Ray's New Higher Arithmetic · Arts. 230–231 · Unit 13: Proportion

60. Proportion and the rule of three

Goal: Your student tests a proportion by its means and extremes, and sets out and solves a simple proportion.

⏱ About 35 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

Proportion defined, its sign, couplets, antecedents and consequents, extremes and means, mean proportionals, simple and compound proportion, three principles, then simple proportion: the statement, the rule, and two worked examples (horses and men).

The lesson, step by step

  1. Warm-up

    Ask: If 3 pencils cost 12 cents, what do 5 cost?

  2. The idea

    Read:

    Proportion is an equality of ratios.

    The Sign of Proportion is the double colon ( : : ).

    The book's example: 4 : 6 :: 8 : 12, read "4 is to 6 as 8 is to 12."

  3. Means and extremes

    Read:

    The first and last terms of a proportion are called the extremes.

    The middle terms are the means. The test:

    the product of the means is equal to the product of the extremes.

    Check: 6 × 8 = 48 and 4 × 12 = 48.

  4. Simple proportion

    Read:

    employed when three terms are given and we wish to find the fourth.

    The Statement is the proper arrangement of the terms of the proportion.

    For the third term, write that number which is of the same denomination as the number required.

    Then put the larger of the other two second if the answer should be larger, and:

    Divide the product of the second and third terms by the first

  5. Work the book's example

    If 6 horses cost $300, what will 15 horses cost? 6 horses : 15 horses :: $300 : answer. 15 × 300 ÷ 6 = $750. Proof: 750 × 6 = 300 × 15.

  6. Inverse case

    The book's second example: 15 men do a piece of work in a number of days (9⅗ by the book's working; the fraction is faint in the scan); how long will 36 men take? More men need fewer days, so the answer must be smaller. Days go third; the smaller of the other two (15 men) goes second: 36 men : 15 men :: 9⅗ days : answer. 15 × 9⅗ ÷ 36 = 4 days.

  7. Wrap-up

    Ask: How do you know which number goes third? (It's the one like the answer.)

Tip

Have your student always ask first: should the answer be bigger or smaller? That one question sets up every proportion.

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