Ray's New Practical Arithmetic · Art. 224 · Unit 22: Ratio and Proportion
104. Simple proportion (the Rule of Three)
Goal: Your child sets up and solves word problems by simple proportion, deciding whether the answer should be larger or smaller.
You'll need
- Paper and pencil, or a slate or small whiteboard
What's in the book
Page 285 introduces the Rule of Three (Art. 224) and works a cloth example. Page 286 works a cellar-digging example (inverse ratio), gives the rule and starts the problems, which run through page 289. Answers are printed.
The lesson, step by step
Warm-up
Ask: If 2 apples cost 10 cents, what do 6 apples cost? If 2 workers take 6 hours, do 3 workers take more or less time?
The idea
Proportion, when applied to the solution of concrete problems, has been styled "The Rule of Three," because three terms are given to find the fourth.
The use of Proportion was formerly so extensive that it was often called "The Golden Rule."
Work the book's examples
If 2 yd. of cloth cost $4, what will 6 yd. cost?
The answer is dollars, so $4 is the third term. 6 yd. cost more than 2 yd., so 6 is the second term: 2 : 6 :: $4 : what? The answer is 6 × 4 ÷ 2 = $12.
If 3 men can dig a cellar in 10 days, in how many days can 5 men dig it?
More men need fewer days, so the smaller number goes second: 5 : 3 :: 10 : what? The answer is 3 × 10 ÷ 5 = 6 days.
The rule
1. For the third term, write that number which is of the same denomination as the number required.
Then put the larger of the other two second if the answer should be larger, and the smaller second if it should be smaller.
3. Divide the product of the second and third terms by the first; the quotient will be the fourth term, or number required.
Guided practice
Do these together, asking "more or less?" first:
If 4 hats cost $14, what cost 10 hats?
If 8 men perform a piece of work in 24 days, in what time can 12 men perform it?
Independent practice and check
Your child works a selection of problems 3–42 on pages 286–289. The book prints the answer at the end of each problem line. Have your child cover the answers with a strip of paper, work the problem, then slide the strip down to check.
Mental-math wrap-up
Aloud: 3 pens cost 30 ct., 7 pens? 4 men take 6 days, 8 men? $2 for 5 lb., $6 for?
Tip
Old customs note: problem 30 is about a 4-cent loaf whose weight changed with the price of flour. Bakers sold bread at fixed prices then and changed the size of the loaf.