Understanding Ratios and Unit Rates
Students describe multiplicative comparisons with ratios and calculate unit rates involving common quantities.
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Identify Ratio Relationships
A ratio compares two quantities by showing how many of one quantity there are in relation to another. Ratios describe multiplicative comparisons, not just differences. Suppose a basket contains 8 red apples and 12 green apples. The ratio of red apples to green apples is 8 to 12. This means that for every 8 red apples, there are 12 green apples. Order matters: the ratio of green apples to red apples is 12 to 8, which describes the relationship in the opposite direction. You can also compare one part with the whole. Because there are 20 apples altogether, the ratio of red apples to all apples is 8 to 20. Always identify the two quantities and place them in the order named.
Write Ratios Three Ways
A ratio can be written in three common ways: with the word “to,” with a colon, or as a fraction. Imagine a box holding 3 pencils and 5 markers. The ratio of pencils to markers can be written as 3 to 5, 3:5, or 3/5. All three forms describe the same comparison. The first number always represents the first quantity named, so the order must stay the same in every form. If the comparison is reversed, the ratio of markers to pencils is 5 to 3, 5:3, or 5/3. When a ratio is written as a fraction, the numerator represents the first quantity and the denominator represents the second quantity. Read the question carefully before choosing which quantity comes first.
Interpret Ratios in Context
Interpreting a ratio means explaining what both numbers represent in the situation. A drink recipe uses 2 cups of juice concentrate for every 5 cups of water. The ratio of concentrate to water is 2:5. In context, this means that each batch combines 2 cups of concentrate with 5 cups of water. It does not mean that the finished drink contains only 5 cups. The total is 7 cups because 2 + 5 = 7. Therefore, the ratio of concentrate to the total drink is 2:7, while the ratio of water to the total drink is 5:7. Be precise when using ratio language. State the quantities, their order, and what one complete group or batch represents.
Find Equivalent Ratios
Equivalent ratios describe the same relationship using different numbers. To create an equivalent ratio, multiply or divide both terms by the same nonzero number. For example, the ratio 3:4 is equivalent to 6:8 because both 3 and 4 were multiplied by 2. It is also equivalent to 9:12 because both terms were multiplied by 3. A ratio table can organize these related pairs. If 3 notebooks cost 4 dollars, then 6 notebooks cost 8 dollars at the same rate, and 9 notebooks cost 12 dollars. Multiplying only one term would change the relationship and would not produce an equivalent ratio. You can check equivalence by simplifying: 6:8 and 9:12 both divide to 3:4.
Calculate Unit Rates
A unit rate is a ratio that compares a quantity with one unit of another quantity. The word “per” often signals a unit rate. Suppose a car travels 180 miles in 3 hours. The rate is 180 miles per 3 hours. To find the distance traveled in 1 hour, divide both quantities by 3: 180 ÷ 3 = 60 and 3 ÷ 3 = 1. The unit rate is 60 miles per hour. This means the car travels 60 miles for each hour when its rate stays constant. Unit rates make comparisons easier because every rate uses the same second quantity of 1. Always include units in the answer; the number 60 alone does not explain what was measured.