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the quotients and the numbers not divided in a line below. Divide this line and set down the results as before ; thus continue the operation till there are no two numbers which can be divided by any number greater than 1. The continued product of the divisors inio the numbers in the last line, will be the least common multiple required. 16. Find the least common multiple of 6, 8, and 12. First Operation.

Second Operation. 2)6 8 12

6)6 " 8

12
2)3
4 6

2)1 8 2
3)3 2 3

1 4 1
1
1

Now 6X2 X4=48. 2 X2 X3 X2=24 Ans.

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Obs. 1. In the first operation, we divide by the smallest numbers which will divide any two of the given numbers without a remainder, and the product of the divisors, &c., is 24, which is the answer required.

In the second operation, we divide by 6, then by 2, which are not the smallest numbers that will exactly divide two of the given numbers, and the continued product of the divisors into the figures in the last line is 48, which is not the least common multiple. Hence,

2. We must divide, in all cases, by the smallest number that will divide any two of the given numbers exactly; otherwise, the divisor may contain a factor common to it and some one of the quotients, or undivided numbers in the last line, and consequently the continued product of them will be too large for the least common multiple.

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.

QUEST.-Obs. Why do you divide by the smallest number that will die vide two or more without a remainder.

SECTION VI.

FRACTIONS.

MENTAL EXERCISES.

Art. 103. When a number or thing is divided into two equal parts, one of those parts is called one half. If the nuniber or thing is divided into three equal parts, one of the parts is called one third ; if it is divided into four equal parts, one of the parts is called one fourth, or one quarter ; two of the parts, two fourths; three, three fourths ; if divided into five equal parts, the parts are called fifths; if into six equal parts, sixths ; if into ten, tenths; if into a hundred, hundredths, &c. That is,

When a number or thing is divided into equal parts, the parts always 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, 6, &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 ?

Quest.–103. What is meant by one half ? How many halves make a whole one ? What is meant by one third? How many thirds make a whole one? What is meant by a fourth? 3 fourths ? What are fourths sometimes called ? How many fourths make a whole one? 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, froin what do the parts take their name? 104. Upon what does the value of one of these equal parts depend? Which is the greater, a half or a third ? A sixth or a fourth? A seventh or a tenth ?

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 12? Of 15 ? 18 ? 21 ? 24 ? 27 ? 30 ? 36 ? 39 ? 45 ? 60 ?

4. What is a fourth of 8 dollars ? Of 12 ? 16 ? 20 ? 24 ? 28 ? 32 ? 36 ? 40 ? 44 ? 48 ?

5. What is a fifth of 5 ? 10 ? 15? 20 ? 25 ? 30 ? 35 ? 40 ? 45 ? 50? 55 ? 60 ? 100 ?

6. What is a sixth of 12 ? 18 ? 24 ? 36 ? 30 ? 48 ? 60 ? 54 ? 42 ? 72 ?

7. What is a seventh of 14 ? 28? 35 ? 21 ? 42 ? 56 ? 49 ? 63 ?

8. What is an eighth of 16 ? 24 ? 40 ? 32 ? 64 ? 48 ? 56 ? 72 ? 88 ?

9. What is a ninth of 9? 18? 36 ? 27 ? 45 ? 54 ? 72 ? 63 ? 81 ? 99 ?

10. What is a tenth of 20 ? 40 ? 60 ? 50 ? 30 ? 100 ? 90 ? 120 ? 11. What part of 2 is 1 ?

Ans. One half. 12. What part of 3 is 1 ? Of 4? 5? 7? 10 ? 15? 19? 37 ? 200 ?

13. What part of 3 is 2 ?

Suggestion. Two is two times the third part of 3, or two thirds of 3.

14. What part of 5 is 2 ? is 3 ? is 4 ? is 5 ? is 6? is 8? is 9 ?

15. What part of 8 is 3 ? is 7 ? is 6 ? is 9 ? is 8 ? 12? 15 ?

16. What part of 17 is 5? 8? 9? 13 ? 15 ? 16? 20 ? 17. What part of 100 is 13? 29 ? 63 ? 75 ? 92 ?

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

Suggestion. If I 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. Ans. 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 i 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 ? What will a. whole yard cost ?

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

23. If I sixth of a bogshead of molasses cost 5 dollars, what will be the cost of a hogshead? Of 4 hogsheads ? Of 10 hogsheads?

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

Suggestion. If 1 pound cost 12 cents, it is plain that 1 half of a pound will cost 1 half of 12 cents. 1 half of 12 cents is 6 cents. Ans. 6 cents.

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

26. If one pound of tea cost 4 shillings, how much will i fourth of a pound cost? How much will 2 fourths cost?

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

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

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

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

Suggestion. If i bushel cost 1 half a dollar, 2 bushels will cost twice as much. 2 times 1 half are 2 halves, or a whole dollar. 4 bushels will cost 4 times 1 half, or 2 whole dollars.

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

32. How many whole ones are 4 halves equal to ? 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 oats a day, how many will he eat in 6 days ? In 8? In 10 ? In 12?

36. If a boy saws 1 eighth of a cord of wood in a day, how much will 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?

Suggestion. 1 is 1 fourth of 4; hence, 1 boy must receive I fourth part of the oranges. 1 fourth of 12 oranges is 3 oranges.

38. A builder employed 6 men to do a job of work, for which he

gave

thein 24 dollars : what part of the money did 1 man receive ? What part did 2 receive ? What part did 4 receive? How many dollars did one man have ? How many

did two have? Three ? Four ? 39 If 5 yards of cloth cost 40 dollars, what part of 40 dollars will 1 yard cost ? 3 yards ? 4 yards ? How many dollars will one yard cost ? 2 yards ? 3 yards? 4 yards?

40. 2 is 1 third of what number?

Solution. If 2 is 1 third of a number, 3 thirds, or the whole number, is three times as many.

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

41. 4 is 1 fisth or 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 ? I twelfth ?

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

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