Rates & Unit Rates

A rate is a ratio that compares two quantities measured in different units. For example, traveling 150 miles in 3 hours compares distance in miles with time in hours. Rates are commonly used to describe speed, cost, wages, recipes, and other real-world relationships.nnA unit rate is a rate in which the second quantity is 1. The phrase per one can help identify a unit rate. To find the unit rate for 150 miles in 3 hours, divide both quantities by 3. The result is 50 miles in 1 hour, or 50 miles per hour.nnUnit rates may be written with words, a fraction bar, or the word per. Examples include 60 miles per hour, $2.50 per pound, 12 pages per minute, and 4 cups of flour per batch. Units must be included because they explain what the numbers represent.nnTo calculate a unit rate, divide the quantity named before the word per by the quantity that should equal one. If 5 notebooks cost $15, divide 15 by 5 to find the cost of one notebook. Each notebook costs $3, so the unit rate is $3 per notebook.nnSome situations require interpreting a fraction as a unit rate. If a runner travels 3/4 of a mile in 1/2 hour, divide 3/4 by 1/2. Multiplying 3/4 by the reciprocal 2/1 gives 3/2, or 1.5 miles per hour.nnRates and unit rates can be represented in tables. If 3 movie tickets cost $36, a table may show 1 ticket for $12, 2 tickets for $24, and 5 tickets for $60. Each pair has the same unit rate of $12 per ticket.nnA double number line can display corresponding quantities and equivalent rates. One number line may represent hours while the other represents miles. If a vehicle travels 45 miles each hour, the points 1 hour and 45 miles, 2 hours and 90 miles, and 3 hours and 135 miles align.nnRates can also be graphed. When a relationship has a constant unit rate, the points form a straight line through the origin. The unit rate appears as the amount the vertical quantity changes for each increase of 1 in the horizontal quantity. This idea will connect to constants of proportionality in the next lesson.nnUnit rates make comparisons fair when the original quantities are different. Suppose one package contains 8 ounces for $3.20 and another contains 12 ounces for $4.56. The first costs $0.40 per ounce, while the second costs $0.38 per ounce. The 12-ounce package has the lower unit price.nnCommon misconceptions and corrections: Students may divide the quantities in the wrong order, omit the units, or assume that the larger package always has the better price. Identify which amount should be expressed per one, divide in the correct order, label the result, and compare the unit rates rather than the original totals.