Ratios and Proportions

• Comparing quantities involves understanding the relative sizes or values of two or more items.
• Ratios and proportions are fundamental tools for comparing quantities.
• Percentages are a special type of ratio that simplifies comparisons, especially when dealing with different base values.

Ratios

• A ratio is a comparison of two quantities by division.
• If 'a' and 'b' are two quantities, the ratio of a to b is written as a:b.
• The terms 'a' and 'b' must be in the same units for the ratio to be meaningful.
• Ratios can be simplified by dividing both terms by their greatest common divisor.
• Equivalent ratios are obtained by multiplying or dividing both terms of a ratio by the same non-zero number.
• Ratios can divide a quantity into parts, with the parts being (a/(a+b)) * Total Quantity and (b/(a+b)) * Total Quantity if a quantity is divided in the ratio a:b.

Proportions

• A proportion states that two ratios are equal.
• If a:b = c:d, then a, b, c, and d are said to be in proportion.
• The first and fourth terms (a and d) are called extremes, while the second and third terms (b and c) are called means.
• In a proportion, the product of the means equals the product of the extremes (ad = bc).
• The unitary method is beneficial when solving proportion problems by finding the value of one unit first and then scaling it.
• Direct Proportion: Two quantities are in direct proportion if an increase in one quantity causes a proportional increase in the other, and vice versa.
• Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity causes a proportional decrease in the other, and vice versa.

Percentages

• A percentage is a ratio expressed as a fraction of 100.
• The term "percent" means "per hundred" or "out of one hundred."
• Percentages are denoted by the symbol %.
• To convert a fraction or decimal to a percentage, multiply by 100.
• To convert a percentage to a fraction or decimal, divide by 100.
• Percentage increase is calculated as [(New Value - Original Value) / Original Value] * 100.
• Percentage decrease is calculated as [(Original Value - New Value) / Original Value] * 100.

Applications of Percentages

• Profit and Loss: Profit percentage is [(Selling Price - Cost Price) / Cost Price] * 100; Loss percentage is [(Cost Price - Selling Price) / Cost Price] * 100.
• Simple Interest: Calculated as (Principal * Rate * Time) / 100, where Principal is the initial amount, Rate is the interest rate per year, and Time is the duration in years.
• Compound Interest involves interest being added to the principal, and subsequent interest is calculated on the new principal.
• Discounts: A discount is a reduction in the marked price of an item, usually expressed as a percentage.
• Sales Tax/Value Added Tax (VAT): A percentage added to the cost of goods or services.

Explore ratios and proportions as essential tools for comparing quantities. Understand how ratios express the division of two quantities, the importance of using the same units, and methods for simplification. Learn how proportions demonstrate the equality of two ratios, forming a foundation for problem-solving.