Equivalent rates describe the same relationship using different quantities.
A proportion can be used to solve equivalent rate problems.
Mixed numbers should often be converted to improper fractions before multiplying or dividing.
You can solve some rate problems by first finding the unit rate and then scaling up.
Always keep the units in the same order when setting up equivalent rates.
✏️ Worked Examples
Problem: A recipe calls for \(2\frac{1}{2}\) teaspoons of cinnamon to make 4 dozen oatmeal cookies. How many teaspoons of cinnamon are needed to make 60 oatmeal cookies?
Important: Since 60 cookies equals 5 dozen cookies, we can compare teaspoons of cinnamon to dozens of cookies.
Notice: The rate stays equivalent because both the cookie amount and cinnamon amount are scaled by the same factor.
🧠 Math Vocabulary
Rate: a ratio that compares two quantities with different units.
Equivalent rate: a rate that describes the same relationship using different values.
Proportion: an equation showing that two ratios or rates are equal.
Unit rate: a rate that compares a quantity to exactly 1 unit.
Scale factor: the number used to multiply quantities to create an equivalent rate.
💡 Main Idea
Equivalent rate problems can be solved in several ways. You can set up a proportion, find the unit rate first, or use a scale factor. When mixed numbers appear in the problem, converting them to improper fractions usually makes the calculations easier.
📚 What You Should Already Know
Students should know how to convert mixed numbers to improper fractions, multiply fractions, divide fractions, and write equivalent ratios or rates in the same order.
🚀 What Comes Next
After solving equivalent rate problems with mixed numbers, students can apply the same reasoning to unit rates, proportions, percent problems, scale drawings, and proportional relationships.
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