The rule of three: direct, inverse and compound proportion with examples

The rule of three is the classic name for solving a proportion with one unknown value. Before setting up any table, one question decides everything: are the quantities directly or inversely proportional?

Directly and inversely proportional quantities

• Direct: when one doubles, the other doubles too. Their ratio is constant: $\dfrac{y}{x} = k$. Example: amount bought and total price.
• Inverse: when one doubles, the other halves. Their product is constant: $x \cdot y = k$. Example: speed and travel time over the same distance.

Not every relationship is one or the other. Age and height grow together, but not proportionally. The rule of three only works when the proportionality is real.

Simple direct rule of three

Worked example: 3 kg of coffee cost $24. How much do 5 kg cost?

1. Quantities: mass and price. More coffee, higher price in the same proportion: direct.
2. Set up the proportion: $\dfrac{3}{24} = \dfrac{5}{x}$.
3. Cross-multiply: $3x = 120$, so $x = 40$.
4. Check: one kilogram costs $24 \div 3 = 8$, and $5 \cdot 8 = 40$.

Step 4 shows the safest shortcut: find the unit value and multiply.

Simple inverse rule of three

Worked example: 6 bricklayers build a wall in 10 days. How many days do 4 bricklayers need for the same wall?

1. Fewer workers, more days: inverse.
2. In inverse proportion, the product is constant: $6 \cdot 10 = 4 \cdot x$.
3. $60 = 4x$, so $x = 15$ days.

Had you set it up as direct, you would get $\dfrac{6}{10} = \dfrac{4}{x}$ and $x \approx 6.7$ days, meaning fewer people finish sooner, which makes no sense. Always check that the answer is reasonable.

Compound rule of three

With three or more quantities, compare each quantity with the one you are looking for, one at a time, assuming the others stay fixed. The unknown equals the known value times one ratio per quantity, inverted when the relationship is inverse.

Worked example: 8 machines make 1,200 parts in 5 days. How many parts do 12 machines make in 3 days?

1. Unknown quantity: parts.
2. Machines vs. parts: more machines, more parts (direct). Ratio $\dfrac{12}{8}$.
3. Days vs. parts: fewer days, fewer parts (direct). Ratio $\dfrac{3}{5}$.
4. Compute:

Worked example: 10 workers, working 6 hours a day, finish a job in 18 days. How many days do 15 workers, working 8 hours a day, need for the same job?

1. Unknown quantity: days.
2. Workers vs. days: more workers, fewer days (inverse). Inverted ratio: $\dfrac{10}{15}$.
3. Hours per day vs. days: more hours per day, fewer days (inverse). Inverted ratio: $\dfrac{6}{8}$.
4. Compute:

Check with total labor hours: $10 \cdot 6 \cdot 18 = 1080$ and $15 \cdot 8 \cdot 9 = 1080$. The total work is the same.

Common mistakes

1. Deciding direct or inverse while looking at all quantities at once instead of one at a time.
2. Using the rule of three for relationships that are not proportional, such as area and side length.
3. Mixing units: hours with minutes, grams with kilograms.