Solve for X in a Proportion Calculator

How to use this calculator

Type the letter x into exactly one of the four boxes of the proportion — it can be either numerator or either denominator. Fill the other three boxes with whole numbers (positive or negative). The calculator cross-multiplies, isolates x, and shows the exact fraction, the mixed number (if x is improper), and the decimal, along with every intermediate step.

The cross-multiplication formula

A proportion sets two fractions equal:

$\frac{n_1}{d_1} = \frac{n_2}{d_2}$

Cross-multiplying gives one universal identity, no matter which term is unknown:

$n_1 \times d_2 = d_1 \times n_2$

Whichever term is x, divide both sides by the number multiplied against it to isolate x. For example, if x is $n_1$: $x \times d_2 = d_1 \times n_2 \Rightarrow x = \dfrac{d_1 \times n_2}{d_2}$.

Why cross-multiplication works

Cross-multiplying is really just multiplying both sides by 1, written as the other side's denominator over itself:

$\frac{n_1}{d_1} \times \frac{d_2}{d_2} = \frac{n_2}{d_2} \times \frac{d_1}{d_1} \;\Rightarrow\; \frac{n_1 d_2}{d_1 d_2} = \frac{n_2 d_1}{d_1 d_2}$

Both sides now share the denominator $d_1 d_2$, so the numerators alone must be equal: $n_1 d_2 = d_1 n_2$. Nothing about the proportion's value changed, since multiplying by $\frac{d_2}{d_2}$ or $\frac{d_1}{d_1}$ is multiplying by 1.

Worked examples

Example 1 — x is a numerator: $\dfrac{x}{4} = \dfrac{3}{8}$

Cross-multiply: $x \times 8 = 4 \times 3 = 12$, so $x = \dfrac{12}{8} = \dfrac{3}{2} = 1\dfrac{1}{2} = 1.5$.

Example 2 — x is a denominator: $\dfrac{2}{x} = \dfrac{4}{6}$

Cross-multiply: $2 \times 6 = x \times 4 \Rightarrow 12 = 4x$, so $x = \dfrac{12}{4} = 3$ — a whole number, so no mixed form or decimal is shown.

Example 3 — matches the source example, $\dfrac{5}{6} = \dfrac{x}{18}$

Cross-multiply: $5 \times 18 = 6 \times x \Rightarrow 90 = 6x$, so $x = \dfrac{90}{6} = 15$.

Example 4 — x stays a fraction: $\dfrac{2}{3} = \dfrac{x}{7}$

Cross-multiply: $2 \times 7 = 3 \times x \Rightarrow 14 = 3x$, so $x = \dfrac{14}{3} = 4\dfrac{2}{3} \approx 4.666667$.

Checking your answer

Once you have a value for x, substitute it back into its original box and cross-multiply once more. If the two cross products come out equal, the proportion holds and the answer is confirmed — for Example 3 above, $\dfrac{5}{6} = \dfrac{15}{18}$ cross-multiplies to $5 \times 18 = 90$ and $6 \times 15 = 90$, which match.

Precision note

Every value is computed as an exact fraction using arbitrary-precision integers (BigInt), never floating-point numbers — so the cross product and the reduced answer are always exact. Only the decimal display, when x isn't a whole number, is limited to 6 digits after the point (with a trailing ellipsis for repeating decimals like $\frac{14}{3} = 4.666666\ldots$).