Three numbers are known and a fourth is missing: 6 litres of fuel take a car 75 km, so how many litres for 200 km? 5 machines make 400 parts in 8 days, so how many parts do 3 machines make in 10 days? The rule of three is the method for these questions. It is the proportion of the previous chapter turned into a routine: decide, quantity by quantity, whether it moves with the unknown or against it, write the ratios, and cross-multiply. With two quantities it is the simple rule of three; with three or more, the compound one.
By the end you can
- Solve a simple rule of three, direct and inverse.
- Decide whether two quantities are directly proportional, inversely proportional, or not proportional at all.
- Set up a compound rule of three with the arrow method.
- Explain why the ratios of a compound rule of three multiply.
- Check an answer with the constant the quantities share.
The name says what the method uses: three known numbers find the fourth. Medieval and Renaissance arithmetic books called it the golden rule, because merchants used it for every price, exchange and share; Portuguese, Spanish and Italian still call it the rule of three, while English school books usually just say proportion problems. Put the two quantities in two columns, the known case in one row and the new case in the next. When the quantities are directly proportional (double one and the other doubles), the ratio of the two values in one column equals the ratio in the other column.
Everything depends on one decision, made before any arithmetic. Imagine doubling one quantity while everything else stays as it was. If the other quantity doubles too, they are directly proportional and their ratio is fixed. If it halves, they are inversely proportional and their product is fixed: twice the workers, half the days. If it does something else (stays the same, quadruples, grows by a fixed amount), they are not proportional at all, and the rule of three gives a wrong answer however carefully it is done. "More gives more" is not enough for direct: it has to be in the same proportion.
| Pair of quantities | Type | Why |
|---|---|---|
| kilograms bought and price paid | direct | every kilogram costs the same |
| machines and parts made (same days) | direct | every machine makes the same number of parts |
| speed and travel time (same distance) | inverse | speed × time = distance, which is fixed |
| workers and days (same job) | inverse | workers × days = the work, in worker-days, which is fixed |
| taps and time to fill a tank | inverse | taps × hours = the tank's volume in "tap-hours" |
| side of a square and its area | neither | double the side and the area is multiplied by 4 (area = side²) |
| taxi kilometres and fare | neither | the fixed starting fee makes 0 km cost more than 0; double the km and the fare less than doubles |
| a child's age and height | neither | a 10-year-old is not twice as tall as a 5-year-old |
At 60 km/h a trip takes 4 hours. At 80 km/h it takes less, so speed and time move against each other. The distance did not change, and distance = speed × time, so the product is what stays fixed: 60 × 4 = 240 = 80 × x, and x = 3 hours. In the two-column layout this means one ratio has to be turned upside down before cross-multiplying. The usual way is to keep the column with x as it is and flip the inverse one.
Brazilian and Portuguese schools teach a layout that makes the decision visible, and it scales to any number of quantities:
- Write one column per quantity, with its unit at the top. The known case goes in the first row, the new case in the second; x goes where the missing number is.
- Draw an arrow down the column with x. It is the reference.
- For each other column, ask the doubling question against x alone, as if every other column were frozen. Same direction: arrow down. Opposite: arrow up.
- Write the ratio of the x column, top over bottom. Set it equal to the product of the other ratios, each read in the direction of its own arrow: a down arrow is top over bottom, an up arrow is bottom over top.
- Solve for x, then check the answer with the constant (next section).
The column with x always gets a down arrow. For every other column, freeze all the rest and ask: if this quantity grows, does x grow too? Then its arrow points down like x's (direct). Does x shrink? Then it points up (inverse), and its ratio is written upside down, in purple. The ratio of x's column equals the product of the others, each read along its arrow. Start with every arrow down and see how far off the answer is.
5 machines make 400 parts in 8 days. How many parts do 3 machines make in 10 days? Now three quantities change together. The key is step 3 of the method: compare each quantity with the unknown on its own. With the days frozen, more machines make more parts: direct. With the machines frozen, more days make more parts: direct. Machines and days are never compared with each other; only with x. Both arrows point down like x's, so 400/x = (5/3) · (8/10) = 40/30, and x = 400 · 30/40 = 300 parts.
With an inverse quantity: the wall
6 workers, working 8 hours a day, build 400 m of wall in 20 days. How many days do 10 workers, working 6 hours a day, need for 600 m? The unknown is days. More workers: fewer days (inverse, arrow up). More hours a day: fewer days (inverse, arrow up). More metres: more days (direct, arrow down).
Why the ratios multiply
The method looks like a recipe, but it comes from one fact: when each quantity is proportional to the unknown on its own, there is a single constant that ties them all together. In the wall, each worker lays the same length per hour, so metres = k × workers × hours a day × days, with the same k for both teams.
A percentage as a rule of three
18 students are 40 % of a class. How many students are in the class? The whole class is 100 %, and students and percentage are directly proportional: 18 goes with 40, x goes with 100. 18/x = 40/100, so 40x = 1800 and x = 45. This is the same as the "whole = part / percentage" of the previous chapter, 18/0.4 = 45, laid out as a rule of three.
Scaling a recipe
A cake for 8 people takes 300 g of flour and 3 eggs. For 12 people: 8/12 = 300/x gives x = 450 g of flour, and 8/12 = 3/y gives y = 4.5 eggs. The baking time does not follow: a bigger cake does not take 1.5 times as long, because time is not proportional to the number of guests. The doubling test catches this before the arithmetic does.
Three inverse quantities
4 identical pumps, running 6 hours a day, empty a reservoir in 15 days. How many days do 5 pumps, running 8 hours a day, take? More pumps: fewer days (inverse). More hours a day: fewer days (inverse). 15/x = (5/4) · (8/6) = 40/24, so x = 15 · 24/40 = 9 days. Check: 4 × 6 × 15 = 360 pump-hours, and 5 × 8 × 9 = 360 pump-hours as well.
| Wrong | Right | Why |
|---|---|---|
| cross-multiplying an inverse pair | flip the inverse column first | otherwise faster gives later and more workers give more days |
| comparing machines with days | compare each column with x only | the arrows say how each quantity moves the unknown, with the others frozen |
| a rule of three on a square's area | A = s² | double the side, four times the area: not proportional |
| a rule of three on a fare with a fixed fee | remove the fixed part first | a fixed part breaks the proportion |
| minutes in one row, hours in the other | one unit per column | the ratios only cancel when both values share a unit |
| an answer that is never checked | recompute the constant | the ratio (direct) or product (inverse) must come out the same for both cases |
A round of 10 questions
Answer as many as you can. Mistakes come back at the end for a review round.
Every exercise is made on the spot, so a new round is never the same. The computer picks the answer first and builds the question around it, working with exact fractions, so the answer key is always right. Answers are read exactly: 0.75, 3/4 and 6/8 count as the same number (where lowest terms are asked for, 6/8 just gets a reminder). A wrong answer shows the worked solution line by line, in the order the lesson does it; the new number in each line is coloured.
Key ideas
- The rule of three finds a fourth number from three, when the quantities are proportional.
- Doubling test: the other doubles → direct (ratio fixed); it halves → inverse (product fixed); anything else → no rule of three.
- Direct: a/c = b/x, so x = bc/a. Inverse: flip one ratio, so x = ab/c.
- Arrow method: arrow down on x; each other column compared with x alone, down if it moves with x, up if against; up-arrow ratios are flipped.
- Compound: the ratio of x's column equals the product of all the other ratios.
- It works because one constant ties all the quantities together, and it cancels when the two cases are divided.
- Check with the constant: parts per machine-day, metres per worker-hour.