Ray's New Higher Arithmetic · Art. 226 · Unit 11: Compound Denominate Numbers
58. Aliquot parts
Goal: Your student finds the cost or amount of a compound quantity at a given rate by splitting it into simple parts.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Aliquot parts defined, worked examples (28 A. 145 sq. rd. 15 sq. yd. of land at $16 an acre; a traveler's distance), the rule with notes, a list of aliquot parts of 100, and examples with printed answers (gold, vinegar, a heart beating, railway cost). A topical outline of operations with compound numbers follows.
The lesson, step by step
Warm-up
Mental: What part of 100 is 25? 12½? 33⅓? 6¼?
The idea
Read:
An aliquot part is an exact divisor of a number.
In all questions in aliquot parts, one of the numbers indicates a rate.
Work the book's example
Read:
Find the cost of 28 A. 145 sq. rd. 15 sq. yd. of land, at $16 per acre.
28 acres at $16 = $448. 80 sq. rd. is half an acre ($8), 40 sq. rd. a quarter ($4), 20 sq. rd. an eighth ($2), 5 sq. rd. a quarter of that (50¢); add the yards in the same way. The book's total: $462.55.
The rule
Read:
Add these different values; the result will be the entire value required.
and Note 2:
Aliquot parts are generally used in examples involving U. S. money,
Guided practice
Together:
What is the cost of 88 gal. 3 qt. 1 pt. of vinegar, at 37 ct. a gallon?
Check the price on the page (the scan shows 37 ct. but the printed answer, $33.33, fits 37½ ct. a gallon).
Independent practice
Your student works the other examples, including:
If the heart should beat 97920 times in each day, how many times would it beat in 8 da. 5 hr. 25 min. 30 sec.?
(805494 times.)
Wrap-up
Look over the topical outline of operations with compound numbers. Your student names each operation and one thing to watch for.
Tip
Aliquot parts were a merchant's mental-math tool. Today the same skill is useful for tips and discounts: 12½% is an eighth, 33⅓% a third.