Rule of Three Calculator

The setup is always the same sentence: A is to B as C is to X. Three values known, one missing, and a fixed relationship between two quantities. When the two quantities move together — more of one means more of the other — the relationship is direct, and the ratios stay equal: A ÷ B = C ÷ X. Cross multiplying gives X = (B × C) ÷ A. A worked example. If 3 apples cost $6, what do 5 cost? Here A = 3 apples, B = $6, C = 5 apples, so X = (6 × 5) ÷ 3 = $10. The layout that prevents most errors is to keep the same kind of quantity in the same column: apples above apples, dollars above dollars. Written that way, A and C are one quantity, B and X are the other, and it becomes very hard to divide by the wrong number. Units matter as much as position. If A is in hours and C is in minutes, convert before you calculate — the rule has no way of noticing.

Some pairs of quantities move in opposite directions: more of one means less of the other. More workers, fewer days. Higher speed, less travel time. More pipes filling a tank, less time to fill it. For these, the ratio is not what stays constant — the product is. The total amount of work, or the total distance, does not change just because you spread it differently. So instead of crossing, you multiply straight across and divide: X = (A × B) ÷ C. If 4 workers take 6 days, how long do 8 workers take? X = (4 × 6) ÷ 8 = 3 days. The product is 24 worker-days either way. That quantity — 24 worker-days, or kilometre-hours, or tank-minutes — is the real invariant, and naming it is the fastest way to check an inverse problem. If your answer does not reproduce it, the setup was wrong.

Choosing the wrong type is the only serious way to get this wrong, and it is easy to test for. Ask: if I double the first quantity, does the second double or halve? Doubles, so direct: quantity and price, hours worked and pay, distance and fuel used, ingredients and servings, time and distance at a fixed speed. Halves, so inverse: workers and time for a fixed job, speed and travel time for a fixed distance, machines and production hours, people sharing a fixed cost, gear teeth and rotation speed. A second check is the direction of the answer. In a direct problem, if C is larger than A then X must be larger than B. In an inverse problem it must be smaller. Glance at your result before you use it — a number moving the wrong way is a mis-set type, not a rounding issue. The intuition is that direct problems share a rate, and inverse problems share a total.

The rule of three is a shortcut for something more transparent: work out the value of one unit, then scale up. It is worth knowing because it explains what the formula is doing and catches setup errors that the formula hides. Direct: if 3 apples cost $6, one apple costs $2, so 5 apples cost $10. Same answer, arrived at through a number you can sanity-check on its own. Inverse: if 4 workers take 6 days, one worker would take 24 days, so 8 workers take 24 ÷ 8 = 3 days. The intermediate figure of 24 worker-days is exactly the invariant from the previous section. If the two methods disagree, the setup is at fault, not the arithmetic. And when the intermediate value comes out absurd — a single worker finishing in an hour, one apple costing $40 — you have caught a direct/inverse mix-up before it propagated.

When three or more quantities vary at once, you get a compound rule of three. If 4 workers build 200 metres of wall in 6 days, how long do 6 workers need for 500 metres? Handle one factor at a time: more wall lengthens the job (direct), more workers shorten it (inverse). Applying them in either order to the original 6 days gives 10 days. The rule fails wherever the underlying relationship is not proportional, and that is worth taking seriously: Anything with a fixed component. A delivery costing a flat fee plus a per-item charge does not double when the order doubles. Bulk discounts, tiered pricing and utility standing charges all break it the same way. Anything non-linear. Doubling a linear dimension multiplies area by 4 and volume by 8, so paint and concrete estimates do not scale with a simple ratio. Doubling the oven temperature does not halve the baking time. And anything with a practical ceiling. Nine women do not produce a baby in one month; twice the fertilizer does not double the harvest. The mathematics is fine — the assumption of unbounded proportionality is not.