Understanding Ratios and Proportions

• Ratio and proportion are fundamental mathematical concepts used to compare quantities and establish relationships between them
• Ratios express the relative sizes of two or more values
• Proportions state that two ratios are equal

Ratios

• A ratio compares two or more quantities of the same kind
• Expressed as a:b, where a and b are numbers
• The order of the terms in a ratio is significant
• Ratios can be simplified by dividing each term by their greatest common divisor
• A compound ratio is obtained by multiplying together the corresponding terms of two or more ratios
• For example, the compound ratio of a:b and c:d is ac:bd
• A duplicate ratio of a:b is a²:b²
• A sub-duplicate ratio of a:b is √a:√b
• A triplicate ratio of a:b is a³:b³
• A sub-triplicate ratio of a:b is ³√a:³√b
• Ratios are used in various applications like scaling recipes, map reading, and financial analysis

Proportions

• A proportion is an equality between two ratios
• If a:b = c:d, then a, b, c, and d are said to be in proportion
• The terms a and d are called extremes, while b and c are called means
• The product of the means equals the product of the extremes (ad = bc)
• This property is used to solve for unknown quantities in a proportion
• If a:b = c:d = e:f = ..., then each ratio is equal to (a+c+e+...)/(b+d+f+...)
• A continued proportion occurs when a:b = b:c
• In this case, b is the mean proportional between a and c, and b² = ac

Variations

• Direct variation: If two quantities x and y are such that y = kx, where k is a constant, then x and y are said to vary directly
• As x increases, y increases proportionally
• Inverse variation: If two quantities x and y are such that xy = k, where k is a constant, then x and y are said to vary inversely
• As x increases, y decreases proportionally
• Joint variation: If a quantity varies directly as the product of two or more other quantities, it is said to be in joint variation
• For instance, z varies jointly as x and y if z = kxy
• Combined variation: This involves a combination of direct and inverse variations
• For example, z varies directly as x and inversely as y if z = k(x/y)

Partnership

• In a partnership, two or more individuals invest money in a business and agree to share the profits or losses in a certain ratio
• The profit or loss is usually divided in the ratio of their investments or a combination of investment and time
• If investments are for the same period, the profit is divided in the ratio of their investments
• If investments are for different periods, the equivalent capitals are calculated by multiplying each investment with its respective time period
• The profit is then divided in the ratio of these equivalent capitals
• If some partners contribute their management skills in addition to their capital, they may receive a salary before the profits are divided

Mixtures and Alligations

• Mixtures involve combining two or more ingredients in a certain ratio
• Alligation is a rule that enables us to find the ratio in which two or more ingredients at given prices must be mixed to produce a mixture of a desired price
• The mean price is the cost price of a unit quantity of the mixture
• The rule of alligation states that if two ingredients are mixed, then (Quantity of cheaper / Quantity of dearer) = ( (Cost of dearer - Mean price) / (Mean price - Cost of cheaper) )

Important Formulas and Rules

• Componendo: If a/b = c/d, then (a+b)/b = (c+d)/d
• Dividendo: If a/b = c/d, then (a-b)/b = (c-d)/d
• Componendo and Dividendo: If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d)
• Invertendo: If a/b = c/d, then b/a = d/c
• Alternendo: If a/b = c/d, then a/c = b/d
• These rules are useful in simplifying complex proportions and solving problems involving ratios

Applications

• Ratios and proportions are applied widely across various fields
• In cooking, they help scale recipes while maintaining the correct taste
• In finance, they are used to analyze financial statements, calculate ratios like debt-to-equity, and determine investment strategies
• In physics, they are used in scaling laws, determining relationships between physical quantities, and modeling phenomena
• In chemistry, they are used in stoichiometry to balance chemical equations and calculate reactant and product ratios
• In engineering, they are used in scaling designs, analyzing stress and strain, and designing structures
• In everyday life, they help in comparing prices, calculating discounts, and making informed decisions

Problem Solving Techniques

• Understand the problem: Identify the quantities being compared and the relationships between them
• Set up ratios or proportions: Express the relationships using ratios or proportions, ensuring units are consistent
• Solve for unknowns: Use properties of proportions (e.g., cross-multiplication) to solve for unknown quantities
• Check the solution: Validate that the solution makes sense in the context of the problem
• Convert between ratios and fractions: Ratios can be expressed as fractions, and vice versa, which can simplify calculations
• Use direct or inverse variation: Identify whether the quantities vary directly or inversely and use the corresponding formulas
• Break down complex problems: Divide the problem into smaller, manageable parts
• Use the alligation rule: Apply the alligation rule when mixing different ingredients at different prices