Compare ratios: word problems

Learning Objectives Extract two or more ratios from a complex word problem. Convert ratios into a common format for comparison, such as unit rates or equivalent ratios with a common term. Apply mathematical methods, including cross-multiplication and unit rate calculation, to determine which ratio is greater. Interpret the result of a ratio comparison to answer a specific question posed in a word problem. Solve multi-step problems that require comparing ratios as an intermediate step. Justify their conclusions by explaining the method used and the meaning of the comparison. Analyze and compare rates of change expressed as ratios in various contexts like speed, density, or concentration. Which is a better deal: a 12-pack of soda for $7.50 or a 20-pack for $12.00? 🛒 Let&#03...

TermDefinitionExample RatioA comparison of two quantities by division, showing their relative sizes. It can be expressed as a:b, a/b, or 'a to b'.In a class with 15 boys and 10 girls, the ratio of boys to girls is 15:10, which simplifies to 3:2. Unit RateA ratio where the second quantity (the denominator or consequent) is one unit. It is used to express a quantity 'per' another quantity.If a car travels 120 km in 2 hours, its unit rate (speed) is 60 km per 1 hour, or 60 km/h. ProportionAn equation stating that two ratios are equivalent.The ratios 2:3 and 4:6 are in proportion because 2/3 = 4/6. Equivalent RatiosRatios that express the same relationship between two quantities. They can be found by multiplying or dividing both parts of a ratio by the same non-zero number...

Method 1: Unit Rate Conversion To compare the ratios \(a:b\) and \(c:d\), convert each to a unit rate by dividing the antecedent by the consequent. Compare \(\frac{a}{b}\) and \(\frac{c}{d}\). This is the most versatile method, especially when the quantities being compared are different. It answers the question 'How much of the first quantity corresponds to one unit of the second quantity?' Method 2: Common Consequent (Denominator) To compare \(a:b\) and \(c:d\), find a common multiple, \(M\), of \(b\) and \(d\). Convert each ratio to an equivalent ratio with \(M\) as the consequent. Then compare the new antecedents. This method is useful when the consequents are small, easily manageable numbers. It's like finding a common denominator for fractions. Method...