Ray's New Higher Arithmetic · Arts. 232–233 · Unit 13: Proportion

62. Compound proportion

Goal: Your student solves problems with several conditions by compound proportion or by cause and effect.

⏱ About 45 minutes · 7 steps

Open it at the Internet Archive ↗

You'll need

  • The book
  • A slate, small whiteboard or scrap paper
  • A notebook and pencil

What's in the book

Compound proportion defined, the rule, the example of men mowing acres, the method of cause and effect with its rule and the fence example, and examples for practice with printed answers (the use of money, travelers, bricks and tiles, pipes filling a cistern).

The lesson, step by step

  1. Warm-up

    Ask: If 2 men dig a ditch in 6 days, how long for 4 men? (3 days.) And if the ditch were twice as long?

  2. The idea

    Compound proportion is

    an expression of equality between two ratios

    when one or both ratios are compound. Work the book's example: If 3 men mow 8 A. in 4 da., how long would 10 men take to mow 36 A.? More acres need more days (8 : 36); more men need fewer (10 : 3). Days = 4 × 36 × 3 ÷ (8 × 10) = 5⅖ days.

  3. Cause and effect

    Read:

    Separate all the quantities contained in the question into two causes and their effects.

    The two causes must be exactly alike in the number and kind of their terms

    First cause : second cause :: first effect : second effect.

  4. Work the fence example

    6 men, working 10 days of 9 hours, build 25 rods of fence. How many hours a day must 8 men work to build 48 rods in 12 days? 6 × 10 × 9 : 8 × 12 × (hours) :: 25 : 48. The page leaves the answer blank; work it out together (below).

  5. Guided practice

    Together:

    The use of $100 for 1 year is worth $8: what is the use of $4500 for 2 yr. 8 mon. worth?

    ($960.)

  6. Independent practice

    Your student works the examples for practice, including the pipes problem (27 pipes).

  7. Wrap-up

    Ask: Why can cancellation save so much work here? (All the terms are multiplied and divided in one line.)

Answers to the book's problems

Fence example: hours = 6 × 10 × 9 × 48 ÷ (8 × 12 × 25) = 10⅘ hours a day.

Worked out for this site from the scan. If a number in your copy differs, trust the book.

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