Cross Multiplication Calculator - Fraction Equation Solver

A cross multiplication calculator is a fast algebra tool that solves equations of the form (ax+b) / (cx+d) = e / f for x, verifies simple a/b = c/d proportions, and compares two fractions a/b and c/d using cross products. The single move it relies on is the same one textbooks teach: multiply the numerator of each side by the denominator of the other side, then solve the resulting linear equation. The tool does that step for you, displays the cross products a*d and b*c, and shows the substituted-back value so you can confirm the answer.

• • Solve rational equations: Enter (ax+b) / (cx+d) = e / f and read x in one line, with the substituted value shown for verification.
• • Compare two fractions: Enter a/b and c/d, read the >, <, or = comparison that the cross products give you.
• • Verify a proportion: Type a/b = c/d and read the cross products a*d and b*c to confirm the proportion balances.
• • Sanity check word problems: Convert a recipe, map, or rate problem into a/b = c/d, then check the missing value with the calculator.

Cross multiplication is the algebra move that turns two equal fractions into a single product equation, the form that is easiest to solve. The technique generalises from the simple proportion a/b = c/d to the rational equation (ax+b) / (cx+d) = e / f, and from solving a missing value to comparing two fractions or verifying that a stated proportion is true.

For the simple a/b = c/d case, the dedicated Proportion Calculator returns the missing value in a slightly shorter form, while the cross multiplication calculator also covers the equation form and the fraction comparison.

The calculator reads the six inputs a, b, c, d, e, and f, treats the left side as (ax+b) / (cx+d) and the right side as e / f, then applies the cross multiplication rule to turn the equation into a linear expression in x. It also reports the cross products a*d and b*c that you would get from the simpler a/b = c/d interpretation.

• a, b: Coefficient and constant in the numerator of the left side (ax+b).
• c, d: Coefficient and constant in the denominator of the left side (cx+d). Set c to 0 if the denominator has no x.
• e, f: Numerator and denominator of the right side e/f. Use f = 1 to keep it a whole number.
• x: The unknown the calculator solves for. The result panel shows the value that satisfies the equation.

The equation form (ax+b) / (cx+d) = e / f is the same idea applied to a rational function. The calculator solves it with the closed-form expression x = (e*d - f*b) / (f*a - e*c), and uses the denominator f*a - e*c to detect parallel lines (no unique solution) or coincident lines (infinite solutions).

According to Wikipedia, cross multiplication is the procedure of clearing denominators in a fractional equation by multiplying each side by every denominator that appears, and the same step turns a/b = c/d into the equivalent a*d = b*c when no denominator is zero.

According to Khan Academy, multiplying both sides of an equation by the common denominator is the standard way to clear fractions, and that step is exactly the cross multiplication rule the calculator applies to isolate x in a rational equation.

When the solved x or the verification value is a fraction that you need to simplify, multiply, or add, the Fraction Calculator handles the next step without leaving the math-conversion category.

Cross multiplication rests on a small set of rules that also explain its limits. These four concepts cover the algebra, the proportion test, the source of extraneous solutions, and the link to ratios.

These four ideas cover most of what trips students up. The cross multiplication rule is the move itself; the cross products are the verification; clearing denominators is the connection to general algebra; and extraneous solutions are the trap to watch for when the denominator contains x.

The cross products a*d and b*c are the same numbers you compare when you simplify a ratio, so the Ratio Calculator is the natural follow-up when you want the reduced form of a/b and c/d.

Type the six coefficients and constants of your equation into the form, choose a decimal precision, and read the solved x, the cross products, and the verification in the result panel.

1. 1 Write the equation as (ax+b)/(cx+d) = e/f: Match each piece of your equation to a, b, c, d, e, f. If a part has no x, set its coefficient to 0.
2. 2 Enter the six coefficients and constants: Fill the form with a, b, c, d, e, and f. The defaults match the example (2x+1)/(3x-1) = 5/3.
3. 3 Choose a decimal precision: Pick 0 to 8 decimal places. 4 is fine for school, 6 or 8 for engineering, 0 for whole numbers.
4. 4 Read the solved x and the cross products: The primary result gives x, the cross products a*d and b*c sit in the next two rows, and the proportion status tells you whether a/b = c/d holds.
5. 5 Confirm the verification value: Substitute x back into (ax+b)/(cx+d) and check the verification row. If it matches e/f, the solution is correct. If the row says extraneous or undefined, reject it.

Type 2, 1, 3, -1, 5, 3 and read x = 0.8889 with verification 5/3.

If your main goal is to see which of two fractions is larger rather than to solve for x, the Comparing Fractions Calculator is the focused tool for the comparison use case.

A dedicated cross multiplication tool removes the manual multiplication and the off-by-decimal mistakes that show up when you do the cross step yourself, and it returns more than just x.

• • Solves the equation form, not just the proportion: Most proportion tools stop at a/b = c/d. This calculator handles (ax+b)/(cx+d) = e/f directly, which is the form textbooks call cross multiplication in algebra.
• • Cross products a*d and b*c for verification: The proportion test is shown next to the solved x, so you can confirm the answer with the same step you would use by hand.
• • Catches the no-solution and infinite-solution cases: When f*a - e*c = 0 the calculator reports 'no unique solution' or 'every x is a solution' instead of returning a wrong numeric answer.
• • Flags extraneous solutions: If the solved x makes the original denominator zero, the result is marked extraneous so you do not accept a value that the original equation rejects.
• • Adjustable precision: Pick 0 to 8 decimal places so the answer matches the workflow: 0 for recipes, 4 for school, 6 or 8 for engineering and scientific work.

These advantages matter most when you are juggling several problems in a row, such as a homework set, a chemistry dilution, or a scaled recipe. The cross products give a single-glance yes/no check, and the verification row makes sure the algebra does not silently introduce an extraneous solution.

When the cross multiplication confirms that a/b = c/d and you need the matching list of equivalent fractions, the Equivalent Fractions Calculator extends the same result into a sequence you can copy into a recipe or worksheet.

A few real-world factors can change what counts as the right answer. Most users only need to think about the first one, but the others matter for signed equations, parallel cases, and large batch jobs.

• • The calculator handles one rational expression on the left and one fraction on the right. Equations that need two or more cross-multiplication steps, such as (x+1)/(x-1) = (x-2)/(x+3), need a different algebraic setup.
• • All inputs are real numbers. If the problem lives in the complex plane or involves trigonometric, exponential, or logarithmic terms, the cross multiplication step is not enough on its own.

When in doubt, compare the two cross products a*d and b*c and the verification row. If a*d equals b*c to the chosen precision and the verification matches e/f, the cross multiplication is correct.

According to Wikipedia, a proportion is the statement that two ratios are equal, and the cross-multiplied form a*d = b*c is the standard algebraic test for whether the two ratios are in the same proportion.