Do the ratios form a proportion: word problems

Learning Objectives Analyze word problems to identify two distinct ratios. Set up ratios correctly from given information in word problems. Apply the cross-multiplication property to determine if two ratios form a proportion. Simplify ratios to their simplest form to compare if they are equivalent. Justify whether two ratios form a proportion based on mathematical evidence. Distinguish between situations that represent proportional relationships and those that do not. Ever wonder if comparing two different situations means they're 'balanced' or 'fair'? 🤔 We'll learn how to check if two comparisons are truly equivalent! In this lesson, you'll learn how to read word problems, pull out the important numbers, and figure out if the ratios they...

TermDefinitionExample RatioA comparison of two quantities by division. It can be written as a:b, a/b, or 'a to b'.If there are 3 apples and 2 bananas, the ratio of apples to bananas is 3:2 or $\frac{3}{2}$. ProportionAn equation stating that two ratios are equivalent. It shows that two comparisons are balanced.$\frac{1}{2} = \frac{2}{4}$ is a proportion because both ratios represent the same relationship. Equivalent RatiosRatios that represent the same relationship between quantities, even if the numbers themselves are different.1:2 and 5:10 are equivalent ratios because both simplify to 1:2. Cross ProductsIn a proportion $\frac{a}{b} = \frac{c}{d}$, the cross products are the results of multiplying the numerator of one ratio by the denominator of the other (a * d and b * c).For...

Definition of a Proportion Two ratios $\frac{a}{b}$ and $\frac{c}{d}$ form a proportion if they are equivalent, written as $\frac{a}{b} = \frac{c}{d}$. This rule states that if two ratios represent the same relationship, they are proportional. You can check this by simplifying both ratios to their simplest form and seeing if they are identical. Cross-Multiplication Property For two ratios $\frac{a}{b}$ and $\frac{c}{d}$, they form a proportion if and only if their cross products are equal: $a \cdot d = b \cdot c$. This is a powerful method to quickly determine if two ratios are equivalent without simplifying them. Multiply the numerator of the first ratio by the denominator of the second, and compare it to the product of the denominator of the first ratio and the numerator o...