Section 5.1 Ratios Rates Proportions and Measurements

Section 5.1 Ratios, Rates, Proportions and Measurements Ratio – a fraction, or quotient of two quantities with the same units. A ratio of length to width would be calculated by finding length/width (also can be calculated by doing length ÷ width) A ratio of 16 oz to 2 lbs would first have to be converted to the same units. To quantities with units that can be converted to the sae unit are called commensurable. 16 oz = 1 lb. 16 oz to 2 lbs = 1 lb to 2 lbs = 1 lb / 2 lbs = ½ Rate – a ratio of two quantities with different types of units. 60 miles/ 2 hours , $500/2 weeks, etc… A unit rate is a rate with 1 as the denominator. Example 1 Find the unit rate of miles per hour. 60miles/2 hours can be converted to a unit rate by doing the division 60/2 = 30. 60 miles /2 hours = 30 miles/1 hour = 30mph Means Extremes Property a c  b d a and d are the Extremes. b and c are the Means. The Means Extremes Property says that the product of the Extremes is equal to the product of the Means. ad = bc PROPERTY 2 fu a proportion. the means or the extremes (or both) may be interchanged· that is if a bd c d b and-=-. (where a. b, c, and dare nonzero), then -= . - = -, d· ·b c a c a f=� When given the proportion � = 182, Property 2 enables u. to draw conclu ions • uch as 2 8 12 2. 3 3, 3. 2 1. --- 12 2 12 8 (means interchanged) (extrem.es interchanged) (both sides inverted) PROPERTY 3 a c lff ,; = d (whereb -:t:- . . a+b c +d . a-b c -d 0 and d # 0), then - =- and - = -. b d b d A "Concrete" Example Ratios can have more than two numbers! For example concrete is made by mixing cement, sand, stones and water. A typical mix of cement, sand and stones is written as a ratio, such as 1:2:6. You can multiply all values by the same amount and you will still have the same ratio. 10:20:60 is the same as 1:2:6 So if you used 10 buckets of cement, you should use 20 of sand and 60 of stones. Example: if you have just put 12 buckets of stones into a wheelbarrow, how much cement and how much sand should you add to make a 1:2:6 mix? Let us lay it out in a table to make it clearer: Ratio Needed: Cement Sand Stones 1 2 6 You have: 12 You can see that you have 12 buckets of stones but the ratio says 6. That is OK, you simply have twice as many stones as the number in the ratio ... so you need twice as much of everything to keep the ratio. The constant of variation is 2, because you will multiply each number in the ratio 1:2:6 by 2 to get the new numbers when you have 12 buckets of stones. Here is the solution: Cement Sand Stones Ratio Needed: 1 2 6 You have: 2 4 12 And the ratio 2:4:12 is the same as 1:2:6 (because they show the same relative sizes) Why are they the same ratio? In the 1:2:6 ratio there is 3 times more Stones as Sand (6 vs 2), and in the 2:4:12 ratio there is also 3 times more Stones as Sand (12 vs 4) ... similarly there is twice as much Sand as Cement in both ratios. That is the good thing about ratios. You can make the amounts bigger or smaller and so long as the relative sizes are the same then the ratio is the same. So the answer is: add 2 buckets of Cement and 4 buckets of Sand. (You will also need water and a lot of stirring....) Section 5.2 Similar Polygons Two polygons are Similar if the only difference is size (and possibly the need to turn or flip one around). These triangles are all similar: (Equal angles have been marked with the same congruency marks) Some of them have different sizes and some of them have been turned or flipped. Similar Polygons have: all their angles equal corresponding sides have the same ratio Corresponding Sides In similar triangles, the sides facing the equal angles are always in the same ratio. For example: Triangles R and S are similar. The equal angles are marked with the same numbers of arcs. What are the corresponding lengths? The lengths 7 and a are corresponding (they face the angle marked with one arc) The lengths 8 and 6.4 are corresponding (they face the angle marked with two arcs) The lengths 6 and b are corresponding (they face the angle marked with three arcs) Calculating the Lengths of Corresponding Sides It may be possible to calculate lengths we don't know yet. We need to: Step 1: Find the ratio of corresponding sides in pairs of similar triangles. Step 2: Use that ratio to find the unknown lengths. Example: Find lengths a and b of Triangle S above. Step 1: Find the ratio We know all the sides in Triangle R, and We know the side 6.4 in Triangle S The 6.4 faces the angle marked with two arcs as does the side of length 8 in triangle R. So we can match 6.4 with 8, and so the ratio of sides in triangle S to triangle R is: 6.4 to 8 Now we know that the lengths of sides in triangle S are all 6.4/8 times the lengths of sides in triangle R. Step 2: Use the ratio a 6.4  a faces the angle with one arc as does the side of length 7 in triangle R. 7 8 b faces the angle with three arcs as does the side of length 6 in triangle R. b 6.4  6 8 8b  6  6.4 a  6  6.4  / 8  4.8 8a  7  6.4 a  7  6.4  / 8  5.6 The triangle has been flipped and rotated as well as resized, but by comparing the angles, we can identify equivalent sides: AC and DF face the smallest angle, so are equivalent. AB and EF face the middle-sized angle, so are equivalent. BC and DE face the largest angle, so are equivalent. We can work out the length of DE by comparing sides we know. The two triangles are similar. What is the length of DE? So DE/BC = DF/AC ⇒ DE/10.8 = 5.5/4.4 ⇒ DE = 10.8 × (5.5/4.4) = 10.8 × 1.25 = 13.5