Unit Rates and Proportional Reasoning

A rate compares two quantities that have different units — like miles and hours, or dollars and pounds. A unit rate is a special rate where the second quantity equals 1. Unit rates make it easy to compare speeds, prices, and densities because every comparison is now "for one unit" of the same thing.

Rate vs. Unit Rate

• A rate compares two quantities with different units, like 240 miles (about 390 km) in 4 hours.
• A unit rate has a denominator of 1 — it tells you the amount per ONE unit.
• The word "per" means "for every one": miles per hour, dollars per pound, calories per serving.
• Common unit rates: speed (miles per hour), price (dollars per item), density (people per square mile).

How to Find a Unit Rate

• Write the rate as a fraction: first quantity over second quantity.
• Divide both the top and bottom by the bottom number to make the denominator 1.
• In short: divide the first quantity by the second quantity.
• Worked example: 240 miles in 4 hours becomes 240 ÷ 4 = 60. The unit rate is 60 miles per hour (about 97 km/h).

Best Buy — Comparing Prices

When two stores sell the same item in different package sizes, you cannot compare the total prices fairly. Convert each price to a unit price (dollars per pound, dollars per ounce, dollars per item) and pick the smaller unit price. The smaller the unit price, the better the buy.

• Store A: $6 for 3 lbs (about 1.4 kg) of apples → $6 ÷ 3 = $2 per lb (about $4.40 per kg).
• Store B: $10 for 4 lbs of apples → $10 ÷ 4 = $2.50 per lb.
• Store A is the better buy because $2 per lb is less than $2.50 per lb.

Proportional Reasoning

Once you know a unit rate, you can scale up or scale down to any quantity. If a car drives 60 miles in 1 hour, it will drive 120 miles in 2 hours, 180 miles in 3 hours, and so on — these rates are all equivalent. Two rates are proportional when they share the same unit rate, and writing them as equal, like 60/1 = 120/2, makes a proportion.