Pre Algebra, Unit 6: Ratio and Proportion

Transcription

1 Unit 6: Ratio and Proportion Pre Algebra, Unit 6: Ratio and Proportion Syllabus Objectives: (2.10) Students will compute unit cost. (2.11) Students will solve proportions. (2.12) Students will find the ratio of corresponding side lengths of similar figures. (2.13) Students will find unknown side lengths of similar figures. Ratios Ratio is a comparison between two quantities. We use ratios everyday; one Pepsi costs 50 cents describes a ratio. On a map, the legend might tell us one inch is equivalent to 50 miles or we might notice one hand has five fingers. Those are all examples of comparisons ratios. 3 Ways to Write a Ratio A ratio can be written three different ways. If we wanted to show the comparison of one inch representing 50 miles on a map, we could write that as: Using the word to 1 to 50 or Using a colon 1:50 or Using a fraction 1 50 Because we are going to learn to solve problems, it s easier to write the ratios using fractional notation. If we looked at the ratio of one inch representing 50 miles, 1, we might determine 2 50 inches represents 100 miles, and 3 inches represents 150 miles by using equivalent fractions. Pre-Algebra Unit 6: Ratio and Proportion Page 1 of 11

2 Simplifying Ratios That just seems to make sense. Look at that from a mathematical viewpoint, it appears that we 3 might also be able to reduce 150 to Does 150 represent the same comparison as 1 50? The answer is yes and if we looked at other ratios, we would see that reducing ratios does not affect those comparisons. 3 We noticed that 150, which means 3 inches represents 150 miles, could be reduced to 1 50 meaning 1 inch represents 50 miles. Mathematically, by setting the ratios equal, we could write Applications of Ratios Rate is a ratio that compares two quantities measured in different units. Example Person A travels 300 miles in 5 hours, find A s rate. A s rate 300 miles 5 hours Unit rate is a rate whose denominator is 1. To convert a rate to a unit rate, divide both the numerator and denominator by the denominator. Example Find the unit rate of person A. 300 miles 5 60 miles ; read 60 miles per hour 5 hours 5 1 hour Example Bob s heart beats 520 times every four minutes, find Bob s heart rate. 520 beats beats ; read 130 beats per minute 4 minutes 4 1 minute Pre-Algebra Unit 6: Ratio and Proportion Page 2 of 11

3 Slope Slope is the ratio of rise to run. slope rise run Example Find the slope of the line that connects the points in the graph. The first point is (2, 1), the second point is (6, 5). How many units did I go up? Count them, up 4. Then we went to the right 4. 4 So the slope is y x Example Find the slope of the line that connects the points on the graph. 8 4 y In this case, I will start at the point furthest to the left in Quadrant II, ( 4, 5). To get to the point on the right in Quadrant IV (3, 2), I need to go down 7, ( 7) and go to the right 7, (+7). 7 So the slope is x Pre-Algebra Unit 6: Ratio and Proportion Page 3 of 11

4 Proportions Proportion is a statement of equality between 2 ratios. Looking at a proportion like, 1 3 we might see some relationships that exist if we take time 2 6 and manipulate the numbers. For instance, what would happen if we tipped both ratios up-side down? and, notice they are also equal, so How about writing the original proportion sideways, will we get another equality? and, notice they are equal also, so If we continued looking at the original proportion, we might also notice we could find cross products and retain an equality. In other words, Makes you wonder whether tipping ratios up-side down, writing them sideways or using cross products only works for our original proportion? Well, to make that determination, we would have to investigate some more proportions. Try some; if our observation holds up, we ll be able to generalize what we saw. Let s try these observations with this proportion Can I tip them up-side down and still retain an equality? In other words, does 3 6? 2 4 How about writing them sideways, does 2 3? 4 6 How about using cross products in the original proportion, does ? Pre-Algebra Unit 6: Ratio and Proportion Page 4 of 11

5 The answer to all three questions is yes. Pre-Algebra Unit 6: Ratio and Proportion Page 5 of 11

6 Since everything seems to be working, we will generalize our observations using letters instead of numbers. If a b c d, then 1. b a d c 2. a c b d 3. ad bc Those 3 observations are referred to as Properties of Proportions. Those properties can be used to help us solve problems. To solve problems, most people use either equivalent fractions or cross products to solve proportions. Generally, you use equivalent fractions when either the numerator or denominator of a fraction is a multiple of the numerator or denominator of the other fraction. If that is not immediately obvious, then use cross products. Example Find the value of x x This problem can be solved by equivalent fractions or cross products. By finding cross products, we have 6x 360 or x 60. Pre-Algebra Unit 6: Ratio and Proportion Page 6 of 11

7 Example If a turtle travels 5 inches every 10 seconds, how far will it travel in 50 seconds? What we are going to do is set up a proportion. How surprising? The way we ll do this is to identify the comparison we are making. In this case we are saying 5 inches every 10 seconds. Therefore, and this is very important, we are going to set up our proportion by saying, inches to seconds for our two ratios. On one side we have 5 describing inches to seconds. On the other side we have to again use the 10 same comparison, inches to seconds. We don t know the inches, so we ll call it n. Where will the 50 go in the ratio, top or bottom? Bottom, because it describes seconds good deal. 5 n So now we have: Now, we can find n by equivalent fractions or we could use Property 3 and find cross products. 5 n n n 250 n 25 The turtle will travel 25 inches in 50 seconds. It is very important to write the same comparisons on both sides of the equal signs. In other words, if we had a ratio on one side comparing inches to seconds, then we must write inches to seconds on the other side. If we compared the number of boys to girls on one side, we would have to write the same comparison on the other side, boys to girls. We could also write it as girls to boys on one side as long as we wrote girls to boys on the other side. The first Property of Proportion, tipping the ratios up-side down, permits this to happen. In the above examples, the fractions could have been simplified before using cross products. By simplifying first, that keeps the numbers smaller. You get the same answers. Pre-Algebra Unit 6: Ratio and Proportion Page 7 of 11

8 Applications Proportions Similar Polygons Similar Polygons have the same shape but not necessarily the same size. Corresponding parts of polygons are in the same relative position Similar Polygons two polygons are similar if: a) the measure of their corresponding angles are equal b) the ratio of the lengths of their corresponding sides are proportional Example Find the value of x, given these two rectangles are similar. 4 in. x 10 in. 5 in. Since the two rectangles are similar, their sides must be in proportion. That is, the left side is to the bottom in the large rectangle as the left side is to the bottom in the small rectangle. Another way of saying that is, the width is to the length in the large rectangle as the width is to the length in the small rectangle. 4 x or 10x x 20 x 2 Pre-Algebra Unit 6: Ratio and Proportion Page 8 of 11

9 Sometimes two polygons are contained in one figure. When that occurs, it might be helpful to redraw the two polygons separately so they don t overlap so you can see the corresponding parts more readily., find BE Example Given ABC DEC Taking DEC out of the ABC, we have the similar triangles shown without any overlapping. Writing the proportion, we have the left side is to the bottom of the large triangle as the left side is to the bottom of the small triangle. AB BC DE EC x ( x ) x x 12 x Pre-Algebra Unit 6: Ratio and Proportion Page 9 of 11

10 We could have reduced to 5, then crossed multiplied. That would allow us to work 12 with smaller numbers and we would still get the same answer. Pre-Algebra Unit 6: Ratio and Proportion Page 10 of 11

11 Note to CCSD teachers: Probability Probability does appear as the last 2 sections of Chapter 6 in your textbook. CCSD benchmark calendar begins the new quarter with these two objectives in combination with Chapter 11. You will find these notes included in the next unit of study (Unit 11) on the website. Pre-Algebra Unit 6: Ratio and Proportion Page 11 of 11