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Connexus lesson 10 applications of dividing rational numbers

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Certainly! In a lesson on applications of dividing rational numbers, you can explore several key concepts, including the definition of rational numbers, how to divide them, and real-world applications that show the importance of dividing rational numbers in various contexts. Here’s a guide that covers these concepts:

Key Concepts

  1. Understanding Rational Numbers:

    • Rational numbers are numbers that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q \neq 0 \). Examples include \( \frac{1}{2}, -\frac{3}{4}, 2, \) and \( 0.75 \).

  2. Dividing Rational Numbers:

    • When dividing rational numbers, you can use the rule: \[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \quad (c \neq 0) \]
    • This means you multiply by the reciprocal of the divisor.

  3. Steps to Divide Rational Numbers:

    • Example: Divide \( \frac{1}{2} \) by \( \frac{3}{4} \).
      1. Write down the problem: \( \frac{1}{2} \div \frac{3}{4} \)
      2. Multiply by the reciprocal: \( \frac{1}{2} \times \frac{4}{3} \)
      3. Multiply the numerators and denominators: \( \frac{1 \times 4}{2 \times 3} = \frac{4}{6} \)
      4. Simplify: \( \frac{4}{6} = \frac{2}{3} \)

Applications of Dividing Rational Numbers

  1. Cooking and Recipes:

    • If a recipe requires \( \frac{2}{3} \) of a cup of sugar and you want to make \( \frac{1}{2} \) of the recipe, you would need to calculate how much sugar is required: \[ \frac{2}{3} \div 2 = \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3} \text{ cup of sugar} \]

  2. Finance:

    • If you are dividing \( $200.00 \) among four people, you can find out how much each person will get by: \[ 200 \div 4 = 50 \text{ dollars per person} \]

  3. Distances and Speed:

    • If a car travels \( \frac{150}{2} \) miles in \( 5 \) hours, calculating the average speed involves dividing the distance traveled by time: \[ \frac{75}{5} = 15 \text{ miles per hour} \]

  4. Conversions:

    • Dividing by rational numbers is useful in unit conversions. For instance, converting \( \frac{3}{4} \) of a liter into milliliters (knowing \( 1 \text{ liter} = 1000 \text{ milliliters} \)): \[ \frac{3}{4} \div 1 = \frac{3 \times 1000}{4} = 750 \text{ milliliters} \]

  5. Splitting Costs:

    • If a group of friends buys a pizza costing \( \frac{15}{2} \) dollars and want to share it equally, and there are \( 3 \) friends: \[ \frac{15}{2} \div 3 = \frac{15}{2} \times \frac{1}{3} = \frac{15}{6} = \frac{5}{2} \text{ or } 2.5 \text{ dollars each} \]

Practice Problems

  1. \( \frac{3}{5} \div \frac{1}{2} \)
  2. \( \frac{7}{4} \div \frac{14}{5} \)
  3. A \( \frac{1}{3} \) mile long track is to be divided among \( \frac{1}{6} \) mile segments. How many segments can be made?
  4. You have \( \frac{8}{9} \) of a gallon of paint, and you want to paint in walls that take \( \frac{1}{3} \) gallon each. How many walls can you paint?

Conclusion

Dividing rational numbers is not only a fundamental skill in mathematics but also has practical applications in everyday life. Understanding how to divide these numbers can aid in problem-solving across various subjects, whether in cooking, finance, or science.

Stay curious and keep practicing to become proficient in working with rational numbers!