Ray's New Intellectual Arithmetic · Lesson LXVII · Unit 8: Percentage and interest
67. What per cent? Gain and loss
Goal: Your child finds what per cent one number is of another, and the per cent gained or lost.
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
Fourteen problems, with models: 2 is ⅖ of 5, which is 40 per cent; and a $2 gain on a $5 cost is 40 per cent.
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
SayThe gain is $2 on a cost of $5. $2 is two-fifths of $5, and two-fifths is 40 per cent. So the gain is 40 per cent.
The per cent gain is always worked on the cost.
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 9 of 36 pupils are absent, what per cent are absent?
A merchant buys cloth at $5 a yard and sells it at $7: what per cent does he gain?
Henry buys a horse for $15 and sells it for $24: what per cent does he gain?
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) 40%. 2) 60%; 25%. 3) 50%; 12½%. 4) 25%; 16⅔%. 5) 5%. 6) 25%. 7) 33⅓%. 8) 66⅔%. 9) 20%. 10) 40%. 11) 25%. 12) 20% loss. 13) 50%. 14) 60%.
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 ↗.