Proportion Calculator

What Is a Proportion?

A proportion is a mathematical statement that two ratios are equal. Written as a:b = c:d or equivalently a/b = c/d, it is read as "a is to b as c is to d." Proportions are one of the most fundamental concepts in mathematics and science, describing how quantities scale together.

The four quantities in a proportion — a, b, c, and d — are called the terms. The outer terms (a and d) are the extremes, and the inner terms (b and c) are the means. A key property: the product of the means equals the product of the extremes, i.e., b × c = a × d. This is the basis of cross-multiplication.

Cross Multiplication Method

To solve a proportion a/b = c/d for an unknown value, cross-multiply to eliminate the fractions:

• a × d = b × c
• To find d: d = (b × c) / a
• To find a: a = (b × c) / d
• To find b: b = (a × d) / c
• To find c: c = (a × d) / b

Cross multiplication works because multiplying both sides of a/b = c/d by b × d gives a × d = b × c. This technique lets you solve any proportion with one unknown in a single step.

Direct Proportion (y = kx)

Two quantities are in direct proportion if they increase or decrease at the same rate, maintaining a constant ratio. If y is directly proportional to x, then:

• y = kx, where k is the constant of proportionality
• k = y/x (the unit rate)
• Doubling x doubles y; tripling x triples y
• The graph is a straight line through the origin

Examples of direct proportion: distance = speed × time (if speed is constant); total cost = price per item × number of items; earnings = wage rate × hours worked.

Inverse Proportion (y = k/x)

Two quantities are in inverse proportion (indirect proportion) if one increases as the other decreases, keeping their product constant. If y is inversely proportional to x:

• y = k/x, or equivalently x × y = k
• Doubling x halves y; tripling x gives y/3
• The graph is a rectangular hyperbola

Examples of inverse proportion: more workers finish a job in fewer days (workers × days = constant work); more pipes fill a tank faster; higher speed means shorter travel time for a fixed distance.

Constant of Proportionality

The constant of proportionality (k) captures the fixed relationship between two quantities. For direct proportion, k = y/x is the unit rate — how much y changes per unit of x. For inverse proportion, k = x × y is the constant product. The constant k has units that depend on the context: km/h for speed, $/item for price, person-days for work.

Scale Factors and Ratio Simplification

A ratio a:b is in its simplest (lowest) form when a and b share no common factor greater than 1. To simplify, divide both terms by their Greatest Common Divisor (GCD). For example, 12:18 simplifies to 2:3 because GCD(12,18) = 6.

Applications of ratios and proportions:

• Recipes and cooking: Scale ingredients up or down while maintaining flavor balance
• Map scales: Convert map distances to real distances (1:50,000 means 1 cm = 500 m)
• Currency exchange: If 1 USD = 83 INR, how many INR for 250 USD?
• Similar triangles: Corresponding sides of similar triangles are proportional
• Shadow problems: A 6 m pole casts a 4 m shadow; how tall is a tree with an 18 m shadow?
• Mixtures: Maintaining the right concentration ratio when scaling
• Scale models: Architectural models, model trains, and engineering prototypes
• Pharmacy: Drug dosage proportional to body weight

Types of Proportion at a Glance

Type	Relation	Formula	Graph shape	Real-world example
Direct	Both increase together	y = kx	Straight line through origin	Distance vs. time at constant speed
Inverse	One increases, other decreases	y = k/x	Rectangular hyperbola	Workers vs. days to finish a job
Cross ratio	Two ratios equal	a/b = c/d	—	Map scale, recipe scaling
Compound	Multiple quantities involved	y = k × x1 / x2	—	Work = workers × time / output