How to do a rule of three: direct, inverse and compound

What a rule of three is

A rule of three solves a proportion: it relates two quantities of which you know three values, and returns the fourth. If 3 kg of oranges cost €12, what do 5 kg cost?

What makes that work is not the cross you were taught to draw at school. It is that something does not change between the two pairs, and that something always has a real name:

• With the oranges, what does not change is the price per kilo: €12 ÷ 3 kg = €4/kg. That is a quotient.
• If 4 workers finish a job in 6 days, what does not change is the total work: 4 × 6 = 24 worker-days. That is a product.

That is the only distinction that matters on this page. When the constant is a quotient, the rule of three is direct. When it is a product, it is inverse. The two formulas people memorise are consequences of that, not independent rules, and if you ever forget them you can rebuild them in five seconds by asking which quantity stays the same.

How to do a rule of three step by step

The method has four steps, and the first one prevents almost every error.

Step 1. Write each value with its unit. Not 3, 12 and 5, but 3 kg, €12 and 5 kg. With the units in front of you it is impossible to put the kilos where the euros belong, which is where most bad set-ups come from.

Step 2. Put the pairs in two rows, with the same quantity always in the same column:

Step 3. Decide the type. If the first quantity doubles, does the second double or halve? Six kilos cost more than three, so it is direct.

Step 4. Solve.

• Direct: X = B × C ÷ A = 12 × 5 ÷ 3 = €20.
• Inverse: X = A × B ÷ C.

You can run all four steps, and check the answer, in the rule of three calculator: it asks for the three values and the type, and returns the proportion it set up as well as the number.

Direct rule of three: what is preserved is a quotient

In a direct relationship both quantities rise and fall together, and what stays the same is B ÷ A, which almost always has a name of its own: euros per kilo, euros per litre, kilometres per hour, grams of flour per portion.

With the oranges that quotient is €4/kg, so five kilos are 4 × 5 = €20. Cross-multiplying gives exactly the same thing, because the equality of quotients 12 ÷ 3 = X ÷ 5 rearranges into 3 × X = 12 × 5. The cross is an arithmetic shortcut for that rearrangement, not a separate principle.

One practical consequence: in a direct relationship zero sits at zero. Zero kilos cost zero euros. That is the requirement whose failure invalidates a rule of three, and it is the reason for the section on fixed charges below.

Inverse rule of three: what is preserved is a product

In an inverse relationship one quantity rises as the other falls, and what stays the same is A × B.

Four workers finish a job in 6 days. How long will 8 workers take? The total work is 4 × 6 = 24 worker-days and does not depend on how many people share it, so with eight people it is 24 ÷ 8 = 3 days. As a formula, X = A × B ÷ C.

Notice what would happen if you cross-multiplied out of habit: 6 × 8 ÷ 4 = 12 days. Twice the workers taking twice the time. That nonsense is the best check that you picked the wrong type, and it always works: if the answer moves in the opposite direction to common sense, you applied the wrong formula.

Inverse relationships turn up more often than they seem. On a journey the constant is the distance (speed × time), when filling a tank it is the litres (flow × hours), and in a rectangle of fixed area it is the area (base × height).

The question that decides direct or inverse

One sentence settles it: if the first quantity doubles, does the second double or halve? If it doubles, direct. If it halves, inverse.

And if you are still unsure, find the constant and look at its unit. A quotient means direct, a product means inverse.

That the constant has a name is not a curiosity: it is the proof that the proportion exists. When you cannot name it, usually there is not one.

Compound rule of three: one quantity at a time

A compound rule of three relates three or more quantities. The classic exam wording is this: 4 workers on 8-hour days finish a job in 9 days; how long will 6 workers on 7-hour days take?

No new formula is needed. The method is to compare each quantity with the unknown separately, holding the others still, and then multiply the resulting factors:

• Workers. More workers, fewer days: inverse. Factor = old value ÷ new value = 4 ÷ 6.
• Hours per day. More hours a day, fewer days: inverse. Factor = 8 ÷ 7.

X = 9 × (4 ÷ 6) × (8 ÷ 7) = 6.86 days.

The mechanical rule that follows is easy to remember: for an inverse quantity the factor is old ÷ new, and for a direct one it is new ÷ old.

Add a third quantity, this time a direct one, and nothing changes. If on top of the new crew and the new hours the wall grows from 100 to 150 metres, more metres mean more days, so the factor is 150 ÷ 100:

X = 9 × (4 ÷ 6) × (8 ÷ 7) × (150 ÷ 100) = 10.29 days.

The long way is the one that never fails

If you do not trust the sign of one of the factors, there is a slower and far safer alternative: reduce to the unit. Work out the named constant and divide.

The original job is 4 workers × 8 hours × 9 days = 288 worker-hours for 100 metres of wall, which is 2.88 worker-hours per metre. A hundred and fifty metres are 432 worker-hours. The new crew supplies 6 × 7 = 42 worker-hours a day, so 432 ÷ 42 = 10.29 days.

Same answer by a route where every intermediate number means something you can check. When both methods agree, the set-up is right.

When a rule of three does not apply

This is the part that gets told least and costs the most. A rule of three is only valid when the relationship between the two quantities is proportional, meaning that multiplying one by a number multiplies the other by the same number. Four very common families break that.

Banded tariffs

A car under 8 fiscal horsepower pays €12.62 of Spanish road tax under the national tariff in article 95.1 of the consolidated Local Finances Act. One of 8 to 11.99 pays €34.08, and one of 16 to 19.99 pays €89.61.

Doubling fiscal horsepower from 8 to 16 does not double the charge: it multiplies it by 2.63. A rule of three from the first band would give €68.16 and the real charge is €89.61, twenty-one and a half euros apart. Every stepped tariff breaks proportionality at each jump, and in Spain income tax, inheritance tax, property transfer tax and this one are all stepped.

Amounts with a fixed component

A 1% fee with a €3 minimum is not proportional. On €100 it costs €4.00, which is 4%. On €200 it costs €5.00, which is 2.5%. The charge grows with the amount, but it does not start at zero, and that is precisely the requirement of a direct relationship. Delivery charges, maintenance fees and any tariff with a minimum tier have the same problem.

Compound interest

Ten thousand euros at 3% a year become €13,439.16 after ten years and €18,061.11 after twenty. The interest is not €3,439.16 and then twice that, but €3,439.16 and then €8,061.11: a rule of three on the term falls short by €1,182.79, because each year the interest already earned starts earning interest itself. The relationship between term and money is exponential, not proportional, and over a long term the gap stops being a detail. It is set out with the full formula in what compound interest is and can be run on your own figures in the compound interest calculator.

Percentages applied on top of each other

Raising a price by 20% and then cutting it by 20% does not return the original price: 100 becomes 120 and then 96. The second percentage applies to a different figure from the first.

The everyday version of this trap is VAT. At the 21% standard rate of article 90.Uno of Act 37/1992, an item worth €100.00 sells for €121.00. To recover the net amount you must divide by 1.21, not subtract 21%: subtracting gives €95.59 instead of €100.00, an error of €4.41 that grows with the amount. The full breakdown is in how to calculate Spanish VAT.

And the physical limit

Even where the relationship really is inverse, proportionality assumes the work divides without friction. A hundred workers in a sixty-square-metre flat do not finish in a hundredth of the time, and the classic illustration is that nine people cannot produce a baby in one month. Before applying an inverse rule of three, check that the split is possible.

Price per unit of measure: the rule of three already done

In Spain the most useful everyday rule of three, comparing two packs of different sizes, is usually solved before you get there, and by law.

Royal Decree 3423/2000, which transposes Directive 98/6/EC, requires in article 3.2 that the price per unit of measure be shown on every product that must carry a quantity indication and on those sold by the unit. Article 2(b) defines that figure as the final price per kilogram, litre, metre, square metre or cubic metre, including VAT and every other tax, so it is directly comparable between brands.

Article 4.1 is what makes it usable: the selling price and the price per unit of measure must be unambiguous, legible, in the same field of vision, and visible without the consumer having to ask for them. Article 3.5 extends the duty to any advertising that mentions the price. For goods sold loose, article 3.4 says only the price per unit of measure is shown.

Where the sum is left to you is in the seven exceptions of Annex I:

• products sold in quantities under 50 g or 50 ml,
• packs combining different products that are not sold individually,
• vending machines,
• individual portions of ice cream,
• table wines with a geographical indication and wines with a designation of origin,
• spirits with a geographical designation,
• novelty food products.

The first is the one you meet most: spices, baking powder, saffron sachets and much of the baking aisle sit under 50 grams, and there is no price per kilo to read.

There is a second thing worth knowing, because comparing by weight can lead you to the opposite answer. Annex II standardises the unit for some products: eggs are measured by the dozen, cosmetics and food supplements per 100 g or 100 ml, pipe tobacco per 100 g, and laundry detergents by the amount needed for one wash, not by the kilo. Two boxes of detergent are compared per wash precisely because the dose per wash is not the same in both.

Compliance is supervised by the consumer authorities of the autonomous communities (article 5), and article 6 refers penalties to the general consumer protection rules.

A worked example with real numbers

An ordinary shopping trip, with four situations in a row. The prices are an assumption; the arithmetic is what you would do in the aisle.

1. Two jars of coffee. One of 400 g at €8.60 and one of 250 g at €5.60. The rule of three is direct, because zero grams cost zero euros:

• 400 g are €8.60, so 1,000 g are 8.60 × 1,000 ÷ 400 = €21.50/kg.
• 250 g are €5.60, so 1,000 g are 5.60 × 1,000 ÷ 250 = €22.40/kg.

The large jar is €0.90 cheaper per kilo even though its shelf price is higher. Both of those figures should already be printed under the price, under article 3.2.

2. A 1 g sachet of saffron at €2.95. It is under 50 g, so Annex I excuses the shop from showing a price per unit of measure. You do the rule of three: 2.95 × 1,000 = €2,950/kg. That is not sharp practice, it is what saffron costs, but it is exactly the kind of figure the label will not hand you.

3. Two detergents. A box of 40 washes at €12.00 and one of 55 washes at €11.00. Annex II sets the unit as the wash, not the kilo:

• 12.00 ÷ 40 = €0.30 per wash.
• 11.00 ÷ 55 = €0.20 per wash.

The second is a third cheaper per wash and also cheaper on the shelf. Comparing them by weight could have given the opposite answer, because concentrated detergents weigh less per dose.

4. The offer at the till. A €4.00 item on a three-for-two against the same item at 30% off. Three units for the price of two is €8.00 for three, that is 8.00 ÷ 3 = €2.67 a unit, an effective discount of 33.33%. At 30% off each unit would cost €2.80. The three-for-two wins by €0.13 a unit, €0.40 across the three, provided you need all three. If you only wanted one, the 30% discount is better and the three-for-two saves nothing. You can compare offers like these in the discount calculator.

Four rules of three in one shopping trip, and none of them needs anything beyond knowing what the constant is.

The rule of three and percentages are the same sum

A percentage is a direct rule of three in which the first quantity is 100. 20% of 250 is set up as 100 is to 20 as 250 is to X, and is solved the same way: X = 20 × 250 ÷ 100 = 50.

That is why the two tools overlap and it is worth picking the comfortable one. When the question arrives in percentage terms, the percentage calculator gets there sooner, and the method is set out in how to calculate a percentage. When it arrives with real units, kilos, euros, days or hours, the rule of three is the natural set-up, because it forces you to name the constant, and that is what prevents the error.