Question 1

What is the simple rule of three?

Accepted Answer

It is the method for finding a fourth value when two quantities are proportional and you know three numbers. You set up a table with two rows (two situations) and two columns (the quantities), leave one value as the unknown x and solve. If 4 kg cost $20, how much do 7 kg cost? The price per kilo is 20 ÷ 4 = 5, so 7 kg cost 7 × 5 = $35.

Question 2

What is a direct rule of three? Give an example.

Accepted Answer

It is when the quantities vary in the same direction: doubling one doubles the other. The proportion is A ÷ B = C ÷ D, and cross multiplying gives A × D = B × C. Example: a recipe uses 2 eggs for 150 g of flour. For 5 eggs, x = 150 × 5 ÷ 2 = 375 g. Another: a car covers 120 km on 10 liters, so for 420 km it needs 10 × 420 ÷ 120 = 35 liters.

Question 3

What is an inverse rule of three? Give an example.

Accepted Answer

It is when the quantities vary in opposite directions: doubling one halves the other, and the product of the two stays constant. The equality is A × B = C × D. Example: 6 workers finish a job in 10 days. With 15 workers, x = 6 × 10 ÷ 15 = 4 days. Another: at 60 km/h the trip takes 3 h, so at 90 km/h it takes 60 × 3 ÷ 90 = 2 h.

Question 4

How do I know whether the rule of three is direct or inverse?

Accepted Answer

Run the quick test: if one quantity goes up and the other goes up too, it is direct. If one goes up and the other goes down, it is inverse. More kilos, more price: direct. More workers, fewer days: inverse. You decide, the calculator does not guess, because the same table of numbers can be direct or inverse depending on the problem. When in doubt, imagine an extreme case, like doubling one of the quantities, and see whether the other doubles or drops by half.

Question 5

What is the compound rule of three?

Accepted Answer

It is the rule with three or more quantities at once, where the value you want depends on several of them. Each known quantity is compared with the one holding the unknown and marked as direct or inverse. Example: 3 workers build 12 m in 4 days; how many meters do 6 workers build in 8 days? Workers and days are both direct to the meters, so x = 12 × (6 ÷ 3) × (8 ÷ 4) = 48 m. This calculator accepts 2 to 4 known quantities besides the unknown.

Question 6

How do I solve the compound rule of three step by step?

Accepted Answer

First, choose the quantity that holds the unknown. Second, for every other quantity, decide whether it is direct or inverse relative to it. Third, work out each ratio: direct is new value ÷ old value, inverse is old value ÷ new value. Fourth, multiply all the ratios and then multiply by the known value of the unknown. Example with inverses: 8 workers, 6 h a day, 10 days; with 5 workers and 8 h a day, x = 10 × (8 ÷ 5) × (6 ÷ 8) = 12 days.

Question 7

How do I check the result with the cross product?

Accepted Answer

In the direct case, A ÷ B = C ÷ D implies A × D = B × C: with 4 ÷ 20 = 7 ÷ 35, both sides give 140. In the inverse case, A × B must equal C × D: 3 × 8 = 6 × 4 = 24. In the compound case, x times the denominators of the ratios must equal the known value times the numerators. The calculator runs this check with the exact values and marks it with a check mark on the last step.

Question 8

Can I use the rule of three to calculate a percentage?

Accepted Answer

Yes, a percentage is a proportion with 100. To find 20% of 80, set up 100 is to 80 as 20 is to x: x = 80 × 20 ÷ 100 = 16. To find what percent 45 is of 300, set up 300 is to 100 as 45 is to x: x = 100 × 45 ÷ 300 = 15%. The direct rule covers both cases, only the position of the numbers changes. For many percentage problems, the site's percentage calculator is more direct.

Question 9

How do I use the rule of three for map scale, exchange rates and recipes?

Accepted Answer

All of these are direct. Scale: if 1 cm on the map is 25 km, use A = 1, B = 25, C = 7.5 and x = 25 × 7.5 ÷ 1 = 187.5 km. Exchange at a fixed rate: if 1 dollar is worth 5 units of another currency (sample rate only), 250 dollars are worth 250 × 5 = 1,250; the real rate changes every day. Recipe: if 2 eggs call for 150 g of flour, 5 eggs call for 375 g. Name the quantities so the explanation reads well.

Question 10

Can I type decimals, commas and dots? Is the math exact?

Accepted Answer

Yes. It accepts 1.5 and 1,5, and thousands like 1,234.56 too. Numbers become exact fractions and the math is done without floating point error: 0.1 ÷ 0.3 = 0.2 ÷ x gives exactly 0.6. Rounding happens only when displaying, and you choose the decimal places and the mode (standard, banker's, truncate, round up). When the result is rounded, the ≈ sign appears with the exact value, for example 10/3 for 3.33.

Question 11

When does the rule of three not work, and what happens with division by zero?

Accepted Answer

It only works if the relation really is proportional. It does not hold with a fixed cost (delivery fee plus price per item), volume discounts, nonlinear growth or compound interest. Mathematically, if the divisor of the calculation is zero there is no solution. The tool tells you which value cannot be zero: A in the direct case with the unknown in D, C in the inverse case with the unknown in D, and so on depending on the unknown you chose. Zero in the numerator is allowed and gives x = 0.

Question 12

Can any value be the unknown? Can I name the quantities?

Accepted Answer

Yes to both. In the simple rule, use the Solve for this one button on any of the four cells (A, B, C or D). In the compound rule, choose the quantity that holds the unknown. The names are free (workers, days, liters, km) and show up in the headers, the unit rate and the step by step, which makes the explanation readable to paste into an assignment or a message.

Rule of Three Calculator