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Proportion calculator - solve a:b = c:x with the rule of three

Enter three known values and the calculator finds the fourth. Choose a direct proportion (more means more) or an inverse one (more means less) and follow the step-by-step working.

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If a corresponds to b, then c corresponds to d. Enter three values and the calculator finds the fourth.

20

Example: If 3 kg cost $12, how much do 5 kg cost? Enter a = 3, b = 12, c = 5.

Result

Unknown d
20
Ratio b/a
4
the value of b per one unit of a

Step-by-step solution

  1. 1. Write the proportion
    3 : 12 = 5 : x
  2. 2. Cross-multiply (product of extremes = product of means)
    3 · x = 12 · 5
  3. 3. Solve for the unknown
    x = (12 · 5) / 3 = 60 / 3 = 20
  4. 4. Check the result
    3 · 20 = 60, 12 · 5 = 60

How to solve a proportion

  1. 1.

    Choose the type

    Direct when both quantities grow together (price and weight); inverse when one grows as the other shrinks (workers and days).

  2. 2.

    Pick the unknown

    In a direct proportion, select which of the four values you are looking for (in an inverse one it is always x). Enter the other three; decimals and negative numbers are fine.

  3. 3.

    Read the result

    The answer appears right away with the cross-multiplication and a check that both sides are equal. For a direct proportion you also see the b/a ratio.

Direct proportion - formula

Two quantities are directly proportional when their ratio is constant: a : b = c : x. Cross-multiply: a · x = b · c, so x = b · c / a.

Example: 3 kg of apples cost $12. How much do 5 kg cost? x = 12 · 5 / 3 = $20. Check: 3 · 20 = 60 = 12 · 5. The calculator also shows the ratio b/a, here $4 per kilogram. This is the unitary method: find the value of one unit first, then multiply.

Inverse proportion - formula

Two quantities are inversely proportional when their product is constant: a · b = c · x, so x = a · b / c.

Example: 4 workers finish a job in 6 days. How many days will 3 workers need? x = 4 · 6 / 3 = 8 days. Likewise, a car at 60 mph covers a route in 2 hours, so at 75 mph it takes 2 · 60 / 75 = 1.6 hours (1 hour 36 minutes).

How to tell which type you need

  • Direct: price and quantity, distance and time at constant speed, recipe ingredients, map scale, currency conversion, fuel and distance.
  • Inverse: number of workers and time, speed and travel time, number of pumps and time to fill a pool, share of a fixed bill and number of people.

Ask yourself: if one quantity goes up, does the other go up too? If yes, it is direct; if it goes down, it is inverse. Percentages are a special case of direct proportion: 100% corresponds to the whole, p% corresponds to x. For everyday percentage questions the percentage calculator is quicker.

Worked examples and common mistakes

ProblemSet-upResult
A recipe for 4 needs 300 g of flour. How much for 6?4 : 300 = 6 : xx = 300 · 6 / 4 = 450 g
Map scale 1 : 50,000. How far is 3.2 cm on the map?1 : 50,000 = 3.2 : xx = 160,000 cm = 1.6 km
A car uses 6.5 l per 100 km. How much for 340 km?100 : 6.5 = 340 : xx = 22.1 l
2 pumps fill a pool in 9 h. How long for 3 pumps?2 · 9 = 3 · xx = 6 h (inverse)

Common mistakes include using a direct proportion where the relationship is inverse (more pumps means less time), mixing up the order of terms (a and c must be the same kind of quantity, e.g. kilograms, and b and d the other, e.g. dollars), and mixing units (minutes with hours, grams with kilograms). Convert to common units first; the weight converter helps with kilograms, pounds and ounces. Also remember that not everything is proportional: bulk prices with volume discounts or a team that needs time to coordinate do not scale in a straight line.

Frequently asked questions

What is the rule of three?

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It is a method for finding a fourth value from three known ones when the quantities are proportional. For a direct proportion x = b · c / a; for an inverse one x = a · b / c.

What is the difference between direct and inverse proportion?

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In a direct proportion both quantities rise together (more goods, higher price). In an inverse proportion one rises while the other falls (more workers, less time).

How does cross-multiplication work?

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In a : b = c : d you multiply the outer and inner terms: a · d = b · c. Then you solve for the unknown by dividing by the number next to it.

Can I use decimals and negative numbers?

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Yes. Enter numbers with a decimal point, including negative values. The result is rounded to up to 8 decimal places.

How do I calculate a percentage with a proportion?

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Treat the whole as 100%. What is 15% of 240? 100 corresponds to 240, 15 corresponds to x, so x = 240 · 15 / 100 = 36. Enter a = 100, b = 240, c = 15.

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