Ray's New Intellectual Arithmetic · Lesson LXIII · Unit 7: Reasoning puzzles, ratio and proportion
63. Partnership
Goal: Your child shares rent or profit fairly when partners put in different amounts or for different times.
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
About fifteen problems on pasture rent (oxen and sheep) and partners' gains by capital and time. The model counts 10 sheep as 1 ox, then divides the rent in proportion.
The lesson, step by step
Warm up
Ask three or four problems from the last lesson again, out of order. Your child answers and says why.
Show the reasoning
Say180 sheep eat as much as 18 oxen. So A has 27 oxen and B 18: 45 shares in all. $25 divided into 45 shares: A pays 27 of them, $15, and B 18, $10.
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:
A and B rent a pasture for $25; A puts in 27 oxen and B 180 sheep; an ox eats as much as 10 sheep: what should each pay?
E and F were partners for a year: E put in $1000, F three times as much, but F took out $1000 after 8 months; the gain was $770: what is each share?
A gained $70 and B $80; A's money was in 10 months, B's 8; together they put in $1700: what did each invest?
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.
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) A $15, B $10. 11) E $210, F $560. 12) A $1300, B $1100. 13) C $420, D $560. 14) A $700, B $1000. 15) E $300, F $540.
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 ↗.