The cross multiplication calculator above solves a proportion for whichever term is missing. Give it three of the four values in a/b = c/d, tell it which one is unknown, and it returns the missing number together with the cross-multiplied working written out with your own figures substituted in. It also confirms the answer by checking that both cross products are equal, which is the definitive test that two ratios really are in proportion.
Arb Digital publishes this as part of a free arithmetic set because proportion problems are everywhere in commercial work: scaling a recipe, converting a unit price, resizing an image without distorting it, projecting a monthly figure from a partial month, or working out how much budget a channel needs to hit the same efficiency as another. The maths is small, but getting a term the wrong way up is one of the most common and most expensive arithmetic errors people make.
What This Cross Multiplication Calculator Does
It solves the equation a/b = c/d for any single unknown. Choose which of the four positions is missing, fill in the other three, and the answer appears immediately along with the derivation. Solving for d gives d = (b × c) ÷ a. Solving for c gives c = (a × d) ÷ b. Solving for b gives b = (a × d) ÷ c. Solving for a gives a = (b × c) ÷ d. Each of those falls straight out of the same first step: multiply both sides by both denominators to get a × d = b × c, then divide by whichever coefficient sits next to your unknown.
The results grid adds three things a bare answer will not tell you. The two ratio cells show what each side of the proportion actually equals as a decimal, so you can see at a glance whether the relationship is sensible. The scale factor shows what the left-hand ratio was multiplied by to produce the right-hand one — useful when you are scaling a recipe or a design and want to know the multiplier rather than the result. The last cell verifies that a × d and b × c are equal, which is the definition of a valid proportion.
This tool is deliberately different from our ratio calculator, which simplifies and compares ratios such as 16:9 or 3:2. Use the ratio calculator when you want to reduce, scale or compare whole ratios; use this page when you have an equation with one unknown term and want it solved with the algebra shown.
How to Use It
- Write your problem as two fractions. Put the known pair on the left and the pair containing the unknown on the right, keeping the same quantity on top in both.
- Select the unknown position. Most word problems put the unknown at d, the bottom right, but the calculator handles all four.
- Enter the three known values. Decimals and negatives are both accepted; the unknown box is ignored.
- Read the working panel. It shows the cross multiplication substituted with your numbers, then the division that isolates the unknown.
- Check the cross products cell confirms a match. If it does not, one of your inputs is inconsistent with the others.
The Formula and How It's Calculated
A proportion states that two ratios are equal: a/b = c/d. Multiplying both sides by b and then by d clears both denominators and leaves a × d = b × c. The two products are called the cross products because on the page they run along the two diagonals. Once you have that equation, isolating the unknown is a single division.
Take the default values. Three items cost $8, and we want the cost of 21 items, so 3/8 = 21/d. Cross multiplying gives 3 × d = 8 × 21, which is 3d = 168, so d = 168 ÷ 3 = 56. Twenty-one items cost $56. Checking: the left ratio is 3 ÷ 8 = 0.375 and the right ratio is 21 ÷ 56 = 0.375, so the two sides genuinely match, and the cross products are 3 × 56 = 168 and 8 × 21 = 168, which are equal. The scale factor is 21 ÷ 3 = 7, meaning the right-hand pair is simply the left-hand pair multiplied by seven.
Cross multiplication is valid only because multiplying both sides of an equation by the same non-zero quantity preserves equality. That is why a zero denominator breaks it: the step you performed was never legal. The underlying rule of proportion is set out formally in MathWorld's article on proportion.
Setting the Proportion Up the Right Way Round
The arithmetic is trivial. The setup is where marks and money are lost. The rule is that corresponding quantities must occupy corresponding positions on both sides. If items are on top and dollars are on the bottom on the left, items must be on top and dollars on the bottom on the right too. Writing 3/8 = d/21 instead of 3/8 = 21/d gives d = 7.875, which is not the cost of 21 items and is not the answer to any question you asked.
A reliable habit is to label the units next to each number before you solve. If the units read "items over dollars equals items over dollars", the setup is right. If they read "items over dollars equals dollars over items", you have flipped one side and the answer will be wrong by a factor of the ratio squared. This single check eliminates most proportion errors and takes about three seconds.
A second valid arrangement exists: you can also write 3/21 = 8/d, comparing items to items on the left and dollars to dollars on the right. This gives the same answer, because a proportion can be read either across or down as long as you are consistent. What you cannot do is mix the two arrangements within a single equation.
When Proportional Reasoning Does Not Apply
A ratio, as MathWorld defines it, is simply one quantity divided by another — and cross multiplication assumes a strictly linear relationship through the origin: double the input and the output doubles, and zero input gives zero output. Plenty of real quantities do not behave that way, and applying a proportion to them produces a confident, precise, wrong answer.
Anything with a fixed component breaks it. If a delivery costs $8 for 3 items but includes a $5 flat shipping fee, 21 items do not cost $56 — the variable part is only $3 for 3 items, so 21 items cost $21 plus $5, or $26. Bulk discounts break it in the other direction. Tiered tax bands break it. Volume and area break it too: doubling the radius of a circle multiplies the area by four, not two, because the relationship is quadratic rather than proportional. Before you cross multiply, ask whether the relationship really passes through zero and stays straight. Our linear regression calculator is the right tool when a relationship is linear but has an intercept, and our circle calculator handles the quadratic case for areas.
Everyday Uses Worth Knowing
Unit pricing is the most common. Given a price for one quantity, cross multiplication gives you the price for any other, which is how you compare a 400 g pack against a 750 g pack that carry different prices. Recipe scaling is the same operation: 3 eggs for 8 servings scales to 21 servings by the same factor of seven, needing about 8 eggs when rounded sensibly for an ingredient that only comes in whole units.
Image and layout work runs on proportions too. Resizing an 1600 × 900 image to a width of 1200 requires 1600/900 = 1200/h, giving h = 675 — anything else distorts the picture. Our aspect ratio calculator is purpose-built for that case. Percentages are also proportions in disguise: "what is 15% of 240" is the proportion 15/100 = x/240, which is exactly what our percentage calculator solves. Currency and unit conversions work the same way, since an exchange rate or conversion factor is just a fixed ratio applied to any amount.
Reading the Scale Factor Instead of the Answer
The scale factor cell is more useful than it first looks. When you know the multiplier between the two sides, you can often skip the algebra entirely and do the problem in your head. Going from 3 items to 21 items is a factor of seven, so the cost of $8 becomes $56 immediately. Going from 3 to 12 is a factor of four, giving $32. The proportion is only needed when the factor is awkward.
The scale factor also acts as a sanity check on the direction of the answer. If the right-hand quantity is larger than the left, the missing value should be larger too, and if the factor is less than one, the answer should shrink. An answer that moves the wrong way is nearly always a sign that the proportion was set up upside down. When you want the change expressed as a percentage rather than a multiplier, our percentage change calculator converts between the two.
Arb Digital builds free tools like this one and custom calculators for client sites. Browse the public library, or get in touch about something specific to your business.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Flipping one side of the proportion. Corresponding units must sit in corresponding positions. Label the units before solving and the error becomes obvious.
- Cross multiplying across an equals sign that isn't a proportion. The method only works when both sides are single fractions. Add the fractions first if either side is a sum.
- Ignoring a fixed cost or base fee. Any relationship with a constant term is not proportional, and cross multiplication will overstate or understate the result.
- Allowing a zero denominator. If b or d is zero the ratio is undefined, and the cross multiplication step was never valid to begin with.
- Rounding mid-calculation. Round only the final answer. Rounding a scale factor first can shift the result noticeably on large numbers.
Related Free Tools From Arb Digital
To simplify or scale a whole ratio rather than solve for one term, use the ratio calculator. The fraction calculator adds, subtracts and reduces fractions before you cross multiply, and the percentage calculator handles the special case where one denominator is 100. For screen and image work, the aspect ratio calculator keeps dimensions proportional, and the unit converter applies fixed conversion ratios across measurement systems. The full free online tools hub has the rest.
Frequently Asked Questions
Starting from a/b = c/d, multiply both sides by b and then by d. The denominators cancel and you are left with a times d equals b times c. Dividing by whichever number sits beside the unknown then isolates it in one step.
Cross multiply to get a times x equals b times c, then divide both sides by a, so x equals b times c divided by a. With 3/8 = 21/x, that gives x equals 8 times 21 divided by 3, which is 56.
When the relationship is not proportional. Any situation with a fixed fee, a bulk discount, a tiered rate, or a squared or cubed relationship such as area or volume will give a wrong answer, because proportions assume a straight line passing through zero.
The ratio is undefined and the method breaks down, because clearing denominators means multiplying both sides by that value, and multiplying by zero destroys the equation. The calculator flags a zero denominator rather than returning a value.
Yes. Comparing across, as items to items and dollars to dollars, gives the same answer as comparing down, as items to dollars on both sides. What you cannot do is mix the two arrangements within one equation, which is what produces an inverted answer.
Multiply the two diagonals. If a times d equals b times c, the proportion holds. You can also divide each side out as a decimal: both fractions should reduce to exactly the same value.
They are related but not identical. A ratio calculator simplifies or scales an expression such as 16:9, while cross multiplication solves an equation in which one of the four terms is unknown and the other three are given.
