Ray's New Higher Arithmetic · Arts. 145–146 · Unit 8: Common Fractions
37. G. C. D. and L. C. M. of fractions
Goal: Your student finds the greatest common divisor and least common multiple of fractions and solves practical problems with them.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
The greatest common divisor of fractions (definition, principle, rule, 8 examples, including the largest sacks for shipping grain and fence rails) and the least common multiple of fractions (definition, rule, 4 examples, including walkers who meet again). Answers are printed.
The lesson, step by step
Warm-up
Mental: G. C. D. of 12 and 18; L. C. M. of 4 and 6.
G. C. D.
Read:
The greatest common divisor of two or more fractions is the greatest fraction that will exactly divide each of them.
Find the G. C. D. of the numerators of the fractions, and divide it by the L. C. M. of their denominators.
Try 3/4 and 9/10: G. C. D. of 3 and 9 is 3; L. C. M. of 4 and 10 is 20. Answer 3/20. Check: 3/4 ÷ 3/20 = 5, and 9/10 ÷ 3/20 = 6.
L. C. M.
Read:
The least common multiple of two or more fractions is the least number that each of them will divide exactly.
Divide the L. C. M. of the numerators by the G. C. D. of the denominators.
And the note:
The G. C. D. of several fractions must be a fraction, but the L. C. M. of several fractions may be an integer or a fraction.
Guided practice
Try 3/4 and 9/10 again for the L. C. M.: L. C. M. of 3 and 9 is 9; G. C. D. of 4 and 10 is 2. Answer 9/2 = 4½. Check: 4½ ÷ 3/4 = 6 and 4½ ÷ 9/10 = 5.
Independent practice
Your student works the examples, including the fence-rail problem:
how many rails of the greatest length possible will be needed to fence it in straight lines
Check
Answers are printed.
Wrap-up
Ask: In a word problem, how can you tell G. C. D. from L. C. M.? (Greatest piece that fits evenly into each: G. C. D. First time things line up again: L. C. M.)
Tip
The scan's small printed fractions are often hard to read. Keep the paper book or a clear copy of the page open when your student works the examples.