Ray's New Intellectual Arithmetic · Lesson LXXX · Unit 9: Miscellaneous

80. Miscellaneous problems

Goal: Your child solves a final mixed set using every method in the book.

⏱ About 25 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

Sixty-one mixed problems over seven pages: barter, unknown numbers, partnership, work and rate, chases with steps and leaps, ages, profit, wages with forfeits and mixtures. Three have model solutions. This is the last lesson.

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

    Before each problem, ask your child to name the method: through one, parts of a number, ratio, work together, per cent, or interest. Naming it is half the battle.

    SayIf 12 peaches are worth 84 apples, 1 peach is worth 7 apples, so 5 peaches are worth 35 apples. 8 apples are worth 24 plums, so 1 apple is worth 3 plums, and 35 apples are worth 105 plums.

  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:

    If 12 peaches are worth 84 apples, and 8 apples are worth 24 plums, how many plums for 5 peaches?

    If Mary gives each playmate 5 cherries she has 21 left; if 8 each, none left: how many playmates?

    A, B and C can do a piece of work in 4 days, A and B in 8 days, B and C in 6 days: how long would each take alone?

  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

Spread these over several weeks, five at a time. When your child finishes, the next book is Ray's New Practical Arithmetic.

Answers to the book's problems

1) 105 plums. 2) James 8, Lucy 11, Mary 13. 3) 8. 7) 2½ days. 8) 4 cents a dozen. 10) 8 children. 11) 10 beggars. 12) 3 children. 13) 60 leaps. 14) 140 steps. 15) $60. 16) A 13 cents, B 11 cents. 19) 10 hr. 20) 25%. 21) 7 playmates. 22) 105 steps. 25) 80 pears. 26) 9% a year. 27) 20 yd. 30) 60 leaps. 31) James 5, Thomas 15. 37) A 12 days, B 24 days, C 8 days. 39) 50 cents. 43) man 24 days, woman 40 days. 45) A 50, B 10, C 5. 47) A 24 days, B 48 days. 49) 5 days. 50) 22 days. 51) 16 yd. 52) 100 steps. 54) 10, 40 and 120 sheep. 55) 20 days. 59) 50 lb. 61) John 6 cents, James 3 cents.

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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