Proportion Calculator

Fill in three boxes, click Calculate. This proportion calculator solves $a : b = c : d$ for the missing fourth term, checks whether four numbers form a true proportion, and reports the scale factor between the two ratios.

• First term (a) - the left ratio's first number. The page loads with 3.
• Second term (b) - the left ratio's second number. It loads with 4, so the left side starts as 3 : 4.
• Third term (c) - the right ratio's first number. Try 9.
• Fourth term (d) - the term the proportion calculator solves. Leave it blank, or fill it in to run a check instead.

Click Calculate on the preloaded 3, 4, 9 and three lines come back:

• Missing 4th term d - cross-multiplication gives $d = c \times b \div a = 9 \times 4 \div 3 = 12$, so 3 : 4 = 9 : 12.
• Cross-product check $a \times d - b \times c$ - fill all four boxes and this line judges them: 0 means a true proportion, anything else means the four numbers do not match.
• Scale factor $c \div a$ - how many times bigger the right ratio is: $9 \div 3 = 3$.

This page is more than a static proportion calculator - it is also an AI tutor you can ask questions:

• Ask in plain words. “The map scale is 1 : 50000 and I measured 3 cm - how far is that?” The AI sets up the proportion and solves it step by step.
• Take a photo of your homework. The AI reads the four terms and shows the cross-multiplication.
• Ask about the ideas, not just the number. “Why does cross multiplication work?” gets a real explanation - and you can check every number against the proportion calculator above.
• Work backwards. Give the answer you suspect and let the check line confirm or veto it in one click.
• Keep asking follow-ups. Each answer opens the next question, and the proportion calculator page keeps teaching.

A proportion is two ratios set equal - a statement that two comparisons are the same:

$\dfrac{a}{b} = \dfrac{c}{d}$

Read it “a is to b as c is to d.” The preloaded example says $3 : 4 = 9 : 12$: two batches of the same mix, the second three times the size. Every ingredient tripled together, so the comparison each ratio records never moved.

That is the test hidden inside the equals sign. Two ratios form a true proportion when they simplify to the same comparison - 3 : 4 and 9 : 12 both collapse to 3 : 4 - and fail when they do not. The proportion calculator on this page judges the claim for you: fill all four boxes and the cross-product check line returns 0 for a true proportion, anything else for a false one.

A proportion also carries a scale factor: the number that multiplies the left ratio into the right one. Here $9 \div 3 = 3$ and $12 \div 4 = 3$ - the same 3 on both terms, which is precisely why the equality holds. The proportion calculator reports that factor on its third line.

Cross multiplication turns a proportion into plain arithmetic: multiply the numerator of each side by the denominator of the other, and set the products equal.

$a \cdot d = b \cdot c$

Why it works: multiply both sides of $\frac{a}{b} = \frac{c}{d}$ by $b \cdot d$. On the left the $b$ cancels, leaving $a \cdot d$; on the right the $d$ cancels, leaving $b \cdot c$. Nothing was smuggled in - only a legal multiplication of both sides by the same quantity.

Now solve $3 : 4 = 9 : d$, the proportion calculator's preloaded problem:

1. Cross-multiply: $3 \cdot d = 4 \cdot 9$, pairing the first with the fourth term and the second with the third.
2. Simplify: $3d = 36$.
3. Divide both sides by 3: $d = 12$.

In one line, $d = b \times c \div a$ - the formula behind the proportion calculator's first output. The same rule solves for any missing term, not just d: cross-multiply first, then divide by whatever sits next to the unknown.

When all four terms are known, the question changes from “solve” to “is it true?” The answer is one subtraction:

$a \times d - b \times c$

If the difference is 0, the cross products match and the four numbers form a true proportion. Anything else - positive or negative - means they do not. That is the entire test, and it is what the proportion calculator's second output line computes on every click.

Run it on the preloaded example: $3 \times 12 = 36$ and $4 \times 9 = 36$, so the check reads $36 - 36 = 0$ - a true proportion, as promised.

Two habits make the check more than a formality:

1. Check your own solving. Solved $d$ by hand? Type it into the fourth box and let the proportion calculator confirm the 0 before you copy the answer anywhere.
2. Read a failed check as information. A nonzero difference is not an error message - it tells you how far off a candidate value is, and in which direction.

The scale-factor line offers a second opinion: if $c \div a$ and $d \div b$ agree, the proportion is true. Two independent tests, one click.

A map's scale is 1 : 50000 - 1 cm on the map stands for 50000 cm on the ground. You measure 3 cm between two trailheads. How far apart are they really?

Set up the proportion with map distances on the left and real distances on the right:

$1 : 50000 = 3 : d$

1. Cross-multiply: $1 \cdot d = 50000 \cdot 3$.
2. Simplify: $d = 150000$ cm.
3. Convert: $150000\text{ cm} = 1500\text{ m} = 1.5\text{ km}$.

The trailheads are 1.5 km apart.

Two habits from this example travel well. First, keep the units in matching positions - map over real on both sides - so the proportion compares like with like. Second, notice the scale factor: the right side is $3 \times$ the left, term by term. The proportion calculator reports that directly in its scale-factor line. Type 1, 50000, 3 into the boxes and the missing-term line hands back 150000, ready for your unit conversion.

Not every pair of quantities moves the same way, and the difference decides which tool you reach for.

Direct proportion - the two quantities rise and fall together, and the ratio between them stays constant: $\frac{y}{x}$ is fixed. Map distance and real distance are direct: triple one, triple the other. The whole machinery of this page - $a : b = c : d$, cross multiplication, the proportion calculator's missing-term line - is direct-proportion machinery.

Inverse proportion - one quantity rises while the other falls, and the product stays constant: $x \cdot y$ is fixed. Workers and deadlines work this way: more workers on the same job, fewer days to finish, because workers times days stays put. Doubling the crew halves the calendar.

To tell which one you are holding, ask one question: does more of one mean more, or less, of the other? More flour means more water - direct. More workers means fewer days - inverse.

The form above solves the direct case. An inverse problem does not fit $a : b = c : d$ - describe it in the AI box instead (“how do I set up an inverse proportion?”), and the tutor shows the product setup. Guessing direct for an inverse problem is the costliest of the mistakes below.

1. Cross-multiplying on the wrong diagonal. The products pair the first with the fourth term and the second with the third: $a \cdot d$ and $b \cdot c$. In $3 : 4 = 9 : d$, the 3 pairs with d and the 4 with 9. Pair the neighbors instead and every answer downstream is wrong - the proportion calculator always multiplies the correct diagonal, so match your hand work against its missing-term line.
2. Fitting an inverse problem into the direct formula. “More workers, fewer days” cannot be written as a true $a : b = c : d$; the workers-times-days product is what stays constant. Cross-multiplying an inverse setup produces a confident, wrong number - read the direct-vs-inverse block above before typing.
3. Leaving units mismatched. In the map example, $1 : 50000 = 3 : d$ returns 150000 centimeters; calling that kilometers without converting is off by a factor of 100000. Keep each side of the proportion in one unit, and convert after solving - a proportion calculator multiplies numbers faithfully but cannot attach the labels for you.