1. Proportions 6.2 Solve Proportions Use Proportions to Solve Problems

2. What are we doing? We will define proportions and use them to solve problems! Vocabulary : A proportion is two equivalent ratios

3. WARNING We will write equations to solve problems. In many of these the solution will be obvious. We are learning and practicing the process of writing equations to solve for a missing value in a proportion.

4. Cross products in a proportion are equal. a = c_ b d The product of (a)(d) = (b)(c) Proving this with arithmetic…….. _6_ = _8_ 6(12) = 8(9) 9 12 We will write an equation based on this knowledge about proportion cross-products. 7 By now, you should realize that anything that appears this obvious has additional applications!!!!!! 7 2

5. Testing for a proportion……. We know that a proportion is two equal ratios. If the cross products are equal, it IS a proportion. If they are not, it’s NOT. _3_ = _ 21_ 5 35 ? 3(35) = 5(21) 105 = 105 These ratios DO form a proportion. The cross products are equal. _3.4_ = _ 5.8_ 5.5 7.9 (3.4) 7.9 = 5.5 (5.8) 26.86 ≠ 31.9 These ratios do NOT form a proportion. Their cross products are NOT equal. ? If you don’t feel like multiplying these, you can write the two multiplication problems as a fraction. If both terms cancel to one, it is a proportion! 7 7

6. Using proportions to solve problems……….. Sometimes the solution is NOT obvious. One hundred rods is about 275 fathoms. About how many fathoms is 25 rods? Labeling your information as you set up a proportion is VERY IMPORTANT!! What do we know? 100 rods = 275 fathoms length in rods = length in fathoms 25 rods_____ = d fathoms We are given “25 rods.” This info MUST be aligned with the same unit. 68.75 fathoms = d 100

7. Consumer Application…….. If 3 oz. of yogurt cost $1.65, how much would 5 oz.cost? 3 oz. = 5 oz $1.65 ? $2.75 = 5 oz 3.55

8. Consumer Application…….. Could we use the unit rate to find the same value? $1.65 = _?__ 3 oz 1 oz 3.55

9. Check Understanding Solve each proportion. Show the solving steps for the method you choose. 6 = h 44 = t 2 7 77 = d I promise this process is worth learning!!!!!! I promise this process is worth learning!!!!!! 3 5 5 6 6 3 11

10. Test to see if the two ratios form a proportion………. 6 4 9, 6 (9)4 = 6(6) 36= 36 yes 15 5 20, 7 15(7) = 5(20) 105≠ 100 no 7 17.5 12, 30 30(7) = 12(17.5) 210 = 210 yes

11. Problem Solving: Write an equation to solve. About how many rods is 100 fathoms? 100 fathoms ~ 36.36 … rods 20 lbs. of Goop costs $27.50. How much would 12 pounds cost? A fathom is a unit of length used for measuring the depth of water. A rod was a unit used to measure land... A long time ago $16.50 = cost of 12 lbs. 275 20

12. Time check!!!!

13. Leilani is flying from Los Angeles to Honolulu. If the flight takes 6 hours and the distance is 2,574 miles, what is the average speed of the airplane? Reread for facts we can use……. What do we know???????? We can travel how far in how much time? When “average speed” is asked for, we want to find the UNIT RATE of miles in one hour. Remember, this ratio is a rate. We see that our average rate of speed is 429 miles per hour. The unit rate allows us to see how far we travel in one hour. Finding Rate of Speed

14. Working with Rates of Speed Joe and his family started their vacation last week. They drove 28 miles for one-half of an hour before they stopped for gas. How fast were they driving? We are again looking for the rate of speed. Could we have solved this as a proportion??????? YES!

15. Comparing Rates of Speed We are going to use our ratio/proportion processes again. Mary did well in the bike race. The 31 mile race took her 3 hours to finish. Joe finished a different race. His 25 mile race took 2 hours to finish. Find the rate of speed for both Joe and Mary. Who was faster? WHAT DO WE KNOW???? Mary…….. Joe…………….

16. Using Rates to Predict Distance Let’s take Joe’s rate of speed and predict how far he would travel in 5 hours…. We know what his rate of speed is as a unit rate…….. If we are looking for a new distance with a new time, we line up the values so we can solve this proportion…….

17. Let’s try another problem If we are looking for a new distance with a new time, we line up the values so we can solve this proportion……. Joe’s family is still on vacation. It took them 3 hours to travel 186 miles. If they maintain this rate of speed, how far can they travel in 7 hours? 62 ______ _____ 3 3 62 = n

18. Challenge…Rely on the Process! Find the values across from each other. Now divide by the value that is diagonal from the variable.

19. Did we meet our objectives? Did we find out what a proportion is? Did we use proportions to solve problems?