Algebra unit 2.6

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Transcript of Algebra unit 2.6

  1. 1. UUNNIITT 22..66 RRAATTIIOOSS
  2. 2. Warm UpSolve each equation. Check your answer.1. 6x = 3662.483. 5m = 183.64.635. 8y =18.42.3Multiply.6. 7 7.10
  3. 3. ObjectivesWrite and use ratios, rates, and unit rates.Write and solve proportions.
  4. 4. Vocabularyratio proportionrate cross productsscale scale drawingunit rate scale modelconversion factor
  5. 5. A ratio is a comparison of two quantities bydivision. The ratio of a to b can be written a:bor , where b 0. Ratios that name the samecomparison are said to be equivalent.A statement that two ratios are equivalent, suchas , is called a proportion.
  6. 6. Reading MathRead the proportion as1 is to 15 as x is to 675.
  7. 7. Example 1: Using RatiosThe ratio of the number of bones in a humansears to the number of bones in the skull is 3:11.There are 22 bones in the skull. How manybones are in the ears?Write a ratio comparing bones in earsto bones in skull.Write a proportion. Let x be thenumber of bones in ears.Since x is divided by 22, multiplyboth sides of the equation by 22.There are 6 bones in the ears.
  8. 8. Check It Out! Example 1The ratio of games lost to games won for abaseball team is 2:3. The team has won 18games. How many games did the team lose?Write a ratio comparing games lost togames won.Write a proportion. Let x be thenumber of games lost.Since x is divided by 18, multiplyboth sides of the equation by 18.The team lost 12 games.
  9. 9. A rate is a ratio of two quantities with differentunits, such as Rates are usually written asunit rates. A unit rate is a rate with a secondquantity of 1 unit, such as or 17 mi/gal. Youcan convert any rate to a unit rate.
  10. 10. Example 2: Finding Unit RatesRaulf Laue of Germany flipped a pancake 416times in 120 seconds to set the world record.Find the unit rate. Round your answer to thenearest hundredth.Write a proportion to find an equivalentratio with a second quantity of 1.Divide on the left side to find x.The unit rate is about 3.47 flips/s.
  11. 11. Check It Out! Example 2Cory earns $52.50 in 7 hours. Find the unitrate.Write a proportion to find an equivalentratio with a second quantity of 1.Divide on the left side to find x.The unit rate is $7.50.
  12. 12. A rate such as in which the two quantitiesare equal but use different units, is called aconversion factor. To convert a rate from oneset of units to another, multiply by a conversionfactor.
  13. 13. Example 3A: Converting RatesSerena ran a race at a rate of 10 kilometersper hour. What was her speed in kilometersper minute? Round your answer to thenearest hundredth.To convert the second quantity in arate, multiply by a conversionfactor with that unit in the firstquantity.The rate is about 0.17 kilometer per minute.
  14. 14. Helpful HintIn example 3A , 1 hr appears to divide out,leaving kilometers per minute, which arethe units asked for. Use this strategy ofdividing out units when converting rates.
  15. 15. Example 3B: Converting RatesA cheetah can run at a rate of 60 miles perhour in short bursts. What is this speed infeet per minute?Step 1 2 Convert the speed to feet per hour.minute.To convert the first quantity in arate, multiply by a conversionfactor with that unit in the secondquantity.The speed is 5280 316,800 feet feet per per minute.hour.first
  16. 16. Example 3B: Converting RatesThe speed is 5280 feet per minute.Check that the answer is reasonable. There are 60 min in 1 h, so 5280 ft/min is60(5280) = 316,800 ft/h. There are 5280 ft in 1 mi, so 316,800 ft/his This is the given ratein the problem.
  17. 17. Check It Out! Example 3A cyclist travels 56 miles in 4 hours. What is thecyclists speed in feet per second? Round youranswer to the nearest tenth, and show that youranswer is reasonable.Step 1 Convert the speed to feet per hour.Change to miles in 1 hour.To convert the first quantity in arate, multiply by a conversionfactor with that unit in thesecond quantity.The speed is 73,920 feet per hour.
  18. 18. Check It Out! Example 3Step 2 Convert the speed to feet per minute.To convert the second quantity in arate, multiply by a conversionfactor with that unit in the firstquantity.The speed is 1232 feet per minute.Step 3 Convert the speed to feet per second.To convert the second quantity in arate, multiply by a conversionfactor with that unit in the firstquantity.The speed is approximately 20.5 feet per second.
  19. 19. Check It Out! Example 3Check that the answer is reasonable. The answeris about 20 feet per second. There are 60 seconds in a minute so 60(20)= 1200 feet in a minute. There are 60 minutes in an hour so 60(1200)= 72,000 feet in an hour. Since there are 5,280 feet in a mile 72,000 5,280 = about 14 miles in an hour. The cyclist rode for 4 hours so 4(14) = about56 miles which is the original distance traveled.
  20. 20. In the proportion , the products a d andb c are called cross products. You can solvea proportion for a missing value by using theCross Products property.Cross Products PropertyWORDS NUMBERS ALGEBRAIn a proportion, crossproducts are equal.2 6 = 3 4If and b 0and d 0then ad = bc.
  21. 21. Example 4: Solving ProportionsSolve each proportion.3(m) = 5(9)3m = 45m = 15Use crossproducts.Divide bothsides by 3.Use crossproducts.6(7) = 2(y 3)42 = 2y 6+6 +648 = 2y24 = yA. B.Add 6 tobothDisviiddees b. othsides by 2.
  22. 22. Check It Out! Example 4Solve each proportion.2(y) = 5(8) 4(g +3) = 5(7)y = 2012 124g = 23g = 5.75A. B.Use crossproducts.Divide bothsides by 2.Use crossproducts.Subtract 12from bothsides.Divide bothsides by 4.2y = 404g +12 = 35
  23. 23. A scale is a ratio between two sets of measurements,such as 1 in:5 mi. A scale drawing or scale modeluses a scale to represent an object as smaller orlarger than the actual object. A map is an example ofa scale drawing.
  24. 24. Example 5A: Scale Drawings and Scale ModelsA contractor has a blueprint for a housedrawn to the scale 1 in: 3 ft.A wall on the blueprint is 6.5 inches long. Howlong is the actual wall?blueprint 1 in.Write the scale as a fraction.actual 3 ft.Let x be the actual length.Use the cross products to solve.x 1 = 3(6.5)x = 19.5The actual length of the wall is 19.5 feet.
  25. 25. Example 5B: Scale Drawings and Scale ModelsA contractor has a blueprint for a housedrawn to the scale 1 in: 3 ft.One wall of the house will be 12 feet long whenit is built. How long is the wall on the blueprint?blueprint 1 in.actual 3 ft.Write the scale as a fraction.Let x be the actual length.Use the 12 = 3x cross products to solve.4 = xSince x is multiplied by 3, divideboth sides by 3 to undo themultiplication.The wall on the blueprint is 4 inches long.
  26. 26. Check It Out! Example 5A scale model of a human heart is 16 ft. long.The scale is 32:1. How many inches long isthe actual heart it represents?model 32 in.actual 1 in.Write the scale as a fraction.Convert 16 ft to inches.Let x be the actual length.Use the cross 32x = 192 products to solve.Since x is multiplied by 32, divideboth sides by 32 to undo themultiplication.x = 6The actual heart is 6 inches long.
  27. 27. Lesson Quiz: Part 11. In a school, the ratio of boys to girls is 4:3.There are 216 boys. How many girls are there?162Find each unit rate. Round to the nearesthundredth if necessary.2. Nuts cost $10.75 for 3 pounds. $3.58/lb3. Sue washes 25 cars in 5 hours. 5 cars/h4. A car travels 180 miles in 4 hours. What is thecars speed in feet per minute? 3960 ft/min
  28. 28. Lesson Quiz: Part 2Solve each proportion.5.6.6167. A scale model of a car is 9 inches long. Thescale is 1:18. How many inches long is the carit represents?162 in.
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