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Interpret rates as comparisons of quantities

A rate compares two quantities measured in different units and describes how much of one quantity corresponds to a given amount of another, often expressed as a quotient or “per” relationship. The learner interprets and compares rates such as miles per hour, dollars per item, or cups per serving; identifies unit rates and equivalent rates; and understands that a rate reflects a multiplicative relationship rather than an additive difference. This supports proportional reasoning, tables, graphs, and equations without extending to instantaneous or complex compound rates.

Detailed Explanation: Interpret rates as comparisons of quantities

A rate compares two quantities with different units. To find a unit rate, divide so that one of the quantities is 11. The word “per” tells you to divide.

Example:
Store A sells 8 notebooks for 12.StoreBsells5notebooksfor. Store B sells 5 notebooks for $8$. Which store has the lower cost per notebook?

  1. Find Store A’s cost per notebook:

$128 notebooks=$1.50 per notebook\frac{\$12}{8\text{ notebooks}}=\$1.50\text{ per notebook}

This means each notebook costs 1.50$ at Store A.

  1. Find Store B’s cost per notebook:

$85 notebooks=$1.60 per notebook\frac{\$8}{5\text{ notebooks}}=\$1.60\text{ per notebook}

This means each notebook costs 1.60$ at Store B.

  1. Compare the unit rates:

$1.50 per notebook<$1.60 per notebook\$1.50\text{ per notebook}<\$1.60\text{ per notebook}

Therefore, Store A has the lower rate, so it is the better buy.

The rate is a multiplicative comparison: the number of notebooks is multiplied by the cost per notebook to find the total cost. For example, at Store A,

8⋅$1.50=$12.8\cdot \$1.50=\$12.

Learn by doing: Interpret rates as comparisons of quantities

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    Ratios - Unit Rates, Solve for Rate

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