Ray's New Higher Arithmetic · Arts. 172–177 · Unit 10: Circulating Decimals
45. Working with circulates
Goal: Your student changes a pure circulate to a common fraction and knows how circulates are added, subtracted, multiplied and divided.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Case I, a pure circulate to a fraction (.53 repeating = 53/99), Case II, a mixed circulate, then addition, subtraction, multiplication and division of circulates, each with a rule and examples. Pages 119, 121 and 124 were not checked for this guide; many answers are hard to read in the scan.
The lesson, step by step
Warm-up
Ask: What fraction is .333... ? (1/3.) And .999... ? (Discuss: it equals 1.)
Pure circulate to a fraction
Read:
Write the repetend for the numerator, omitting the decimal point and the dots
and for the denominator as many 9s as there are figures in the repetend. The book's reason: 100 × .5353... − .5353... = 53, so 99 times the number is 53. Answer 53/99.
Adding
Read:
Make the repetends similar, if they be not so; add, and point off as in ordinary decimals.
The book also says to carry into the last column whatever would come from the next repetition.
Multiplying and dividing
Read:
If the multiplier contain a repetend, change it to a common fraction.
Change the terms to common fractions; then divide as in division of fractions, and reduce the result to a repetend.
The book calls fractions the easiest method for division.
Guided practice
Change to fractions: .27 repeating (27/99 = 3/11); .142857 repeating (1/7). Then add 1/3 + 3/11 as fractions and turn the sum back into a circulate.
Independent practice
Your student tries a few of the book's subtraction examples (answers printed), such as:
Subtract .0074 from .26
Wrap-up
Ask: Which is easier, working with circulates directly or changing them to fractions first? Most people choose fractions, as the book does.
Answers to the book's problems
Guided practice: 1/3 + 3/11 = 20/33 = .60 repeating (60).
Worked out for this site from the scan. If a number in your copy differs, trust the book.