1 / 22

Mastering Ratios and Proportions: A Comprehensive Guide

Learn how to solve ratios, scale proportions, cross multiply, and solve equations involving proportions in this detailed math lesson.

rudolphm
Download Presentation

Mastering Ratios and Proportions: A Comprehensive Guide

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Math CC7/8 – Oct. 29 Things Needed Today (TNT) • Pencil/Math Notebook Math Notebook • Topic: Scaling to Solve Proportions • C.S. HW: pg. 25, #19, 20, #24-32 Please have ALL TNT items on your desk!

  2. What’s Happening? • New Unit- C&S 1.4! • Review SS Unit Test • Begin HW?

  3. 3 ways to show RATIOS Using the ":" to separate the values: 3 : 1 Instead of the ":" we can use the word "to": 3 to 1 Or write it like a fraction: 3 1 ratio can be scaled up:

  4. Using Ratios Example There are 5 dogs, 2 are boys, and 3 are girls • The trick with ratios is to ALWAYS multiply or divide the #s by the SAME VALUE to scale up Example 4:5 is the same as 4*2: 5*2 = 8:10 Part to PartPart to Whole Part to Part The ratio of boys to girls is 2:3 or 2/3 The ratio of girls to boys is 3:2 or 3/2 Part to Whole 4 : 5 8 : 10 The ratio of boys to ALL pups is 2:5 or 2/5 The ratio of girls to ALL pups is 3:5 or 3/5 X 2 X 2

  5. What are proportions? • An equation in which two ratios are equal is called a proportion • A proportion can be written using colon notation like this a : b :: c : d • or as the more recognizable (and useable) equivalence of two fractions. =

  6. Proportions Means Extremes In a proportion the product of the means is equal to product of the extremes. 3 : 5 = 6 : 10

  7. Proportions Means Extremes 6 • 5 = 3 • 10 30 = 30

  8. Proportions Determine if the following are proportions. 1) 2)

  9. Proportions 3 • 60 = 5 • 36 4 • 15 = 8 • 8 180 = 180 60 64 Yes, it is a proportion. No, it is not a proportion.

  10. Cross Multiplication • Solving the proportion—Process of using cross products to find a missing term in a proportion. • Solving a proportion is similar to solving an equation.

  11. Example x = 4 9 6 x • 6 = 9 • 4 (Cross multiply) 6x = 36 (Solve for the variable) 6x = 36 (Divide to undo multiplication 6 6 x = 6

  12. Question What strategies can you use to find a missing value in a proportion? What is your preferred strategy and why? ****How do you form a proportion?**** You form a proportion, when you write 2 equivalent ratios in fraction form & set them equal to each other.

  13. Ratios In Stretching & Shrinking, we worked with ratios to find missing lengths in similar figures. There are many other situations in which setting up a proportion can help you solve a problem. For example, suppose that among American doctors men outnumber women by a ratio of 12:5(in other words, for every 12 men there are 5 women) If about 600,000 American doctors are men, how can you figure out how many are women?

  14. There are 4 waysto write this as a proportion. What do you notice about how the proportions are written? Hmmmmm…

  15. Using what you know about equivalent ratios, you can find the # of women doctors from any one of these proportions. Finding the missing value in a proportion is called solving the proportion. • Does one of the proportions seem easier to solve than the others? • How many women doctors are there? 250,000 women doctors

  16. For each of the following questions, set up a proportion that shows the relationship between known and unknown quantities. Then use equivalent fractions, ratios, & scaling to solve each proportion. Imani gives vitamins to her dogs. The recommended dosage is 1 teaspoon per day for adult dogs weighing 10 pounds. She needs to give vitamins to Bruiser, who weighs 80 pounds and to Dust Ball, who weighs 7 pounds. What is the correct dosage for each dog? 80 lbs. x tsp. Dust Ball Bruiser lbs. x tsp. lbs. 1 tsp. lbs. 1 tsp. = = Dust Ball needs0.7teaspoons Bruiser needs8teaspoons

  17. 1) Jogging 5 miles burns about 500 calories. How many miles does Tanisha need to jog to burn off the 1,200-calorie lunch she ate? = Tanisha needs to jog 12 miles. What method did you use to solve this proportion? 2) Tanisha jogs about 8 miles in 2 hours. How long will it take her to jog 12 miles? x miles 1200 cal. 5 miles 500 cal. 8 miles 2 hours 12 miles x hours What method will you use to solve this problem? x = 3 hours It will take Tanisha 3 hours to jog 12 miles. =

  18. The triangles in the picture below are mathematically similar. Find the height of the tree Use proportions & ratios to solve! Or scale factor Cross Multiplication Method 8h = 240 Divide both sides by 8 h = 30ft = Ratios with height over base of the triangles Nested Triangles Again Hint: Try cross (X) multiplication for this problem! 30 ft. Scale factor = 6 From sm to lg Scale factor = From lg to sm

  19. Solve these proportions for the variable x. Use the ratios, cross multiplication or scale factor to solve. Complete in your math notebook! Show ALL WORK 3 = 1 = 2 x = 35 x = 20 x = 5.25 = = 4 5 6 9x = 24 9 x = 8 3 2 5 Or .4 3x = 600 Divide both side by 3 x = 200

  20. Nic was working on the proportion below. = He could not see a way to scale 10 to make 6. Instead, he scaled both sides of the proportion. His work is shown below. How could Nic complete his solution? 3(6) = 10(x) 18= 10x Divide each side by 10 1.8 = x

  21. Kevin think Nic’s idea is great, but he used 30 as a common denominator. Show what Kevin’s version of the proportion would look like. Does Kevin’s scaled-up proportion give the same answer as Nic’s? EXPLAIN your reasoning to your partner. Kevin’s proportions would be 9 = 5x 30 30 Then solve cross multiply. 150x=270 Divide both side by 150 X= 1.8 Kevin’s idea does work Does Kevin’s work help you solve =? EXPLAIN 12x = 63 12 12 x = 5.25 Yes, because you can find a common denominator for 12 & 9 which would be 108. Or you could use cross multiplication to solve the problem.

  22. Homework #19, 20, #24-32

More Related