Lesson 2 · from § of the 1910 text
Common Multiples of Fractions
Common multiples of fractions
Two men walk around an island in the same direction, one in 2\frac12 hours and the other in 3 hours. They are together again after a number of hours that is an integral multiple of each time: of 2\frac12 and of 3. The least such number is the least common multiple (L.C.M.) of the two fractions.
The problems below lead to a rule for the L.C.M. of fractions and mixed numbers. Reduce each mixed number to an improper fraction in lowest terms; then compare the L.C.M. of the numerators and the G.C.D. of the denominators.
Answers that are fractions may be typed as 28/3 or as mixed numbers, 9 1/3.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Two men travel in the same direction around an island, one making the circuit every 2\frac12 hours and the other every 3 hours. If they start together, after how many hours will they be together again?
Three automobiles travel in the same direction around a circular road. They make the circuit in 2\frac34 hours, 3\frac13 hours, and 4\frac25 hours respectively. If they start at the same time, after how many hours are the three together again?
Is the answer to Exercise 10 an integral multiple of 2\frac12 and 3? Is it the least integral multiple?
Is the answer to Exercise 11 an integral multiple of 2\frac34, 3\frac13, and 4\frac25? Is it the least integral multiple?
Reduce 2\frac34, 3\frac13, and 4\frac25 to improper fractions and divide the L.C.M. of the numerators by the G.C.D. of the denominators. Compare the result with the answer to Exercise 11.
The method of finding the L.C.M. of two or more fractions or mixed numbers is hinted at in Exercise 14. State a rule therefor. Find by the rule the L.C.M. of 1\frac16, 2\frac13, and 3\frac19.
Find by the same rule the L.C.M. of \dfrac ab and \dfrac cd; of \dfrac ab, \dfrac cd, and \dfrac ef. (Treat the letters as having no common factors.)