Ray's New Intellectual Arithmetic · Lesson LX · Unit 7: Reasoning puzzles, ratio and proportion

60. Ratio

Goal: Your child finds the ratio of two numbers and divides a quantity in a given ratio.

⏱ About 20 minutes · 5 steps

Open it at the Internet Archive ↗

You'll need

  • The book page (press “📖 Book page”, or use a printed copy)
  • Paper for jotting, though the work is meant to be done in the head

What's in the book

The definition: ratio is the relation one quantity bears to another of the same kind. Twenty-four problems, from “the ratio of 12 to 2 is 6” to sharing a pasture rent in a given ratio.

The lesson, step by step

  1. Warm up

    Ask three or four problems from the last lesson again, out of order. Your child answers and says why.

  2. Show the reasoning

    SayThe ratio of 12 to 2 is 12 divided by 2, which is 6.

    To divide 35 pupils in the ratio 2 boys to 3 girls: every group of 5 has 2 boys and 3 girls; 35 makes 7 such groups, so 14 boys and 21 girls.

  3. Work the problems aloud

    Read each problem on the page aloud. Your child answers, then explains in a full sentence, the way the book's model solutions do. The problems go like this:

    The ratio of 21 to 7 equals the ratio of 36 to what number?

    A school of 35 pupils has 2 boys to every 3 girls: how many of each?

    A and B hire a pasture for $45; A puts in 4 cows, B 5: what should each pay?

  4. Ask “Why?”

    After each answer ask “Why?” or “How do you know?” Ray's wants the reasoning said aloud, not just the number. If your child stalls, go back to the model and say the first half of the explanation together.

  5. Wrap up

    Pick the two problems that were hardest today. Tomorrow, start with those as the warm-up.

Afterwards

Do the rest of the lesson's problems over the next day or two, a few at a time. Move on when your child can explain them without help.

Answers to the book's problems

3) 6. 7) 30. 8) 12. 10) 2. 11) 2½. 12) George 15 cents, John 10 cents. 13) 20 and 28. 14) A 8, B 12. 15) John 12 cents, James 16 cents. 16) 60 apple, 36 peach. 17) 14 boys, 21 girls. 18) 12. 19) 5 yd and 20 yd. 20) 12 and 16. 21) A $20, B $25. 23) A $35, B $21. 24) C loses $18, D $12.

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

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