CA > Foundation > Paper 3 – Skim Notes

Chapter 1: Ratio and Proportion, Indices, Logarithms

Overview

  • Understanding ratios and proportions is crucial for comparing quantities in real-life scenarios.
  • Ratios can be simplified and compared using various mathematical principles and properties.
  • Proportions and their properties help understand relationships between different ratios.
  • Indices (or exponents) allow for the representation of repeated multiplication more concisely.
  • The laws of indices facilitate the manipulation of expressions involving exponents.
  • Logarithms serve to express the exponent necessary for a base to produce a certain number.
  • Mastery of logarithms is essential for solving problems across various fields including finance and science.

Key Topics

Ratios

  • A ratio compares quantities of the same kind.
  • It can be expressed as a fraction (a/b) or in the form of ‘a : b’.
  • The terms a and b are called the antecedent and consequent respectively.
  • Ratios can be simplified to their lowest terms, and their order matters (3:4 is different from 4:3).
  • To compare ratios, convert them into equivalent fractions or decimal forms to evaluate which is greater.

Deep Dive

  • Ratios can be compounded (e.g., 3:4 and 5:6 yielding 15:24).
  • Duplicate and triplicate ratios consider squaring and cubing the terms respectively.
  • Understanding inverse ratios can help solve problems efficiently.

Proportions

  • Proportions indicate equality of two ratios, written as a:b::c:d or a/b = c/d.
  • The cross product rule states that ad = bc when a:b = c:d.
  • Various properties such as alternendo and componendo help manipulate proportions.
  • Continued proportion involves three or more quantities, applicable in advanced mathematical contexts.

Deep Dive

  • Mean proportional relates directly to geometric means in statistics.
  • Proportions facilitate the calculation of unknown quantities in real-world problems like mixing solutions.

Indices

  • Indices represent the power to which a number (the base) is raised, showing repeated multiplication, e.g., a^n = a × a × … (n times).
  • The laws of indices make calculations easier, like multiplication and division of powers.
  • Negative indices represent reciprocals, while zero indices equal unity.
  • Examples include operations with various bases and exponents simplifications.

Deep Dive

  • Understanding the relationship between indices and logarithms is crucial.
  • Applications span from basic algebra to complex equations in calculus.

Logarithms

  • Logarithms are the inverse operations of exponentiation, answering the question: to what power must a number (the base) be raised to achieve another number?
  • The three key formulations are loga b = x (where ax = b), and useful when simplifying multiplication and division of exponential forms.
  • Logarithms follow specific laws, such as log(m * n) = log(m) + log(n).
  • They are critical in fields that involve quadratic and exponential growth.

Deep Dive

  • Logarithmic scales are used in measuring phenomena like sound (decibels) and pH levels.
  • Change of base formula allows for conversion between logarithm bases, enhancing flexibility in calculations.

Summary

This chapter covers fundamental concepts of ratios, proportions, indices, and logarithms, necessary for comparing and manipulating numbers effectively in various contexts. Ratios provide a comparative analysis of quantities, while proportions represent equality between ratios. Understanding indices simplifies mathematical expressions involving powers, and logarithms serve as a powerful tool to reverse exponentiation, crucial for solving complex problems in mathematics and applications in science.