Ratio Calculator

Calculator for ratios and the rule of three with mathematical background

Ratio and Proportion Calculator

What is calculated?

This calculator solves ratio equations using the cross-product method. Enter three known values and the fourth unknown value will be calculated automatically.

Enter proportion equation

:

: x

Solution of proportion equation
x =
Calculation formula: x = (b × c) ÷ a
The solution was calculated using the cross-product method

Ratio Calculator Info

Proportional relationships

The calculator solves proportions:

  • a : b = c : x
  • Three values known → compute x
  • Ratios remain equivalent
  • Cross-product method

Rule of three: This is the classic rule-of-three calculation expressed mathematically.

Quick examples
2:3 = 4:x
x = 6 (double values)
5:10 = 3:x
x = 6 (half ratios)
1:4 = x:20
x = 5 (one quarter)
16:9 = 1920:x
x = 1080 (scaling)
Cross product

a : b = c : x is solved by:
a × x = b × c
Therefore: x = (b × c) ÷ a


Mathematical formulas for ratios

Proportion equation
\[\frac{a}{b} = \frac{c}{x}\] Basic form of a proportion
Cross product
\[a \cdot x = b \cdot c\] Multiplication across
Solution for x
\[x = \frac{b \cdot c}{a}\] Solve for the unknown
General form
\[a:b = c:d \Leftrightarrow a \cdot d = b \cdot c\] Cross-product property
Rule of three direct
\[\text{If } a \rightarrow b, \text{ then } c \rightarrow \frac{b \cdot c}{a}\] Classic rule of three
Percentage change
\[\frac{x}{c} = \frac{b}{a} \Rightarrow \text{factor } = \frac{b}{a}\] Scaling factor

Step-by-step example

Example: solve 1920:1080 = 1200:x

1Set up proportion equation

\[\frac{1920}{1080} = \frac{1200}{x}\]

Proportion as a fraction equation

2Form cross-product

1920 × x = 1080 × 1200
1920 × x = 1,296,000

3Solve for x

\[x = \frac{1080 \times 1200}{1920} = \frac{1,296,000}{1920} = \color{blue}{675}\]

Result: x = 675

4Check

\[\frac{1920}{1080} = 1.778... \quad \text{and} \quad \frac{1200}{675} = 1.778...\]

Both ratios are equal ✓

The classic rule of three

Rule-of-three scheme
Step 1: Known ratio
\[a \rightarrow b\]

a leads to b

Step 2: New base value
\[c \rightarrow ?\]

c leads to what?

Step 3: Proportional calculation
\[c \rightarrow \frac{b \cdot c}{a}\]

Proportional computation

Practical example
Given: 3 apples cost €2
\[3 \text{ apples} \rightarrow 2€\]
Sought: What do 5 apples cost?
\[5 \text{ apples} \rightarrow ?€\]
Solution: Ratio calculation
\[\frac{2€ \times 5}{3} = \frac{10€}{3} \approx 3.33€\]

Practical applications

Scale recipes

Example: cake recipe

For 4 people: 200g flour
For 6 people: x g flour

\[4:200 = 6:x\] \[x = \frac{200 \times 6}{4} = 300g\]
Speed and time

Example: car trip

100 km in 2 hours
150 km in x hours

\[100:2 = 150:x\] \[x = \frac{2 \times 150}{100} = 3h\]
Currency conversion

1 Dollar = 0.85 Euro
50 Dollars = x Euro
x = 50 × 0.85 = 42.50 Euro

Scale calculations

Map 1:50,000
2 cm on map = x m in reality
x = 2 × 50,000 = 100,000 cm = 1 km

Types of ratios

Direct ratio

Both values change in the same direction

\[\text{More A} \rightarrow \text{More B}\]

Example: More time → More distance

Inverse ratio

Values change in opposite directions

\[\text{More A} \rightarrow \text{Less B}\]

Example: More speed → Less time

Compound ratio

Multiple ratios combined

\[a:b \text{ and } b:c \Rightarrow a:c\]

Example: Multi-step conversions

Mathematical foundations

Proportionality

Proportional relationships describe constant ratios between quantities. When one quantity changes, the other changes proportionally. The cross product remains constant.

Historical significance

The rule of three is one of the oldest mathematical methods and was known in ancient Egypt. It is fundamental for trade, crafts and scientific calculations.

Key properties
  • Linearity: Doubling leads to doubling
  • Transitivity: a:b and b:c yield a:c
  • Invertibility: a:b corresponds to b⁻¹:a⁻¹
  • Scaling: Ratios are scale-invariant




More Ratio Functions

Aspect Ratio  •  Compare Ratios  •  Decimal to Ratio  •  Fraction to Ratio  •  Ratio-Calculator  •  Ratio Simplifier  •  Ratio to Decimal  •  Ratio to Fraction  •  Scale Ratio  •