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The proportion formula, with all four arrangements
Find a, b, c, or d without losing track of the diagonal products.
For four values arranged as a/b = c/d, the denominators b and d must both be nonzero. This condition belongs to the original ratios and still matters after rearranging the equation.
provided b ≠ 0 and d ≠ 0
Choose the missing position
| Missing value | Formula | Required divisor |
|---|---|---|
| a | a = bc/d | d ≠ 0 |
| b | b = ad/c | c ≠ 0; result b ≠ 0 |
| c | c = ad/b | b ≠ 0 |
| d | d = bc/a | a ≠ 0; result d ≠ 0 |
These are four rearrangements of the same cross-product equation. They are useful when checking work, but writing the original proportion first is more reliable than memorizing positions.
Example: an unknown denominator
In 5/8 = 15/d, a = 5, b = 8, and c = 15. The fourth formula gives d = (8 × 15)/5 = 24. Substitution confirms that 15/24 simplifies to 5/8.
Where the formula comes from
Multiplying a/b = c/d by bd gives ad = bc because each denominator cancels a matching factor. To solve for d, divide both sides by a. This last step is valid only if a is not zero. The restrictions in the table are therefore part of the reasoning, not optional fine print.
What if the divisor is zero?
Sometimes the equation has no solution. For example, 0/3 = 2/d cannot hold for any allowed finite d. Sometimes there are infinitely many solutions: 0/3 = 0/d holds for every nonzero d. A calculator should report these cases rather than divide by zero or invent a number.
A missing numerator can legitimately be zero: x/5 = 0/7 gives x = 0. A missing denominator can never be zero because the original fraction would be undefined. Read equivalent ratios for another way to verify answers and the step-by-step guide for the full workflow.
Make the next step yours.
Try your own numbers in the calculator, or check your understanding with a short practice set.