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**CCSSM Stage 2 Companion Text**

Lesson 2-F CCSSM Stage 2 Companion Text Rates & Ratios with Complex Fractions

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Warm-Up 1. Donna walked 18 miles in 4 hours at the same rate. What was her speed in miles per hour? 2. Simplify. a. b. 3. Convert the rate to miles per minute. 4.5 miles per hour 0.2 mile per minute

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**Rates & Ratios with Complex Fractions**

Lesson 2-F Rates & Ratios with Complex Fractions Target: Compute rates and ratios that include complex fractions.

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**Vocabulary Complex Fraction**

A fraction that contains a fractional expression in its numerator, denominator or both.

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**Write a mixed number as an improper fraction.**

Good to Know! Sometimes a rate or ratio is a complex fraction. Example If Jean walked miles in hour, her rate would be: There are two ways to simplify a complex fraction: Method 1 ~ Division Method 2 ~ Least Common Denominator Write a mixed number as an improper fraction. This rate is hard to understand when it is written as a complex fraction. It needs to be simplified so the rate makes more sense.

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**Simplifying a Complex Fraction Method 1 ~ Division**

1. Rewrite the fraction using division. 2. Simplify. Remember that a fraction is another way to write the operation division. This means Jean walked at a rate of 6 miles per hour.

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**Simplifying a Complex Fraction Method 2 ~ Least Common Denominator**

1. Find the LCD for each fraction in the numerator and denominator. LCD = 4 2. Multiply the numerator and denominator of the complex fraction by the LCD and simplify. This means Jean walked at a rate of 6 miles per hour.

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**Method 2 ~ Least Common Denominator**

Example 1a Simplify each complex fraction. a. Method 1 ~ Division Method 2 ~ Least Common Denominator Rewrite using division: Simplify: Answer: Find the LCD of and : Multiply the numerator and denominator by the LCD. Simplify. LCD = 10 3 1

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**Method 2 ~ Least Common Denominator**

Example 1b Simplify each complex fraction. b. Method 1 ~ Division Method 2 ~ Least Common Denominator Rewrite using division: Simplify: Answer: Find the LCD of and : Multiply the numerator and denominator by the LCD. Simplify. LCD = 7 1 1

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Example 2 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. Rewrite the complex fraction using division. Simplify. 1

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Example 2 (Cont.) 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. Ryan fills the aquariums at a rate of hour per aquarium.

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Example 3 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 4 : 15. 1 3

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**Explore! A Change of Pace**

Kevin walked 13,200 feet in 30 minutes. Follow the directions below to find Kevin’s rate in miles per hour three different ways. Step 1 a. Fill in the conversion needed to change Kevin’s speed to miles per hour. b. Calculate Kevin’s speed in miles per hour. Step 2 a. Convert 13,200 feet to miles. Write your answer as a decimal. b. Convert 30 minutes to hours. Write your answer as a decimal. c. Find Kevin’s speed in miles per hour.

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**Explore! A Change of Pace (Cont.)**

Kevin walked 13,200 feet in 30 minutes. Follow the directions below to find Kevin’s rate in miles per hour three different ways. Step 3 a. Convert 13,200 feet to miles. Write your answer as a fraction. b. Convert 30 minutes to hours. Write your answer as a fraction. c. Find Kevin’s speed in miles per hour. Step 4 In Step 1 you converted feet per minute to miles per hour in one conversion equation. In Steps 2 and 3, you converted feet to miles and minutes to hours first and then found Kevin’s speed. In Step 2 you used decimals and in Step 3 you used fractions. Which of the three methods did you like best to find Kevin’s speed? Why?

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**Exit Problems 1. Simplify . 2. Find the unit rate of .**

3. Rita swam laps in minutes. How many laps per minute did Rita swim?

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Communication Prompt Explain how finding a unit rate that involves whole numbers is similar to finding a unit rate that involves fractions. For example, how is it similar to find the rate in feet per minute if one bug travels at a rate of 20 feet in 2 minutes and a second bug travels at a rate of feet in minute?

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Example 3 Dividing Mixed Numbers ÷ 3 1 6 2 6 5 – 3 19 = 17 6 – Multiply by the reciprocal of 17 6 – 6 – = 3 () 6 – 19 Use rule for multiplying fractions.

Example 3 Dividing Mixed Numbers ÷ 3 1 6 2 6 5 – 3 19 = 17 6 – Multiply by the reciprocal of 17 6 – 6 – = 3 () 6 – 19 Use rule for multiplying fractions.

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