Unit 2: Ratios & Proportions

Transcription

1 Unit 2: Ratios & Proportions Name Period Score /42 DUE DATE: A Day: Sep 21st B Day: Sep 24th Section 2-1: Unit Rates o Rate- A ratio that compares quantities with different kinds of units. o Unit Rate- A rate that is simplified so that is has a denominator of unit. Example of a Ratio: A classroom has 35 students in it. There are 15 boys and 20 girls. The ratio of boys to girls is. We can write this ratio three different ways15 to 20, 15 :20 or as a fraction If we wanted the ratio of girls to boys we would have 20 to 15, 20:15 or Example of a Unit Rate: If you have $12 for 3 pounds, you can write the ratio $12 3lbs. If you have to pay $12 for 3 pounds of almonds then we need to find how much one pound of almonds would be. $12 $4 $4 per one pound of almonds 3pounds 1pound Solve on your own. Circle your answer and show your work. If I read 180 words in 3 minutes, what is my unit rate? or How many words do I read in 1 minute? 180words 3min If you can solve 90 math problem in 5 days, what is your unit rate? or How many math problems do you solve in 1 day? 90 problems days Section 2-2: Proportional & Non-Proportional Relationships o Proportional- The relationship between two ratios that a constant rate or ratio. o Non-proportional- The relationship between two ratios that have a constant rate or ratio o Equivalent ratios- Two ratios that have the same value. A proportional relationship between two quantities is one in which the two quantities vary directly (direct variation) with one another. Example: If one item is doubled, the other, related item is also.

2 Graphs of Proportional relationships: The equations of proportional relationships are always in the form y=mx. When proportional relationships are graphed, they produce a line that passes through the. In this equation, m is the slope of the line, and it is also called the unit rate, rate of change, or constant of proportionality of the function. Tables of Proportional Relationships: Tables can be used to determine if a relationship is proportional. If gasoline costs $4.24 per gallon, the table below can be created to model the situation. This situation is proportional. First, it contains the origin, (0, 0), and this makes sense: if we buy gallons of gas it will cost zero dollars. Second, if the number of gallons is, the cost is doubled; if it is tripled, the cost is. The equation that will represent this data is y = 4.24x, where x is the number of gallons of gasoline and y is the total cost. An important conclusion: The unit rate for any item in a relationship will always be the same for each entry in a table and every point on a graph. Check for proportionality Example: Tess rides her bike at 12 mph. Create a table and a graph to see if the relationship is proportional. The relationship is proportional because 1. The graph is a line through the 2. The unit rate for every point is the same. is the rate The equation of this line would be.

3 Does the graph and table below show proportional relationships? Why or why not? (2 points) Section 2-3: Solve Proportions o Proportion-An equation stating that two ratios or rates are. 1. To keep a number value the same, it can only be multiplied by. 2. To keep the same value, but change a number, multiply by in fraction form. (Ex. 4 4 ) 3. If fractions are equal and have equivalent denominators, the numerators must also be. 4. If fractions are equal and have equivalent numerators, the must also be equal. 5. If fractions do not have equivalent denominators (or numerators), multiply one or both fractions by one 2 in the form of a fraction (ex. 1 ) to make them. 2 Example: How to solve a proportion Solve 3 x Since the x is in the numerator, we want to get the denominators to be equivalent. The only way we can do this is to multiply by 1, in fraction form, on both sides. (You are familiar with this already because you know how to find a common denominator!) 2 3 x x 2. Then, multiply your fractions Since, the denominators now match, we can simply look at the numerators because we know that 6 x they should match as well The resulting equation (in this case) gives us the value of x or in other cases, you will have an equation to solve in order to find x. So x 6

4 Here s another example. 4 x 5 Solve x 5 1. Multiply by 1, in fraction form, to make the denominators equal x 5 2. Multiply your fractions Since the denominators are equal, I can just look at the resulting equation from the numerators and 8 x 5 solve x or x 13 Solve on your own. Show your work and circle your answer x x x Section 2-4: Scale Drawings o Scale Drawings-A drawing that is used to represent objects that are too large or too small to be drawn at actual size. o Scale Models- A model used to represent objects that are too large or too small to be built at actual size. o Scale- The scale that gives the ratio that compares the of a drawing or model to the measurements of the real object. Example If a model of a bird has a wingspan of 6 inches and the actual bird has a wingspan of 3 feet, then: Model length 6 inches 2 in. or 2 in :1 ft Actual length 3 feet 1 ft. Example If a map has a scale of 4 inches to represent 20 miles, then: Model length 4inches or : Actual length 5miles

5 o Scale Factor- A scale written as a without units in simplest form. Example If a model of a bird has a wingspan of 6 inches and the actual bird has a wingspan of 3 feet, then: Model length inches in inches Actual length 3 feet 1 ft 12 inches 6 Example: A model of the Empire State Building is 15 inches tall. The scale of the model : actual is 3 inch : 250 feet. How tall is the actual Empire State Building in New York City? First, we need the scale factor. 3 inches 3 inches 250 feet inches 1000 We can use this scale to solve the following proportion: 1 15 inches 1000 x 1 15 inches Multiply the left side by 1000 x inches inches Set denominators equal to each other x inches x inches Convert inches to feet x 1250 feet So the actual Empire State Building is 1250 feet tall, making it the 14 th tallest building in the world. Resources: Section 2-2 Proportional Relationships and Slope: