Ray's New Intellectual Arithmetic · Lesson LIX · Unit 7: Reasoning puzzles, ratio and proportion
59. Working together
Goal: Your child solves “how long will it take together” problems.
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
Nineteen problems. Models: if one does ½ of a job a day and another ⅓, together they do ⅚ a day and take 1⅕ days; and finding one worker's time from the team's.
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
SayA digs a trench in 6 days, so he does ⅙ of it a day. B takes 12 days, so ¹⁄₁₂ a day. Together they do 2/12 and 1/12, which is 3/12 or ¼ a day. So together they take 4 days.
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 can dig a trench in 6 days and B in 12 days: how long working together?
A man and his wife drink a keg in 12 days, the wife alone in 30: how long would the man alone take?
A, B and C reap a field in 4 days; A alone takes 8 days and B 12: how long would C take alone?
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
10) 1⅕ days. 11) 4 days. 12) 2 11/12 days. 13) 1 day. 14) 6 days. 15) 20 days. 16) 24 days.
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 ↗.