Proportions and Ratios
A map at scale 1:100,000 means that 1 cm on the map equals 1 km in reality. A cake recipe for 4 servings needs to be scaled up to 6. Proportions are the tool for scaling everything around us.
- **Maps and blueprints:** scale 1:100, 1:1000
- **Cooking:** scaling recipes
- **Design:** the golden ratio in composition
The Ratio of Two Quantities
A class has 15 boys and 10 girls. How to describe this relationship? One way: 'for every 3 boys there are 2 girls'. This is a **ratio**.
A **ratio** is a comparison of two quantities by division. The ratio of a to b is written as: • a : b (read 'a to b') • a/b (as a fraction) Example: 15 : 10 = 3 : 2 (divided by 5)
A ratio can be simplified by multiplying or dividing both parts by the same number. 6 : 9 = 2 : 3 (divided by 3). The value of the ratio does not change.
A mixture has 12 kg of apples and 8 kg of pears. What is the ratio of apples to pears in simplest form?
Proportion and Its Properties
A **proportion** is an equality of two ratios. When two ratios are equal, the unknown term can be found.
**Formula for finding x:** If x/a = b/c, then x = (a × b) / c Example: x/4 = 3/6 → x = (4 × 3) / 6 = 2
Proportions are the foundation of scaling: maps, blueprints, recipes. Given three of the four terms, the fourth can always be found.
Find x: 4 : x = 6 : 9
Direct and Inverse Proportionality
Sometimes quantities grow together (**direct** proportionality), and sometimes when one grows the other shrinks (**inverse** proportionality).
**How to identify the type?** The key question: 'If one quantity increases, the other...' • ...also increases → DIRECT • ...decreases → INVERSE
It's important to correctly identify the type of relationship! The mistake 'more workers - more time' will lead to an absurd answer.
A car traveled 300 km in 5 hours. What is the relationship between time and distance covered (at constant speed)?
The Golden Ratio
Among all proportions there is one special one - the **golden ratio**. Ancient Greeks considered it the ideal of beauty and harmony.
**Connection to Fibonacci:** 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... 8/5 = 1.6 13/8 = 1.625 21/13 ≈ 1.615 34/21 ≈ 1.619 The ratio of consecutive terms converges to φ!
The golden ratio is an example of how a mathematical proportion connects to beauty and nature. Designers, architects, and artists use it.
The golden ratio is just a pretty number with no mathematical meaning
The golden ratio is the only number for which φ - 1 = 1/φ
φ is defined by the equation x² - x - 1 = 0. This is not a coincidence: this proportion arises naturally in problems of growth and optimization. That is why it appears in nature.
What is the approximate value of the golden ratio φ?
Key Ideas
- Ratio a : b - comparison by division
- Proportion: a : b = c : d, product of extremes = product of means
- Direct proportionality: y = kx (grow together)
- Inverse proportionality: xy = k (one grows, the other shrinks)
Related Topics
Proportions are the foundation for many topics:
- Percentages — A percentage is a proportion with base 100
- Similar Figures — Proportional sides
- Fibonacci Numbers — Ratios converging to the golden ratio
Вопросы для размышления
- Why is the golden ratio considered a 'beautiful' proportion?
- How to distinguish direct proportionality from inverse in a word problem?
- Where in everyday life does scaling appear?