LESSON 15
RATIOS & PROPORTIONS
Comparing Quantities and Scaling
What is a Ratio?
A ratio is a comparison of two quantities by division. It tells us how much of one thing there is relative to another. Ratios can be written three ways: using a colon (3:4), as a fraction (3/4), or in words ('3 to 4'). All three forms mean the same thing.
A proportion is a statement that two ratios are equal. Proportions are one of the most powerful tools in mathematics — they allow us to scale recipes, convert units, read maps, calculate speeds, and solve countless real-world problems.
Writing and Simplifying Ratios
A ratio should always be written in simplest form by dividing both quantities by their greatest common factor. The order of a ratio matters — 3:4 and 4:3 are different comparisons. Always pay attention to which quantity is mentioned first in the problem.
Setting Up Proportions
A proportion sets two equivalent ratios equal to each other. To set one up, write the known ratio as a fraction, then write the unknown ratio as a fraction with a variable. Make sure the same units are in the same positions (numerator or denominator) on both sides.
Cross Multiplication
Cross multiplication is the standard method for solving proportions. Multiply the numerator of each fraction by the denominator of the other, set the products equal, and solve for the variable. It works because multiplying both sides of a proportion by both denominators produces the same result.
Unit Rates
A unit rate is a ratio with a denominator of 1. It tells us the amount of one quantity per single unit of another. Unit rates make comparisons easy — miles per hour, price per item, and calories per serving are all unit rates. To find a unit rate, divide the first quantity by the second.
Key Facts
The golden ratio (approximately 1:1.618) appears in art, architecture, and nature. The Parthenon, Leonardo da Vinci's paintings, and nautilus shells all exhibit this proportion.
Map scales are ratios. A scale of 1:100,000 means 1 cm on the map equals 100,000 cm (1 km) in real life.
Proportional reasoning is used in cooking, construction, medicine dosing, currency exchange, and photography — it is one of the most applied math skills in daily life.
Similar figures in geometry have proportional side lengths. This allows architects and engineers to work from scale models before building full-size structures.