Proportions Cheat Sheet

Proportions are equations stating that two ratios are equal, used to solve problems involving unknown quantities by cross-multiplication or by dividing a total into specific ratios.

Core Principles

• A proportion is an equation where two ratios are equal (e.g., a:b = c:d).
• Proportions can be set up as two equal fractions (e.g., a/b = c/d).
• Solving generic proportions often involves cross-multiplication.
• Word problems require careful identification of 'complete pairs' or 'categorical pairs' to set up the proportion correctly.
• Split proportions divide a total amount based on a given ratio, requiring the sum of ratio parts to be calculated first.

Action Steps

• Identify the known and unknown quantities.
• Determine if the problem involves a direct proportion or a split proportion.
• For direct proportions: Set up two equal ratios (as fractions or using colons). Ensure units are consistent.
• For split proportions: Sum all parts of the ratio to find the 'Total Parts'.
• Cross-multiply to solve for the unknown variable.
• For split proportions: Calculate the amount for each part using (Total Amount / Total Parts) * Individual Part.
• Check your answer to ensure it makes sense in the context of the problem.

Formulas

• For a proportion a:b = c:d, cross-multiplication yields ad = bc.
• Setting up a proportion from word problems: (Quantity A / Unit A) = (Quantity B / Unit B)
• For split proportions: (Total Amount / Total Parts) = (Individual Part Amount / Individual Part)
• $ \frac{a}{b} = \frac{c}{d} \implies ad = bc $
• $ \frac{\text{Part 1}}{\text{Total Parts 1}} = \frac{\text{Part 2}}{\text{Total Parts 2}} $

Key Terms

• Proportion: An equation stating that two ratios are equal.
• Ratio: A comparison of two quantities.
• Cross-Multiplication: The method of multiplying the numerator of one fraction by the denominator of another and setting the products equal.
• Complete Pairing: In word problems, two numbers connected in a sentence representing different items (e.g., $8 for 23 apples).
• Categorical Pairing: In word problems, two numbers not directly connected but representing the same kind of thing (e.g., 23 apples and 18 apples).
• Split Proportion: Dividing a total quantity into parts according to a specific ratio.

Real World Examples

• Grocery store pricing: Calculating the cost of 5 dozen oranges when the price for 2 dozen is known.
• Fuel consumption: Determining how long a generator will run on a certain amount of gasoline based on its known consumption rate.
• Dividing profits: Distributing a total profit among partners based on a given ratio.
• Tax assessment: Calculating property tax based on a given tax rate per assessed valuation.
• Shadow length: Finding the length of a shadow cast by a building given the shadow length of a flagpole of known height.

People

• Mr. Brown: Individual whose estate is divided according to a specific ratio in an example.