The rule of three finds a fourth number when you know three and the quantities are linked by proportion. If 3 kilos of apples cost $12, what do 5 kilos cost? Three known numbers, one unknown. It is the tool behind recipes, currency exchange, speed problems and scaled drawings.
There are three versions: direct, inverse and compound. The only real decision is whether the quantities move together or in opposite directions.
Direct rule of three
Two quantities are directly proportional when doubling one doubles the other. Their ratio never changes. Put the known pair in one row and the new value in the second row, with the unknown x under its partner:
$$ \frac{a}{b} = \frac{c}{x} \quad\Longrightarrow\quad x = \frac{b \times c}{a} $$
Example 1. 3 kilos of apples cost $12. What do 5 kilos cost? More apples, more money: direct.
$$ x = \frac{12 \times 5}{3} = 20 $$
Five kilos cost $20. Another way to see it: the weight was multiplied by $latex \frac{5}{3}$, so the price is too.
Example 2. A recipe for 4 people uses 300 g of rice. For 10 people: $latex x = \frac{300 \times 10}{4} = 750$ g.
Inverse rule of three
Two quantities are inversely proportional when doubling one halves the other. Their product stays constant. Now the new value divides the answer, so the cross multiplication changes:
$$ a \times b = c \times x \quad\Longrightarrow\quad x = \frac{a \times b}{c} $$
Example 3. 4 workers finish a job in 6 days. How long will 8 workers need? More workers, fewer days: inverse.
$$ x = \frac{4 \times 6}{8} = 3 $$
Example 4. A car at 90 km/h takes 2 hours for a trip. At 60 km/h: $latex x = \frac{90 \times 2}{60} = 3$ hours. Slower speed, more time.
Try it: the fourth number fills in as you type
Choose direct or inverse and change the three numbers. The graph shows the relation: a straight line through the origin for direct problems, a falling curve for inverse ones. The known pair is blue and your answer is orange.
How to tell direct from inverse
Ask one question: if the first quantity goes up, does the second go up or down?
| If more of A means... | The relation is | Formula | Typical examples |
|---|---|---|---|
| more of B | Direct | $latex x = \frac{b \times c}{a}$ | price and weight; pay and hours |
| less of B | Inverse | $latex x = \frac{a \times b}{c}$ | workers and days; speed and time |
Check the result with common sense. In a direct problem the answer should grow when the new value is larger than the old one. In an inverse problem it should shrink.
Compound rule of three
When the unknown depends on two or more quantities, handle them one at a time. Decide for each quantity whether it moves the unknown in the same direction (direct) or the opposite way (inverse), turn it into a ratio, and multiply the starting value by every ratio. A direct quantity gives the ratio new over old. An inverse quantity gives old over new.
Example 5. 6 workers who work 8 hours a day build 120 m of wall in 10 days. How many days will 8 workers working 6 hours a day need to build 180 m?
- More workers means fewer days (inverse): multiply by 6/8.
- Fewer hours per day means more days (inverse): multiply by 8/6.
- More wall means more days (direct): multiply by 180/120.
$$ x = 10 \times \frac{6}{8} \times \frac{8}{6} \times \frac{180}{120} = 15 \text{ days} $$
The first two ratios cancel here, which makes sense: 8 workers for 6 hours is the same work per day as 6 workers for 8 hours. Only the extra wall adds time, by a factor of 1.5.
Each row is a quantity that changes: its name, its old value, its new value, and whether it moves the unknown in the same direction (direct) or the opposite way (inverse).
The unit method and percents
The rule of three is a shortcut for the unit method: find the value of one unit, then scale up. In Example 1, 1 kilo costs $latex 12 \div 3 = 4$, so 5 kilos cost $latex 5 \times 4 = 20$. Use whichever one you trust.
Percents are direct proportions too. If 250 is 100%, what percent is 40? $latex x = \frac{40 \times 100}{250} = 16\%$. See how to calculate percentages for more.
Common mistakes
- Treating an inverse problem as direct. Doubling the workers does not double the days. Ask the up-or-down question first.
- Mixing units. If the first row is in minutes, the second row must be in minutes too. Convert hours to minutes before computing.
- Assuming everything is proportional. If one shirt dries in 2 hours on a line, 5 shirts hung side by side also dry in about 2 hours, not 10. Cooking time, age and prices with volume discounts are not proportional either.
- Putting the numbers in the wrong places. The unknown pairs with the value of the same type as the known partner. Label each column before you multiply.
- Forgetting a quantity in a compound problem. Every quantity that changes needs its own ratio.
Practice problems
Problem 1
7 notebooks cost $42. How much do 12 notebooks cost?
Answer
Direct: $latex x = \frac{42 \times 12}{7} = 72$.
Answer: $72.
Problem 2
2 pipes fill a pool in 9 hours. How long will 6 pipes take?
Answer
More pipes, less time: inverse. $latex x = \frac{2 \times 9}{6} = 3$.
Answer: 3 hours.
Problem 3
A trip takes 5 hours at 80 km/h. How long does it take at 100 km/h?
Answer
Faster means less time: inverse. $latex x = \frac{80 \times 5}{100} = 4$.
Answer: 4 hours.
Problem 4
15 workers working 8 hours a day finish a job in 12 days. How many days will 20 workers working 5 hours a day need?
Answer
More workers: inverse, $latex \frac{15}{20}$. Fewer hours per day: inverse, $latex \frac{8}{5}$.
$$ x = 12 \times \frac{15}{20} \times \frac{8}{5} = 14.4 $$
Answer: 14.4 days.
Problem 5
A printer prints 240 pages in 8 minutes. How long do 2 printers need for 600 pages?
Answer
More pages: direct, $latex \frac{600}{240}$. More printers: inverse, $latex \frac{1}{2}$.
$$ x = 8 \times \frac{600}{240} \times \frac{1}{2} = 10 $$
Answer: 10 minutes.
Problem 6
250 g of flour make 15 cookies. How many cookies can you make with 1 kg of flour?
Answer
Convert 1 kg to 1000 g, then direct: $latex x = \frac{15 \times 1000}{250} = 60$.
Answer: 60 cookies.
Frequently asked questions
Is the rule of three the same as a proportion?
It is a quick way to solve one. Writing a/b = c/x and cross multiplying gives the same result as the rule of three.
How do I know if a problem is direct or inverse?
Ask whether more of the first quantity gives more or less of the second. More gives direct. Less gives inverse.
Can the rule of three be used for percents?
Yes. Treat the whole as 100% and solve for the part, or the other way around. Percents are direct proportions.
Does the compound rule need a table?
A table helps, but all it does is list each quantity with its old and new value and mark whether it is direct or inverse. Then you multiply the ratios.
Related calculators
- Rule of three calculator (direct and inverse proportion)
- Percentage calculator
- Speed, distance and time calculator
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