Ray's New Practical Arithmetic · Arts. 221–223 · Unit 22: Ratio and Proportion
103. Proportion: terms and finding the fourth term
Goal: Your child reads a proportion, names the extremes and means, and finds any missing term using "product of extremes equals product of means".
You'll need
- Paper and pencil, or a slate or small whiteboard
What's in the book
Page 282 defines Proportion (Art. 221). Page 283 names extremes and means and gives the principle (Art. 222) and Art. 223. Page 284 works examples and gives problems with printed answers.
The lesson, step by step
Warm-up
Ask: What is the ratio of 2 to 4? Of 3 to 6? They are equal. That is a proportion.
The idea
Proportion is an expression for the equality of two ratios.
The proportion is indicated by writing : : between the ratios.
"2 : 4 :: 3 : 6" is read "2 is to 4 as 3 is to 6".
The first and fourth terms of a proportion are called the extremes.
The second and third terms of a proportion are called the means.
The principle
In every proportion the product of the extremes is equal to the product of the means.
Check it: 2 × 6 = 12 and 4 × 3 = 12.
So a missing term is easy to find:
1st. If the product of the means be divided by one of the extremes, the quotient will be the other extreme.
Work the book's example
What : 6 :: 4 : 8?
The product of the means 6 and 4 is 24; 24 ÷ 8 = 3.
The book's rule: "Divide the product of the terms of the same name by the other given term."
Guided practice
Do these together:
2 : 8 :: 6 : what?
5 : what :: 6 : 12?
Independent practice and check
Your child works the rest of the problems on page 284. 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 : 6 :: 5 : ?; 4 : ? :: 8 : 10; ? : 9 :: 2 : 3.