Ray's New Intellectual Arithmetic · Lesson XXXIV · Unit 4: Working with fractions

34. Rates with fractional answers

Goal: Your child solves price and work problems where the answer comes out as a fraction.

⏱ 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-two problems. Models: if 3 bushels cost $1, 1 bushel costs $⅓ and 2 cost $⅔; and 4 men for 8 days is 32 days for 1 man, so 5 men take a fifth of 32 days.

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

    SayIf 3 barrels of cider cost $8, 1 barrel costs one-third of $8, which is $2⅔, and 2 barrels cost twice that, $5⅓.

    It's the same “through one” reasoning as before, but now the one is a fraction.

  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 3 barrels of cider cost $8, what will 2 barrels cost?

    If 1 barrel of flour feeds 8 persons 20 days, how long will it last 11 persons?

    If 18 bushels of oats last 5 horses a week, how many will 7 horses need?

  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

1) $⅔. 2) $1⅓. 3) $5⅓. 4) $4½. 5) 1⅗ cents. 6) 4⅔ cents. 7) 14⅖ cents. 8) 22½ cents. 18) 6⅖ days. 19) 14 6/11 days. 20) 4⅜ days. 21) 3 3/7 days. 22) 8⅘ days. 23) 50⅞ cents. 24) 11⅓ min. 25) 25⅕ bu. 26) 7 6/7 bu. 27) $14⅖.

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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