RATES AND RATIOS WITH COMPLEX FRACTIONS. Complex Fractions. Fraction in the denominator

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1 RATES AND RATIOS WITH COMPLEX FRACTIONS LESSON.6 A complex fraction is a fraction that contains a fractional expression in its numerator, denominator or both. The following are examples of complex fractions. Fraction in the numerator _ 6 Complex Fractions Fraction in the denominator Fraction in the numerator AND fraction in the denominator 8 Sometimes a rate or ratio is a complex fraction when it is first written. For example, if Jean walked _ miles in hour, her rate would be: _ miles hour What does this rate mean? Although accurate, this rate is hard to understand when it is written as a complex fraction. The complex fraction needs to be simplified so the rate makes more sense. There are two ways to simplify a complex fraction. Both methods show Jean walked at a rate of 6 miles per hour. Method - Division Method - Least Common Denominator Simplify _. Simplify. Rewrite the fraction using division: _ _.. Find the least common denominator (LCD) for each fraction in the numerator and denominator: LCD =. Simplify: _ = _ _ = = 6. Multiply the numerator and denominator of the complex fraction by the LCD and simplify: _ _ = _ _ _ = 6_ = 6 This means _ is equal to 6. This means _ is equal to 6. Each method shows Jean walked at a rate of 6 miles per hour. Lesson.6 ~ Rates and Ratios with Complex Fractions 7

2 EXAMPLE Simplify each complex fraction. a. b. _ 7 Solutions Method - Division Method - Least Common Denominator a. Rewrite using division: a. Find LCD of and _. LCD = Simplify: Multiply the numerator and denominator by the LCD. Simplify. = 0 = Answer: Answer: b. Rewrite using division: _ 7 b. Find the LCD of and 7. LCD = 7 Simplify: _ 7 7_ Multiply the numerator and denominator by the LCD. Simplify. _ 7 7_ 7 = 8 = 7 Answer: 7 Answer: 7 Anytime a rate or ratio problem involves a complex fraction, simplify the complex fraction to best answer the question. EXAMPLE Solution Ryan has many aquariums. He spent _ hour filling _ of one of his aquariums. Find the unit rate of hours per aquarium to find how long it takes Ryan to fill each one. Write the rate. hour _ aquarium Rewrite the complex _ fraction using division. Simplify. _ = _ hour This can be written as which means it takes Ryan hour to fill aquariums aquarium hour at this rate. But, as a unit rate, this is aquarium = hour or _ hour per aquarium aquarium. The simplified complex fraction of _ can be written as the unit rate. Ryan fills the aquariums at a rate of _ hour per aquarium. 8 Lesson.6 ~ Rates and Ratios with Complex Fractions

3 EXPLORE! A CHANGE OF PACE Kevin walked,00 feet in 0 minutes. Follow the directions below to find Kevin s rate in miles per hour three different ways. Step : a. Fill in the conversion needed to change Kevin s speed to miles per hour. 0 feet mile min miles = 0 min feet hours hours b. Calculate Kevin s speed in miles per hour. Step : a. Convert,00 feet to miles. Write your answer as a decimal.,00 feet = miles b. Convert 0 minutes to hours. Write your answer as a decimal. 0 minutes = hour c. Find Kevin s speed in miles per hour. Step : a. Convert,00 feet to miles. Write your answer as a fraction.,00 feet = miles b. Convert 0 minutes to hours. Write your answer as a fraction. 0 minutes = hour c. Find Kevin s speed in miles per hour. Step : In Step you converted feet per minute to miles per hour in one conversion equation. In Steps and, you converted feet to miles and minutes to hours first and then found Kevin s speed. In Step you used decimals and in Step you used fractions. Which of the three methods did you like best to find Kevin s speed? Why? EXERCISES Simplify each complex fraction.... 8_ 5. _ _ 5_ 6 6. _ 8_ Lesson.6 ~ Rates and Ratios with Complex Fractions

4 7. Trevon insists the two ratios below are equivalent. Pedro disagrees. Who is correct? Explain your reasoning. 6 6 _ 8 Find the unit rate. _ inches 5_ 7 feet 8.. minute 5. seconds miles hour pages.. minutes 7 cookies _ 5 hour. Solve each problem. Show all work necessary to justify your answer.. Luke wrote entries in his journal. It took him _ hours to write them all. Assume each entry took the same amount of time. How many entries did he write per hour? 5. During a snowstorm, _ inches of snow fell in 5 hours. Assume the snow fell at the same rate throughout the storm. How much snow fell per hour? 6. Sasha walked 6 _ miles at a constant rate in _ hours. Trilia walked at a rate of miles per hour. Which person walked at a faster rate? 7. Victor read _ books over days last summer. Assume it took him the same amount of time to read each book. How many books did he read each day? innings games 8. Rodrigo and his family drove to Disneyland for their vacation. In the first _ hour of the trip, they drove 0 miles. If they drive at the same rate for 5 _ hours total, how far will they travel?. Lucy spent _ hour shooting baskets. She made 5 baskets. At that rate, how many hours will it take Lucy to make 0 baskets? 0. A car traveled 5 miles in 0 minutes. Corin and Alejandro found the speed of the car in miles per hour. One of them made a mistake. Identify who made the mistake and fix his solution. Corin 5 miles = mile per minute 0 min miles hour = mile per hour min 60 min 80 Alejandro 5 miles hour = 5 = 5 = 5 miles per hour 0 Lesson.6 ~ Rates and Ratios with Complex Fractions

5 . Lynette wanted to run one mile in eight minutes. To reach her goal, how far will she need to run in _ minutes? Show all work necessary to justify your answer.. Margo rode her bike 5 miles in hour and 0 minutes. How fast did she ride in miles per hour? Use mathematics to justify your answer. REVIEW Complete each rate conversion.. meters per minute = centimeters per second. tons per hour = pounds per minute 5. 5 gallons per week = cups per hour Write the ratio for each situation in simplest form. 6. The Bobcats played 0 baseball games. They won 5 of the games. a. Write the ratio of games won to the number of games played. b. Write the ratio of games lost to the number of games played. c. Write the ratio of games won to games lost. 7. Several dogs at a breeder had puppies. There were 8 black puppies, 5 yellow puppies and brown puppies. a. Write the ratio of black puppies to the total number of puppies. b. Write the ratio of brown puppies to the total number of puppies. c. Write the ratio of black puppies to yellow puppies. d. The next year, the same ratios of puppies were born, but there were only 5 yellow puppies. How many black and brown puppies were born that year? Tic-Tac-Toe ~ Un it R ate s Find 5 different unit rates in life around you. Some examples are dollars per gallon for gas, price per pound for an item at the store, miles per hour for a car or dollars per song download.. Record each of the five different rates.. For each of the five different rates, write a word problem another student could solve.. Solve each word problem. Show all work necessary to justify your answer. Lesson.6 ~ Rates and Ratios with Complex Fractions