Section 3.6 ratios and proportions (Algebra)

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Section 3.6 ratios and proportions (Algebra)

  1. 1. Section 3.6 Ratios and Proportions
  2. 2. Outline: <ul><li>Ratios! </li></ul><ul><li>What is a Ratio? </li></ul><ul><li>How to Use Ratios? </li></ul><ul><li>How to Simplify? </li></ul><ul><li>Proportions! </li></ul><ul><li>What is a proportion? </li></ul><ul><li>Properties of proportions? </li></ul><ul><li>How to use proportions? </li></ul><ul><li>Mysterious Problems… </li></ul>
  3. 3. What is a Ratio? <ul><li>A ratio is a comparison of two numbers. </li></ul><ul><li>Ratios can be written in three different ways: </li></ul><ul><li>a to b </li></ul><ul><li>a:b </li></ul>Because a ratio is a fraction, b can not be zero Ratios are expressed in simplest form
  4. 4. How to Use Ratios? <ul><li>The ratio of boys and girls in the class is </li></ul><ul><li>6 to 11 . </li></ul>4cm 1cm <ul><li>This means, for every 6 boys you can find 11 girls to match. </li></ul><ul><li>There could be just 6 boys, 11 girls. </li></ul><ul><li>There could be 12 boys, 22 girls. </li></ul><ul><li>There could be 60 boys, and ____girls…a huge class </li></ul>What is the ratio if the rectangle is 8cm long and 2cm wide? Still 4 to 1, because for every 4cm, you can find 1cm to match <ul><li>The ratio of length and width of this rectangle </li></ul><ul><li>is 4 to 1 . </li></ul><ul><li>. </li></ul><ul><li>The ratio of cats and dogs at my home is 2 to 1 </li></ul>How many dogs and cats do I have? We don’t know, all we know is if they’d start a fight, each dog has to fight 2 cats.
  5. 5. How to simplify ratios? <ul><li>The ratios we saw on last slide were all simplified. How was it done? </li></ul>Ratios can be expressed in fraction form… This allows us to do math on them. The ratio of boys and girls in the class is The ratio of the rectangle is The ratio of cats and dogs in my house is
  6. 6. How to simplify ratios? <ul><li>Now I tell you I have 12 cats and 6 dogs . Can you simplify the ratio of cats and dogs to 2 to 1? </li></ul>= = Divide both numerator and denominator by their Greatest Common Factor 6 .
  7. 7. More examples… = = = = =
  8. 8. Now, on to proportions! What is a proportion? A proportion is an equation that equates two ratios The ratio of dogs and cats was 3/2 The ratio of dogs and cats now is 6/4 =3/2 So we have a proportion :
  9. 9. Properties of a proportion? 2x6=12 3x4 = 12 3x4 = 2x6 Cross Product Property
  10. 10. Properties of a proportion? <ul><li>Cross Product Property </li></ul>ad = bc means extremes
  11. 11. Properties of a proportion? Let’s make sense of the Cross Product Property… For any numbers a, b, c, d:
  12. 12. Properties of a proportion? If Then <ul><li>Reciprocal Property </li></ul>Can you see it? If yes, can you think of why it works?
  13. 13. How about an example? Solve for x: 7(6) = 2x 42 = 2x 21 = x Cross Product Property
  14. 14. How about another example? Solve for x: 7x = 2(12) 7x = 24 x = Cross Product Property Can you solve it using Reciprocal Property? If yes, would it be easier?
  15. 15. Can you solve this one? Solve for x: 7x = (x-1)3 7x = 3x – 3 4x = -3 x = Cross Product Property Again, Reciprocal Property?
  16. 16. Now you know enough about properties, let’s solve the Mysterious problems! If Mr. Yu’s car gets 30 miles/gallon , how many gallons of gas does Mr. Yu needs to commute to school everyday? 5 miles to school 5 miles to home Let x be the number gallons we need for a day: Can you solve it from here? x = Gal
  17. 17. So he uses up 1/3 gallon a day. How many gallons does he uses for a week? 5 miles to school 5 miles to home Let t be the number of gallons we need for a week: Gal What property is this?
  18. 18. So Mr. Yu uses up 5/3 gallons a week (which is about 1.67 gallons ). Consider if the price of gas is 3.69 dollars/gal , how much would it cost for a week? Let s be the sum of cost for a week: 5 miles to school 5 miles to home 3.69(1.67) = 1s s = 6.16 dollars
  19. 19. So what do you think? 10 miles Mr. Yu pays about 6 bucks a week just to get to school! What about weekends? If he travels twice as much on weekends, say drives 10 miles to the Mall and 10 miles back, how many gallons does he need now? How much would it cost totally? How much would it cost for a month? 5 miles Think proportionally! . . . It’s all about proportions!
  20. 20. ON YOUR HOMEWORK! PAGE: 158 # 20 – 44 EVEN

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