Ray's New Practical Arithmetic · Arts. 138–141 · Unit 12: Decimal Fractions (continued)
65. Reduction of decimals: ciphers, and decimals to common fractions
Goal: Your child knows that adding or dropping ciphers (zeros) at the right of a decimal does not change it, and can change any decimal to a common fraction in lowest terms.
You'll need
- Paper and pencil, or a slate or small whiteboard
What's in the book
Page 175 opens Reduction of Decimals (Art. 138), which has four cases. Page 176 gives Cases I and II (annexing or omitting ciphers) and Case III (decimal to common fraction) with the worked example .75 = 3/4. Page 177 has the practice problems for Case III.
The lesson, step by step
Warm-up
Write .5, .50 and .500 on paper.
SayAre these the same amount or different?
Let your child explain using money: 5 dimes, 50 cents, 500 mills.
The line just above this section (Art. 137) lists the operations still to come:
The operations with decimals are Reduction, Addition, Subtraction, Multiplication and Division.
The idea
Read the definition together:
Reduction of Decimals is changing their form without altering their value.
Then the two rules about ciphers (zeros):
Annexing ciphers to a decimal does not change its value.
Omitting ciphers from the right of a decimal does not change its value.
Note the words: in Ray's day a zero was called a "cipher", and "annex" means "write on the end".
Work the book's example
Case III:
Reduce .75 to a common fraction.
Say.75 is 75 hundredths, so we write 75/100. Now reduce it to lowest terms.
Divide both terms by 25 to get 3/4. The rule is just these two steps:
Write the decimal as a common fraction.
Reduce the fraction to its lowest terms.
Guided practice
Do two of the problems on page 177 together, .25 and .375, saying each step aloud.
For .375, the denominator is 1000 (three decimal places). The greatest common divisor of 375 and 1000 is 125.
Independent practice
Your child works the rest of the Case III problems on page 177 (.6, .035, .5625, .34375, .1484375), then the two that ask for an integer and a common fraction (4.02 and 8.415).
Check
These problems have no printed answers in the book, so the answers are in the answers box below. If a fraction is right but not in lowest terms, ask your child to divide again.
Mental-math wrap-up
Say a decimal and have your child answer with a fraction, quickly: .5, .25, .75, .2, .125, .05.
These few are worth knowing by heart.
Afterwards
Next day, say five fractions you made today and ask for the decimals back. That leads into the next guide.
Tip
If reducing is slow, review the greatest common divisor first. The book uses it again and again from here on.
Answers to the book's problems
Art. 141 problems (page 177; the book prints no answers): .6 = 3/5; .25 = 1/4; .375 = 3/8; .035 = 7/200; .5625 = 9/16; .34375 = 11/32; .1484375 = 19/128; 4.02 = 4 1/50; 8.415 = 8 83/200.
Worked out for this site from the scan. If a number in your copy differs, trust the book.