Ray's New Practical Arithmetic · Arts. 213–217 · Unit 22: Ratio and Proportion

101. Ratio: finding the ratio and a missing term

Goal: Your child finds the ratio of two numbers as their quotient, finds a missing antecedent or consequent, and finds the value of a compound ratio.

⏱ About 35 minutes · 7 steps

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You'll need

  • Paper and pencil, or a slate or small whiteboard

What's in the book

Page 276 defines Ratio (Art. 213). Page 277 gives the rules about terms and Case I (Art. 214). Page 278 has its problems and Case II (Art. 215). Page 279 gives Case III (Art. 216) and compound ratio (Art. 217), and page 280 the compound-ratio problems. Answers are printed.

The lesson, step by step

  1. Warm-up

    Ask: How many times larger is 12 than 3? Than 4? Than 6?

  2. The idea

    Ratio is the relation of two numbers expressed by their quotient.

    The ratio of two numbers is indicated by writing the sign ( : ) between them.

    The first term is called the antecedent, and the second term the consequent.

    So 6 : 2 is 3, because 6 ÷ 2 = 3. Read 2 : 6 as "the ratio of 2 to 6".

  3. Same kind only

    1st. The terms of the ratio may be either abstract or concrete. 2d. When the terms are concrete, both must be of the same denomination. 3d. The ratio is always an abstract number.

    Ray's example: 2 yd. : 2 ft. means 6 ft. : 2 ft., which is 3. There is no ratio between 2 ft. and $2.

  4. The three cases

    Find the ratio:

    Divide the antecedent by the consequent.

    Find the antecedent:

    Multiply the consequent by the ratio.

    Find the consequent:

    Divide the antecedent by the ratio.

    Compound ratio:

    Divide the product of the antecedents by the product of the consequents.

  5. Guided practice

    Do these together:

    What is the ratio of 6 to 3?

    7 is the ratio of what number to 4?

    5 is the ratio of 45 to what number?

  6. Independent practice and check

    Your child works a selection from each case on pages 278–280, including the ones with money and measures ("54 days to 9 days", "1 ft. 9 in. to 3 in."). The book prints the answer at the end of each problem line. Have your child cover the answers with a strip of paper, work the problem, then slide the strip down to check.

  7. Mental-math wrap-up

    Aloud: 20 : 5; 5 : 20; 3 is the ratio of what to 8? 4 is the ratio of 36 to what?

Tip

Ray treats a ratio as a single number (the quotient), so 5 : 10 is 1/2. Some modern books treat a ratio as a pair. For this book, always divide.

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