Ray's New Elementary Algebra · Arts. 141–144 · Unit 3: Chapter III: Algebraic fractions

23. Dividing fractions; complex fractions; series

Goal: Your student divides by a fraction (invert and multiply), simplifies complex fractions, and turns 1/(1 − x) into a series.

⏱ About 45 minutes · 8 steps

Open it at the Internet Archive ↗

You'll need

  • The book, open to the page
  • Pencil and plenty of paper (or a whiteboard)
  • A card to cover printed answers

What's in the book

Case VII: dividing a fraction by an entire quantity, then any quantity by a fraction, with the reasoning ("how many times is 2/3 contained in 4?"); complex fractions; and resolving fractions into infinite series. A general review ends the chapter.

The lesson, step by step

  1. Warm-up

    Ask: How many halves are in 3? How many quarters in 2?

  2. Divide by an entire quantity

    Read Art. 141.

    Divide the numerator by the divisor, if it can be done without a remainder; if not, multiply the denominator.

    Your student does exercises 1 and 2 on p. 85.

  3. Divide by a fraction

    Read Art. 142 and the reasoning for 2/3 in 4: 1/3 is in 4 twelve times, so 2/3 is in it half as often, 6 times. Then the rule:

    Invert the divisor, and proceed as in multiplication of fractions.

    SayFlip the second fraction, then multiply, canceling first.

  4. Guided practice

    Exercises 1 to 4 of Art. 142 while you watch. Check the answers on the page.

  5. Complex fractions

    Read Art. 143. Say: a complex fraction is just a division written on top of itself. Work the book's numerical example two ways: as a division, and by multiplying both terms by the L.C.M. of the small denominators, as the note on p. 88 shows.

  6. Series

    Read Art. 144.

    An Infinite Series is an unlimited succession of terms, which observe the same law.

    Do the long division 1 ÷ (1 − x) together for four terms: 1 + x + x² + x³ + .... Then your student does 1 ÷ (1 + x) (exercise 2).

  7. Independent practice

    The remaining exercises of Arts. 141 and 142 and the complex fractions on p. 88. Then use the General Review on p. 89 as an oral quiz.

  8. Wrap-up

    Ask: Why does dividing by 1/4 give four times as much?

Tip

This finishes fractions. If any case felt shaky, spend one more session on mixed review before equations.

Answers to the book's problems

The book prints the answer after each exercise.

Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.

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