Proportion Calculator

What is a Proportion and How to Solve It

A proportion is an equation stating that two ratios are equal. Mathematically it is written as:

$$\frac{a}{b}=\frac{c}{d}$$

or

$$a:b=c:d$$

The numbers a and d are called the extremes, and b and c are called the means of the proportion.

The Fundamental Property of Proportions (Cross-Multiplication Rule)

The key mathematical rule of proportions states: the product of the extremes equals the product of the means.

$$a·d=b·c$$

This rule allows you to verify if a proportion is correct and to find any unknown term if the other three are known.

How to Find an Unknown Term of a Proportion

To find the unknown value (usually denoted as x), multiply the two known numbers that are diagonal from each other (crosswise), and divide by the third known number.

• If the unknown is a: $$a=\frac{b·c}{d}$$
• If the unknown is b: $$b=\frac{a·d}{c}$$
• If the unknown is c: $$c=\frac{a·d}{b}$$
• If the unknown is d: $$d=\frac{b·c}{a}$$

Worked Examples

Example 1 (solving for unknown): Find x in the proportion:

$$\frac{4}{8}=\frac{5}{x}$$

Here, d is the unknown. We multiply the means (8 and 5) and divide by the known extreme (4):

$$x=\frac{8·5}{4}=\frac{40}{4}=10$$

Example 2 (checking proportion): Check if the proportion is correct:

$$\frac{2}{3}=\frac{8}{12}$$

Cross-multiply:

$$2·12=24$$

and

$$3·8=24$$

Since the products are equal (24 = 24), the proportion is correct.

Practical Applications

• Percentage Calculations. Any percentage problem can be formulated as a proportion. For instance, if 150 grams is 100%, then 30 grams is x%: $$\frac{150}{100}=\frac{30}{x}\Rightarrow x=20\%$$
• Cooking (scaling recipes). If a cake recipe requires 500g of flour and 4 eggs, but you only have 300g of flour, the number of eggs is calculated via proportion: $$\frac{500}{4}=\frac{300}{x}\Rightarrow x=2.4$$ (approx. 2-3 eggs).
• Map Scaling. A scale of 1:100,000 means that 1 cm on the map corresponds to 100,000 cm (1 km) in reality. If map distance is 5.5 cm, real distance is solved using a proportion: $$\frac{1}{100000}=\frac{5.5}{x}\Rightarrow x=550000\text{ cm = 5.5 km}$$
• Chemistry and Pharmacology. Medication dosages are calculated proportionally to the patient's weight. If 2 ml of a drug is prescribed for 10 kg of weight, a patient weighing 25 kg requires: $$\frac{10}{2}=\frac{25}{x}\Rightarrow x=5\text{ ml}$$

Frequently Asked Questions (FAQ)

Can a proportion have two unknown terms?

No. To find an unknown term using the cross-multiplication rule, you need to know at least three of the four terms of the proportion. If there are two unknowns, one proportion is not enough—you need another equation or an additional condition.

How does a proportion differ from a simple ratio?

A ratio is a comparison of two numbers (e.g., a : b), while a proportion is an equation stating that two ratios are equal (a : b = c : d). In other words, a proportion always consists of two ratios joined by an equals sign.

What if the calculated unknown term is not an integer?

This is a normal result—proportions do not have to yield integer answers. For example, in the recipe scaling example, the number of eggs was calculated as 2.4—in practice, this is rounded to a convenient value (2 or 3 eggs) depending on the context.

Can a proportion be solved with negative numbers?

Yes, the cross-multiplication rule works for negative numbers in the exact same way as for positive numbers—you just need to maintain the correct signs during multiplication and division.