Rule of Three Calculator

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direct rule of three

Direct Rule of Three

In a direct proportion, two quantities grow together at the same rate: A/B = C/X, so X = B×C/A. It works for any pair of quantities linked by a fixed rate, from scaling a recipe to converting a map measurement into a real distance.

From recipe to map: where direct proportion applies

A recipe using 4 cups of flour yields 12 cookies; to find how many cookies come from 7 cups, set A=4, B=12, C=7, giving X=(12×7)/4=21 cookies. The cross-multiplication behind the formula, A×X=B×C, guarantees the flour-per-cookie ratio stays equal at both scales: 4/12 and 7/21 are exactly the same fraction, 1/3.
Topographic maps are usually printed at a fixed scale, such as 1:50,000, meaning 1 cm on paper corresponds to 50,000 cm (500 m) on the actual terrain. Measuring 3.4 cm between two points on the map, the direct proportion A=1, B=50000, C=3.4 gives X=(50000×3.4)/1=170,000 cm, which still needs converting to 1700 m or 1.7 km before it becomes a usable distance; the proportion solves the scale, but the unit conversion is left to whoever reads the result.
Direct proportion only works when the relationship passes through the origin: zero of one quantity corresponds to zero of the other. A taxi ride with a fixed flag-drop fee plus a per-km rate does not follow that rule, because doubling the distance does not double the total fare, since the flag-drop fee is still charged only once. Typing text, on the other hand, follows direct proportion without exceptions: someone who types 3 pages in 18 minutes types 5 pages in (18×5)/3=30 minutes, because there is no initial "fixed time" to subtract.

Examples

Scaling a recipe

Input: A=4, B=12, C=7
Expected output: X = 21

4 cups yield 12 cookies; 7 cups yield 21, keeping the same ratio of 1 cup to 3 cookies.

1:50,000 map scale

Input: A=1, B=50000, C=3.4
Expected output: X = 170000

170,000 cm equal 1.7 km; the proportion solves the scale, but converting cm to km still needs a separate step.

Pages typed per minute

Input: A=3, B=18, C=5
Expected output: X = 30

With no fixed fee involved, typing more pages takes proportionally more time, with no initial discount or bonus.

Full tool FAQ

What is the simple rule of three?

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.

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

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

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

What is the compound rule of three?

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

How do I check the result with the cross product?

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

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

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

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

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

Frequently asked questions

Does this rule work for any "if X, then Y" scaling question?

Only when the relationship is genuinely proportional, passing through the origin: zero of one quantity must correspond to zero of the other. A taxi ride with a fixed flag-drop fee plus a per-km rate is not a direct proportion, because doubling the distance does not double the total fare, since the flag-drop fee is charged only once.

How do I use the same map scale in reverse, from real distance to paper measurement?

Just swap B and C in the setup: instead of multiplying the map measurement by 50,000, divide the real distance (already in cm) by 50,000. A real distance of 1700 m (170,000 cm) divided by 50,000 returns exactly the 3.4 cm that would appear on the map.

Does the calculator require both quantities to already be in the same unit?

Yes, within each quantity. In the map example, A and C need to be in centimeters to match the 1:50,000 scale; the result also comes out in centimeters, and converting to meters or kilometers is a manual step done after the calculation.

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direct rule of three

Direct Rule of Three

From recipe to map: where direct proportion applies

A recipe using 4 cups of flour yields 12 cookies; to find how many cookies come from 7 cups, set A=4, B=12, C=7, giving X=(12×7)/4=21 cookies. The cross-multiplication behind the formula, A×X=B×C, guarantees the flour-per-cookie ratio stays equal at both scales: 4/12 and 7/21 are exactly the same fraction, 1/3.

Topographic maps are usually printed at a fixed scale, such as 1:50,000, meaning 1 cm on paper corresponds to 50,000 cm (500 m) on the actual terrain. Measuring 3.4 cm between two points on the map, the direct proportion A=1, B=50000, C=3.4 gives X=(50000×3.4)/1=170,000 cm, which still needs converting to 1700 m or 1.7 km before it becomes a usable distance; the proportion solves the scale, but the unit conversion is left to whoever reads the result.

Direct proportion only works when the relationship passes through the origin: zero of one quantity corresponds to zero of the other. A taxi ride with a fixed flag-drop fee plus a per-km rate does not follow that rule, because doubling the distance does not double the total fare, since the flag-drop fee is still charged only once. Typing text, on the other hand, follows direct proportion without exceptions: someone who types 3 pages in 18 minutes types 5 pages in (18×5)/3=30 minutes, because there is no initial "fixed time" to subtract.

Examples

Scaling a recipe

Input: A=4, B=12, C=7
Expected output: X = 21

4 cups yield 12 cookies; 7 cups yield 21, keeping the same ratio of 1 cup to 3 cookies.

1:50,000 map scale

Input: A=1, B=50000, C=3.4
Expected output: X = 170000

170,000 cm equal 1.7 km; the proportion solves the scale, but converting cm to km still needs a separate step.

Pages typed per minute

Input: A=3, B=18, C=5
Expected output: X = 30

With no fixed fee involved, typing more pages takes proportionally more time, with no initial discount or bonus.

Full tool FAQ

What is the simple rule of three?

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

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

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

What is the compound rule of three?

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

How do I check the result with the cross product?

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

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

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

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

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

Frequently asked questions