Definition of Ratio: A ratio is a comparison of two quantities. It is only defined for quantifiable things and cannot be used for qualitative comparisons.
Usage: Ratios are used in various places for comparisons, such as wealth comparisions, scores, salaries, heights, and weights.
Notable Ratios
Golden Ratio (1.618): A famous and ubiquitous ratio found in nature and art, like the proportions of a butterfly's wings, the Mona Lisa painting, and the structure of an egg.
Definition and Examples
Basic Definition: Ratio is the comparison between two quantities. For example, if A's height is 20 cm and B's height is 10 cm, the ratio of A to B is 20/10 or 2:1.
Notation: Ratios can be written as A/B or A is to B. B (denominator) cannot be zero.
Properties of Ratios
Simplest Form: Ratios are generally expressed in their simplest form. For example, 2:4 can be simplified to 1:2.
Multiplication: Multiplying both terms by the same non-zero number doesn't change the ratio, e.g., 3/5 = 6/10.
Division: Dividing both terms by the same non-zero number doesn't change the ratio, e.g., 4/6 = 2/3.
Constant Ratio: If A/B = C/D, then A and C, and B and D are proportional.
Comparison of Ratios
Cross Multiplication Method: Used to compare two ratios. For example, comparing 2/3 and 4/7: 27 (14) > 34 (12), thus 2/3 > 4/7.
LCM Method: Another method to compare ratios, by making their denominators the same using the Least Common Multiple (LCM).
Proper Fractions: If the difference between the numerator and denominator is the same, the fraction that looks larger is larger.
Improper Fractions: For improper fractions with the same difference between numerator and denominator, the fraction that looks smaller has the highest value.
Advanced Comparison Techniques
General Comparison: For fractions not following a common pattern, use the formula: numerator / (denominator - numerator). This simplifies to an easy comparison.
Example: For fractions 29/34, 37/43, 13/17, 11/14, compute 29/(34-29), 37/(43-37), etc., and compare the resulting values.
Mixed Fractions: Separate fractions into proper and improper, compare within each group, and then combine the results for final ordering.
Summary
Ratios provide a method for comparing quantifiable quantities and have specific properties and methods for comparison.
Proper understanding and application of these comparison techniques can simplify and expedite the process of working with ratios.
Stay tuned for the next part of the ratio lecture series.