Scale Ratio

Calculator and formulas to scale ratios with mathematical background

Ratio Scaling Calculator

What is calculated?

This function scales a ratio by a given factor. Both sides of the ratio are multiplied by the same value, preserving the proportional relationship.

Enter ratio and scale factor

:
Original ratio

Multiplier

Scaled ratio
Original
:
→
Scaled
:
The ratio has been scaled proportionally by the given factor

Scaling Info

Scaling principle

Scaling is done proportionally:

  • a : b → (a×k) : (b×k)
  • Both sides multiplied by k
  • Ratio remains equal
  • Shape does not change

Important: The scale factor affects absolute sizes but not the ratio between values.

Scale types
k > 1: Enlargement
e.g. k=2 → double size
k = 1: No change
Original remains
0 < k < 1: Reduction
e.g. k=0.5 → half size
k < 0: Inversion
Direction change
Applications

Technical drawings: change scale
Recipes: adjust portion sizes
Model building: size ratios


Mathematical formulas for ratio scaling

Basic formula
\[a:b \xrightarrow{k} (a \cdot k):(b \cdot k)\] Proportional scaling
Individual components
\[a' = a \cdot k, \quad b' = b \cdot k\] Scaled values
Ratio invariance
\[\frac{a}{b} = \frac{a \cdot k}{b \cdot k} = \frac{a'}{b'}\] Ratio remains equal
Inverse
\[a':b' \xrightarrow{1/k} a:b\] Unscaling
Chaining
\[a:b \xrightarrow{k_1} \xrightarrow{k_2} (a \cdot k_1 \cdot k_2):(b \cdot k_1 \cdot k_2)\] Multiple scaling
Decimal factor
\[k = 0.5 \Rightarrow a':b' = \frac{a}{2}:\frac{b}{2}\] Reduction by factor 2

Step-by-step example

Example: scale 4:6 by factor 2.5

1Scale first element

\[a' = a \times k = 4 \times 2.5 = 10\]

First component multiplied

2Scale second element

b' = b × k = 6 × 2.5 = 15

Second component multiplied

3Scaled ratio

\[4:6 \xrightarrow{2.5} \color{blue}{10:15}\]

Result: The scaled ratio is 10:15

4Ratio verification

\[\frac{4}{6} = \frac{2}{3} ≈ 0.667 \quad \text{and} \quad \frac{10}{15} = \frac{2}{3} ≈ 0.667\]

Ratio remains equal ✓

Different scale factors

Enlargement (k > 1)
k = 2: Double size
\[3:4 \xrightarrow{2} 6:8\]
k = 3.5: 3.5× size
\[2:5 \xrightarrow{3.5} 7:17.5\]
k = 10: Tenfold size
\[1:3 \xrightarrow{10} 10:30\]
Reduction (0 < k < 1)
k = 0.5: Half size
\[8:12 \xrightarrow{0.5} 4:6\]
k = 0.25: Quarter size
\[20:16 \xrightarrow{0.25} 5:4\]
k = 0.1: Tenth size
\[100:50 \xrightarrow{0.1} 10:5\]

Practical applications

Scale recipes

Example: cake recipe for more people

Original: 2:3 (flour:sugar for 4 people)
For 10 people: factor = 10/4 = 2.5

\[2:3 \xrightarrow{2.5} 5:7.5\]

5 parts flour, 7.5 parts sugar

Technical drawings

Example: change scale

Drawing 1:100 → 1:50 (double size)
Factor = 2

\[20:30 \text{ mm} \xrightarrow{2} 40:60 \text{ mm}\]

All dimensions doubled

Image editing

Change image size: 1920:1080 → 960:540
Scale factor: 0.5 (halving)

Model building

Real airplane: 30:20 m (length:span)
Model 1:72: Factor = 1/72 ≈ 0.0139

Special cases

Factor = 0
\[a:b \xrightarrow{0} 0:0\]

Collapse to zero ratio
(mathematically undefined)

Negative factor
\[4:6 \xrightarrow{-1.5} -6:-9\]

Change sign
Direction change

Factor = 1
\[a:b \xrightarrow{1} a:b\]

Identity transformation
No change

Mathematical foundations

Similarity transformation

Scaling is a special geometric transformation known as a similarity transformation. It changes size but not the shape or the proportional relationships.

Linear transformation

Scaling is a linear operation: f(x) = k·x. This property makes it useful in mathematics because it can be combined with other linear operations.

Key properties
  • Homogeneity: All components scaled equally
  • Reversibility: Invertible by 1/k
  • Commutativity: k₁ × k₂ = k₂ × k₁
  • Proportionality: Ratios remain preserved




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  •