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

56. Number puzzles: sums, differences and ages

Goal: Your child solves classic “two numbers” and age puzzles by reasoning, without algebra.

⏱ 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

Thirty-one problems. Models: a number added to itself is twice the number; for two numbers whose sum is 16 and difference 4, twice the greater is 20; for three boys' ages, take away the differences first.

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 sum of two numbers is 16 and their difference is 4. If the smaller were as big as the larger, the sum would be 16 and 4, or 20. That is twice the larger, so the larger is 10 and the smaller 6.

  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:

    Divide 45 into three parts so the second is 3 times the first and the third 5 times the first.

    The sum of two numbers is 31, and the greater exceeds the less by 7: what are they?

    A watch, chain and ring cost $62; the chain cost $5 less than the ring, and the watch $12 more than the chain: find each price.

  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.

Tip

Drawing the numbers as bars of different lengths makes these much easier.

Answers to the book's problems

1) 7. 2) 8. 3) 4 and 12. 4) 6 and 42. 5) 4, 8, 12. 6) 5, 15, 25. 8) 5. 9) 9 and 3. 10) 12. 11) 4. 12) 13. 13) 10 and 6. 14) 15 and 10. 15) 19 and 12. 24) Henry 10, James 14, Oliver 13. 27) ring $20, chain $15, watch $27. 31) harness $25, horse $75, buggy $125.

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 ↗.

@teddyrookbook GitHub