RATES & RATIOS WITH COMPLEX FRACTIONS. Complex Fractions. Fraction in the denominator
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1 RATES & RATIOS WITH COMPLEX FRACTIONS LESSON -F 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 0 _ 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. Simplify Method - Division. Rewrite the fraction using division: Simplify Method - Least Common Denominator. 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 -F ~ Rates & Ratios With Complex Fractions
2 EXAMPLE Simplify each complex fraction. a. 0 b. Solutions Method - Division Method - Least Common Denominator a. Rewrite using division: 0 a. Find LCD of 0 and _ : LCD = 0 Simplify: 0 Multiply the numerator and denominator by the LCD. Simplify. 0 0 = = Answer: 0 Answer: 0 b. Rewrite using division: Simplify: b. Find the LCD of _ and _ : LCD = Multiply the numerator 8 and denominator by the = = LCD. Simplify. Answer: Answer: 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. _ _ Lesson -F ~ Rates & Ratios With Complex Fractions Simplify. _ _ = _ hour This can be written as which means it takes Ryan hour to fill aquariums aquariums hour hour at this rate. But, as a unit rate, this is = aquarium aquarium or _ hour per aquarium. The simplified complex fraction of _ can be written as the unit rate. Ryan fills the aquariums at a rate of _ hour per aquarium.
3 EXAMPLE Solution Find the scale factor of the similar squares. Write the ratio of the sides of the squares as a complex fraction. Simplify the complex fraction. The scale factor is or :. _ yard _ yards = = 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. 00 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? Lesson -F ~ Rates & Ratios With Complex Fractions
4 EXERCISES Simplify each complex fraction.. 0. _. 8_. 0. Trevon insists 6 reasoning. _ 8 Find the unit rate. _ inches. is equivalent to 6 _ foot Pedro disagrees. Who is correct? Explain your 8.. minute 0. seconds _ pages.. minutes Solve each problem. Show all work. 0 cookies _ hour.. 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?. During a snowstorm, _ feet of snow fell in 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. How fast did she walk in miles per hour? _ 8 miles _ hour innings _ games. 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? 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 _ hours total, how far will they travel?. Lucy spent _ hour shooting baskets. She made baskets. At that rate, how many hours will it take Lucy to make 0 baskets? 6 Lesson -F ~ Rates & Ratios With Complex Fractions
5 0. A car traveled 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 miles = mile per minute 0 min miles hour = mile per hour min 60 min 80 Alejandro miles hour = = = miles per hour. Find the scale factor of the similar rectangles. _ 8 inch _ inches. Find the scale factor of the similar triangles.. Find the ratio of the areas of the squares. _ 6 foot _ feet _ cm _ cm Lesson -F ~ Rates & Ratios With Complex Fractions
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