Ratios and Unit Rates

Understand the idea

A ratio compares two quantities in a stated order. Equivalent ratios multiply or divide both quantities by the same nonzero factor. A unit rate compares a quantity with one unit of another quantity. Keep units attached: miles per hour and dollars per notebook describe different relationships. A proportional relationship has a constant ratio; a starting fee can prevent proportionality.

Worked example

Four notebooks cost $12. Divide both quantities by 4: one notebook costs $3. At that constant price, seven notebooks cost 7 × $3 = $21. A second shop charges $14 for five notebooks: $14 ÷ 5 = $2.80 per notebook, so it has the lower unit price.

Try it yourself

1. A cyclist travels 18 miles in 1.5 hours. Find the average speed.
2. A recipe uses 2 cups of rice for 6 servings. How much rice for 9 servings?
3. A taxi charges $5 plus $2 per mile. Is total cost proportional to miles? Explain.

Check your reasoning

1. 18 ÷ 1.5 = 12 miles per hour.
2. 9/6 = 1.5 times as many servings; 2 × 1.5 = 3 cups.
3. No. A proportional relationship would have zero total cost at zero miles; the $5 starting fee breaks that condition.

Make it work for your learner

For support, model the example one step at a time and ask the learner to explain each choice in their own words. Use a sketch, organizer or physical model where helpful. For a challenge, create a new example, test an alternative explanation or describe when the model would not apply.

Exit check

Use equivalent ratios and unit rates to compare quantities with units. Explain your reasoning with a specific example before moving on.