16. Find the least common multiple of 6, 8, and 12.

Note 1.-In the first operation we divide by the smallest numbers which will divide any two of the given numbers without a remainder, and 24 the product of the divisors, and the numbers in the last line, is the true answer.

2. In the second operation, we divide by 6, then by 2. But 6 is a composite number and contains the factor 2, which is common to the 4 in the last line; consequently, the continued product, though a common multiple, is not the least common multiple of the given numbers. It is twice too large. Hence,

3. It will be seen that "dividing by any number, which will divide two or more of the given numbers without a remainder," according to the rule sometimes given, does not always produce the least common multiple of the given numbers.

17. Find the least common multiple of 4, 9, and 12. 18. Find the least common multiple of 16, 12, and 24. 19. Find the least common multiple of 15, 9, 6, and 5. 20. Find the least common multiple of 10, 6, 18, 15. 21. Find the least common multiple of 24, 16, 15, 20. 22. Find the least common multiple of 25, 60, 72, 35. 23. Find the least common multiple of 63, 12, 84, 72. 24. Find the least common multiple of 54, 81, 14, 63. 25. Find the least common multiple of 12, 72, 36, 144. 26. Find the least common multiple of the nine digits. 27. Find the least common multiple of 17, 29, and 53. 28. Find the least common multiple of 8, 9, 55, and 49. 29. Find the least common multiple of 720, 336, and 1736. 30. Find the least com. mul. of 8, 12, 16, 24, 36, 48, 72, 144. 31. Three men start from the same place at the same time, to go round a circular field; one of which can travel the distance in 8 hours, another in 10 hours, and the other in 12 hours. In what time will they all meet at the starting place?

32. At the time of meeting, 3 others joined the company, one of whom could perform the distance in 6 hours, another in 16 hours, and the other in 18 hours: when will the six Ans. 720 h. meet at the place from which they started?

SECTION VI.

FRACTIONS.

MENTAL EXERCISES.

ART. 103. When a number or thing is divided into two equal parts, one of these parts is called one half. If divided into three equal parts, one of the parts is called one third; if divided into four equal parts, one of the parts is called one fourth or one quarter; if into ten, tenths; if into a hundred, hundredths, &c. That is,

When a number or thing is divided into equal parts, the parts take their name from the number of parts into which the thing or number is divided.

104. The value of one of these equal parts manifestly depends upon the number of parts into which the given number or thing is divided. Thus, if an orange is successively divided into 2, 3, 4, 5, &c. equal parts, the thirds will be less than the halves; the fourths than the thirds; the fifths than the fourths, &c.

Ex. 1. What is one half of 2 cents? Of 4 cents? 6? 8? 16? 18? 20? 24? 30? 40? 50? 60? 70? 80? 100? 2. What is one third of 6 cents? Of 9 12? 15?

OBS. A half of any number, it will be perceived, is equal to as many units, as 2 is contained times in that number; a third of a number is equal to as many units, as 3 is contained times in the given number; a fourth is equal to as many, as 4 is contained in it, &c.

3. What is a third of 18? 21? 24? 27? 80? 36? 39? 45? 60? 42? 54? 75?

QUEST.-103. What is meant by one half? What is meant by one third ? What is meant by a fourth? 3 fourths? What are fourths sometimes called? What is meant by fifths? By sixths? Eighths? How many sevenths make a whole one? How many tenths? What is meant by twentieths? By hundredths? When a number or thing is divided into equal parts, from what do the parts take their name? 104. Upon what does the value of one of these equal parts depond? Which is the greater, a sixth or a fourth? A seventh or a tenth?

4. A fourth of 20? 24? 28? 32? 36? 40? 44? 48? 5. A fifth of 25? 30? 35? 40? 45? 50? 55? 60? 100? 6. A sixth of 18? 24? 36? 30? 48? 60? 54? 42? 72? 7. A seventh of 28? 35? 21? 42? 56? 49? 63? 8. An eighth of 24? 40? 32? 64? 48? 56? 72? 88? 9. A ninth of 18? 36? 27? 45? 54? 72? 63? 81? 99? 10. A tenth of 40? 60? 50? 30? 100? 90? 120? 11. What part of 2 is 1? 12. What part of 3 is 1? 13. What part of 3 is 2?

Ans. One half.

Of 4? 5? 7? 10? 15? 19?

Suggestion. Since 1 is 1 third part of 3, 2 must be two times 1 third part of 3, or two thirds of 3.

14. What part of 5 is 2? 3? 4? 5? 6? 8? 9? 15?
15. What part of 8 is 3? 7? 6? 9? 8? 12? 15? 19?
16. What part of 17 is 5? 8? 9 13? 15? 16? 20?

17. What part of 100 is 13? 29? 63? 75? 92? 99

18. If 1 half an orange cost 2 cents, what will a whole orange cost?

Analysis.—If 1 half of an orange cost 2 cents, 2 halves or a whole orange, will cost twice as much; and 2 times 2 cents are 4 cents. A whole orange will, therefore, cost 4 cents.

19. If 1 third of a pie cost 4 cents, what will 2 thirds cost? What will a whole pie cost?

20. If 1 fourth of a pound of ginger cost 3 cents, what will 2 fourths of a pound cost? 3 fourths? What will a whole pound cost?

21. If 1 eighth of a yard of cloth cost 2 shillings, what will 3 eighths cost? 5 eighths? 7 eighths? A whole yard?

22. If 1 third of a barrel of flour cost 3 dollars, how much will a whole barrel cost? 5 barrels ? 8 barrels ?

23. If 1 sixth of a hogshead of molasses cost 5 dollars, what will be the cost of 1 hogshead? Of 10 hogsheads?

24. If 1 pound of sugar cost 12 cents, what will 1 half a pound cost?

Analysis. If 1 pound cost 12 cents, it is plain that 1 half a pound will cost 1 half of 12 cents; and 1 half of 12 cents is 6 cents. Therefore, 1 half a pound of sugar, at 12 cents a pound, will Cost 6 cents.

25. If 1 yard of ribbon cost 15 cents, how much will 1 third

of a yard cost?

26. If 1 pound of tea cost 4 shillings, how much will 1 fourth of a pound cost? 2 fourths? 3 fourths?

27. If a ton of hay cost 15 dollars, how much will 1 fifth of a ton cost? 2 fifths? 3 fifths?

28. What will 1 tenth of an acre of land cost, at 30 dollars per acre? 2 tenths? 6 tenths?

i 29. What will 1 eighth of a ton of iron cost, at 48 dollars per ton? 3 eighths? 5 eighths? 7 eighths?

30. If 1 bushel of corn cost 1 half a dollar, what will 2 bushels cost? 4 bushels? 8 bushels?

Analysis. If 1 bushel cost 1 half a dollar, 2 bushels will cost twice as much; and 2 times 1 half are 2 halves, which are equal to a whole dollar. Again, 4 bushels will cost 4 times as much as 1 bushel, and 4 times 1 half dollar are 4 halves, equal to 2 whole dollars.

31. If one man eats 1 half a loaf of bread at a meal, how many loaves will 3 men eat?

32. How many whole ones in 4 halves? 5 halves? 6 halves? 8 halves? 9 halves?

33. If I burn 1 third of a ton of coal in a week, how much shall I burn in 3 weeks? 4 weeks? 6 weeks? 10 weeks? 12 weeks?

34. How many whole ones in 4 thirds, and how many over? In 6 thirds? 8 thirds? 11 thirds? 14 thirds? 35. If a horse eat 1 fourth of a bushel of nany will he eat in 6 days? In 8? In 10?

oats a day, how In 12 ? 36. If a boy can saw 1 eighth of a cord of wood in a day, how much can he saw in 6 days? In 12 days? In 15 days? In 24 days?

37. If 12 oranges were divided equally among 4 boys, what part of them would each boy receive; and how many oranges would each have?

Analysis.—1 is 1 fourth of 4; hence, 1 boy must receive 1 fourth part of the oranges. Now 1 fourth of 12 oranges is 8 oranges. Therefore, each boy would receive 3 oranges.

88. A builder employed 6 men to do a job of work, for

which he gave them 24 dollars: what part of the money did 1 man receive? 2 men? 3 men? 4 men? How many dollars did 1 man receive? 2 men? 3 men? 4 men ?

39. If 5 yards of cloth cost 40 dollars, what part of 40 dollars will 1 yard cost? 2 yards? 3 yards? 4 yards? How many dollars will 1 yard cost? 2 yards? 3 yards? 4 yards?

40. 2 is 1 third of what number?

Analysis.—If 2 is 1 third of a number, 3 thirds or the whole number, must be 3 times as many; and 3 times 2 are 6. Therefore 2 is 1 third of 6.

Or thus, 2 is a third of 3 times 2; and 3 times 2 are 6.

41. 4 is 1 fifth of what number? 1 sixth of what number? 1 third? 1 eighth? 1 fourth? 1 seventh?

42. 6 is 1 third of what number? 1 fourth? 1 seventh? 1 tenth? 1 ninth? 1 twelfth ?

43. 5 is 1 fourth of what number? 1 sixth? 1 eighth? 1 eleventh? 1 twelfth?

44. 8 is 1 seventh of what number? 1 sixth? 1 tenth? 1 ninth? 1 twelfth ?

45. 4 is 2 thirds of what number?

Suggestion. First find 1 third. is 1 half of 4, which is 2; and 3 Therefore 4 is 2 thirds of 6.

Now if 4 is 2 thirds, 1 third thirds is 3 times 2, or 6.

46. 9 is 3 fourths of what number?
47. 8 is 4 fifths of what number?
48. 16 is 4 ninths of what number?
49. 20 is 5 eighths of what number?
50. 32 is 8 twelfths of what number?

51. What is 2 thirds of 24?

Analysis.-1 third of 24 is 8, and 2 thirds is 2 times as

many, which is 16. Therefore 16 is 2 thirds of 24.