 is the same arithmetic as working out the price of one item, and this page will solve it, while the unit rate page is the one that insists the second term is exactly 1.
Worked examples
3 : 4 = 12 : ?
- The fourth term is the one left blank, so that is what gets solved
- Cross-multiply: 3 × D = 4 × 12, which is 3 × D = 48
- Divide both sides by 3: D = 48 ÷ 3 = 16
- Check by the scale factor: 12 ÷ 3 = 4, and 4 × 4 = 16, so both ratios are the same
The default pair, and the shape most people arrive with: the first three terms given and the last one wanted. The two supporting readings are the two halves of the check — the cross product 48 is the number on both sides of the equation, and the scale factor 4 is what the second ratio multiplies the first by. Notice that the solved term is the one that makes both ratios equal to 0.75: 3 ÷ 4 and 12 ÷ 16. That equality is the whole content of a proportion, and this page is just rearranging it.
? : 4 = 12 : 16
- The first term is blank this time, and nothing else about the method changes
- Cross-multiply: A × 16 = 4 × 12, which is A × 16 = 48
- Divide both sides by 16: A = 48 ÷ 16 = 3
- The cross product is still 48, because it comes from the pair that was filled in
The same proportion with the answer in the first slot instead of the last, and every reading is identical — 3, 48, 4. This is the question the four-slot layout answers that a form with a fixed output slot cannot: the unknown is wherever you left the hole. It is worth watching the cross product here, because it is 4 × 12 rather than A × 16 — the page multiplies the pair you filled in, not the pair that happens to contain the term it just solved for. Had it used the solved value, the reader checking by hand would be multiplying by a rounded number.
3 : ? = 12 : 16
- The blank is in the second slot, and the method is the same one again
- Cross-multiply: 3 × 16 = B × 12, which is 48 = B × 12
- Divide both sides by 12: B = 48 ÷ 12 = 4
- The cross product comes from the complete pair this time, 3 × 16 = 48
The blank moved to a second position, which is where the two pairs swap over: with the first term missing the complete pair is B and C, and with the second term missing it is A and D. The printed product is 48 either way here, but that is a coincidence of these four numbers rather than a rule — the rule is that the page multiplies whichever pair you filled in, so the number on the panel can always be reproduced by hand from what you typed. Dividing by 12 rather than multiplying by it is the step people most often get backwards.
12 : 5 = ? : 1
- The blank is the third term: 12 : 5 should equal C : 1
- Cross-multiply: 12 × 1 = 5 × C, which is 12 = 5 × C
- Divide both sides by 5: C = 12 ÷ 5 = 2.4
- The scale factor is 2.4 ÷ 12 = 0.2, below one because the second ratio is smaller
The same arithmetic as working out the price of a single item when five of them cost twelve, and the answer is a decimal rather than a whole number. This page prints that decimal and does not turn it into a fraction, which is a deliberate boundary: reducing a result to lowest terms belongs to the fraction pages, and a proportion whose terms are decimals may not have a tidy fraction at all. The scale factor below one is the useful signal here — it says the second ratio is a fraction of the first, which is easy to misread as an error when the answer is supposed to come out larger.
-3 : 4 = -12 : ?
- The blank is the fourth term, and the terms may be negative
- Cross-multiply: -3 × D = 4 × (-12), which is -3 × D = -48
- Divide both sides by -3: D = -48 ÷ -3 = 16
- The cross product is negative and the scale factor is positive, and both are correct
Negative terms are legal, because a ratio may be negative and -0.75 = -0.75 is a true proportion. The two supporting readings point in opposite directions here and that is not a contradiction: the cross product -48 is the product of the two terms on one side of the equation, while the scale factor 4 is -12 ÷ -3, and dividing one negative by another gives a positive. This is the case to look at if the page ever seems to be returning a sign at random — the sign belongs to whichever product or quotient is printed, not to the proportion as a whole.
Limitations
Exactly one slot is left blank. Fill all four and the page refuses, even if the four numbers do form a correct proportion such as 3 : 4 = 12 : 16 — with no blank there is no term to solve for, and printing one of them as the answer would be the page quietly picking which. That also makes the way to check a proportion you believe holds: leave one term blank and see whether the solved value is the number you had in mind. Fewer than three filled slots are refused as well, since there is nothing to solve. Zero cannot appear in any of the four slots, and the four cases come down to one reason: in A × D = B × C a zero term either makes the equation unsolvable or makes it true for every value, so the page rejects it rather than choosing an answer. A zero as the second or fourth term is the clearest case, since a ratio written against zero is not a ratio at all. Negative terms are legal and the answer may be negative, so do not read a minus sign as an error. The four readings are rounded to four decimal places, which matters when a term does not divide evenly: 1.5 : 4.5 = 2.5 : 7.5 solves to 2.5 with a scale factor of 1.6667. Results are printed as decimals, never as fractions — this page does not reduce anything, and a proportion whose terms are decimals may have no exact fraction to reduce. None of the four terms carries a unit, because a proportion compares quantities of the same kind and attaching a dimension would shut out every other kind; if the ratio is a rate with units on each side, the unit rate page is the one that handles it. The page solves one proportion at a time and offers no compound or chained proportions.
Frequently asked questions
- How do you solve a proportion?
- Cross-multiply, then divide. For 3 : 4 = 12 : ? the formula gives 3 × D = 4 × 12, so 3 × D = 48 and D = 16. In words: multiply the pair you filled in, then divide by the diagonal partner of the term you are solving for. The same single method covers all four positions.
- Can I solve for the first term rather than the last?
- Yes, and for any of the four. Leave the slot you want solved blank and fill the other three — there is no separate mode and no menu asking which term you mean. For ? : 4 = 12 : 16 the answer is 3, and the panel looks the same as when the blank is at the end.
- What happens if I fill in all four terms?
- The page refuses, even when the four numbers do form a correct proportion. With no blank there is no term to solve for, and printing one of them would be the page choosing for you. If you want to check a proportion you believe holds, leave one term blank and see whether the solved value matches the number you had in mind.
- Can the terms be negative or zero?
- Negative terms are fine: -3 : 4 = -12 : 16 is a true proportion. Zero is refused in any of the four slots. In A × D = B × C a zero term either makes the equation unsolvable or makes it true for every value, and a ratio such as 3 : 0 is not a ratio at all. The solved term itself is never zero.
- What is the cross product line showing?
- The product of the pair you filled in — 48 for 3 : 4 = 12 : ?, from 4 × 12. Cross-multiplying a proportion says the two products are equal, so this is the number sitting on both sides of the equals sign and the one worth checking by hand. Which pair is printed depends on which slot was blank, so it is always a pair you can multiply from what you typed.
- What does the scale factor mean?
- How many times the second ratio scales the first: 4 for 3 : 4 = 12 : 16, since 12 is 3 × 4 and 16 is 4 × 4. It is not the value of the ratio itself — the two ratios are equal by definition, which is what a proportion says. It is the quickest one-glance check that the answer is the right size.
References
- Proportional — the relation between two quantities whose ratio is constant, which is what a proportion states and what this page solves for its missing term — Wolfram MathWorld (United States)
- Linear Equation — an equation in which each term is a constant or a constant multiple of a single variable, which is the form cross-multiplication reduces a proportion to before the missing term is divided out — Wolfram MathWorld (United States)
- Ratio — the quotient of two numbers, which is what each side of a proportion is and what the scale factor line measures the change in — Wolfram MathWorld (United States)
- Least Common Multiple — used on this page's neighbour, the ratio reducer; listed here because reducing a ratio and solving a proportion are the two operations most often confused with each other — Wolfram MathWorld (United States)
- Greatest Common Divisor — the divisor behind reducing a ratio to its simplest form, which this page does not do because it returns decimals rather than a reduced ratio — Wolfram MathWorld (United States)