8. A man gave of a bushel of oats to some horses, giving to each of a bushel; to how many did he give it? and what was the remainder?

How many times will go in 5? In? How many times will go in ?

9. A man has of a dollar; he gives of a dollar to one person, and of a dollar to a second, what part of a dollar has he left?

How many cents had he at first? How many cents did he give away? How many cents had he left?

10. If 13 pounds of figs cost of a dollar, what is that a pound?

11. If 5 lbs. of figs cost of a dollar, what is that a pound? Find first what one half pound will cost.

12. If of a cwt. of iron cost 4 dollars, what will a hundred weight cost?

13. If 341 lbs. of tea cost 11

cost?

dollars, what will 1 pound

69
2

pounds cost

of a dollar: therefore 69

Here you find pounds must cost of a dollar.

14. If of a barrel of flour cost 33 dollars, what is that a barrel?

15. If wood is 5

cost? What will 4

dollars a cord, what will of a cord cords cost?

16. If 33 gals. of molasses cost 11 dollars, what is that a gallon?

17. If 31 gals. of vinegar cost 4ğ dollars, what is that a gallon?

18. If a bottle of wine containing 1 pints cost of a dollar, what would a barrel of wine come to at that rate?

19. In a pile of wood there a 13 cords; how many loads of of a cord each are there in the pile?

20. How many times will 24 go in 7? In 9? In 11? 21. How many loaves, of 8 oz. of flour each, can be made from 7 pounds of flour?

22. If a family consume 3 pounds of flour a day, how long will a barrel of flour, that is 196 pounds, last them?

How long will it last if they consume 24 lbs. a day?

23. If a barrel of flour last a family 40 days, how long will 14 pounds last them?

24. A garrison of 100 men is allowed 12 oz. of flour a day to each man; how long will 10 barrels last them?

25. Two men hire a horse, a week, for 5 dollars; one travels with him 30 miles, the other 45 miles; what ought each to pay?

26. Two men hire a pasture in common for $4,80; one pastures his horse in it 7 weeks; the other pastures his horse 9 weeks; what ought each to pay?

27. A boy bought 3 doz. of oranges for 37 cents, and sold them for 1 cents apiece; what did he gain?

28. A man bought 7 yds. of cloth for 16 dollars, and sold it for 3 dollars a yard; what did he gain on each yard?

29. A man worth 1690 dollars, left of his property to his wife; how much did she receive? The remainder he divided equally among 3 sons; what did each one receive?

30. A man bequeathed his estate of 14,000 dollars, one third to his wife, and the remainder to be divided equally among four sons; what did the wife and what did each son receive.

31. In an orchard one third of the trees bear apples, two fifths as many bear plums, and the rest bear cherries; what portion of the trees bear plums? What portion bear cherries? The number of cherry trees is 40; what is the whole number of trees in the orchard?

32. What is of 549? What is 8 of 374?

33. What is of 175?

What is of 198?

34. What is of of 1640? What is of 972 ?

35. If 2 barrels of flour cost 11 dollars, what will 17 bar

rels cost? What will 22 barrels cost?

36. If 21 cords of wood cost 15 dollars, what will 68 cords cost? What will 200 cords?

37. If a horse eat 2 tons of hay in 30 weeks, what part of a ton will he eat in 1 week?

38. What is the cost of 23 yard?

39. What is the cost of 31 dollar a gallon?

yds. of cloth at of a dollar a

gallons of molasses at of a

40. A grocer drew from a cask containing 31 gallons, of its contents. Now how much did he draw out? How much remained?

SECTION X.

THE LEAST COMMON MULTIPLE.

The method stated in the foregoing section for finding the smallest common denominator, serves to introduce a topic which requires some more extended and careful study.

It often becomes desirable to ascertain, with respect to several numbers, what number there is which contains them all in itself as factors. A number which contains another number as a factor of itself is a multiple of that number. Thus 6 is a multiple of 2, and also of 3. A number which contains several numbers as factors of itself, is a common multiple of those numbers. Thus 12 is a common multiple of 2 and 3.

The smallest number which contains several numbers as factors of itself, is the least common multiple of those numbers. Thus, though 12 is a common multiple of 2 and 3, it is not the least common multiple; for 6 contains them both as its factors; 6 is therefore a smaller common multiple of 2 and 3 than 12 is; and as no number smaller than 6 does contain 2 and 3 as its factors, 6 is the smallest common multiple of 2 and 3.

Suppose now we wish to find the smallest common multiple of 3 and 5. The number, it is clear, must he a certain number of 3s, and also a certain number of 5s. Now by multiplying 3 and 5 together we evidently obtain such a number; for it will be 3 times 5, and it will be 5 times 3. Multiplying the two numbers together then, will always give their common multiple. The next question is, will this product of the two numbers be their least common multiple? This will depend on the character of the numbers. If the numbers are prime to each other their product will be their least common multiple. For example, in the numbers 3 and 5, if we take any number of 5s less than 3, as 3×5, the factor 3 has disappeared, and the number is no longer a multiple of 3. If we take any number of 3's less than 5, as 4X3, the factor 5 has disappeared, and the number is no longer multiple of 5. The product, therefore, of numbers prime to each other, is their least common multiple. In the above example, the numbers of 3 and 5 were prime in themselves, and not merely prime to each other. To make the principle more clear, we will

take two numbers that are not prime in themselves, but are only prime to each other.

What is the least common multiple of 8 and 9? Multiplying them together we have 72. 72 is, then, a common multiple of 8 and 9. The question is, is it their smallest common multiple? Writing the numbers with their factors they are 2X2X2 and 3X3. Now if we erase one of the 2's we have no longer the factors of 8, and the product of the factors will not be divisible by 8. In the same way, if we erase one of the 3's the product will not be divisible by 9.

If, then, the numbers are either prime, or prime to each other, the product is their least common multiple.

Next let us inquire, what is the least common multiple of 4 and 6? Their product is 24, but this is evidently not their least common multiple, for 12 contains both 4 and 6 as factors. To show why it is, that in this case, something less than the product of the numbers is their least common multiple, we will express each by its factors, thus, 2×2, 2×3. Now it is clear that any number of times which you take 2X2 as a factor will be a multiple of 2×2. If then we throw out the 2 in the 2×3, and multiply by the remaining 3, the product will be a multiple of 2X2, or 4. Looking now at the 2×3, or 6, it is evident that any number of times which you may take that as a factor will be a multiple of 2X3. But the 2 we may take from the 2×2, throwing away that in the 2X3; this leaves us to multiply the 2×3 by 2; as we before multiplied the 2×2 by 3, making 12 as the least common multiple. The rule, therefore, is: Retain of each prime factor the highest power which appears in any of the given numbers; erase the rest, and multiply together what then remain.

Find the least common multiple of 8, 24 and 36. Expressed by the factors they are 2×2×2. 2×2×2×3. 2X2X3X3. Now 2X2X2 is common to 8 and 24; it may be thrown out of the latter, leaving only 3. Examining again you observe that 2X2 is common to 8 and 36; we throw this out of 36, leaving 3×3. Finally 3, we find, is common to 24 and 36; throwing this out of 24, we find the numbers appear as follows: 2X2X2. 2×2×2×3. 2×2×3×3.

These multiplied together give for the least common multiple, 72. This conforms to the rule; for 2×2×2 is the highest power of the factor 2, and 3X3 of the factor 3. What is the

least common multiple of 24, 60 and 100? These factors are 2×2×2×3; 2×2×3×5; 2×2×5×5. We see that 2X2 is common to them all; expunge it in the second and third number. Next, 3 is common to the 1st and 2d; expunge it in the 2d. Lastly, 5 is common to the 2d and 3d; expunge it in the 2d, and the numbers will stand, 2×2×2×3. 2×2×3×5. 2×2×5×5. These multiplied together, give 600.

To multiply these most easily, first take 2×2×5×5=100; then the remaining factors, 2×3, multiplied by 100, give 600. What is the least common multiple of 24, 40 and 72? What is the least common multiple of 18, 54, 81? What is the least common multiple of 15, 4, 7? of 15, 40, 27? of 16, 14, 6? of 60, 12, 18?

From the foregoing reasoning and examples you will perceive that the least common multiple of several numbers is the product of all their prime factors, each taken in the highest power in which it appears in any of the numbers.

SECTION XI.

PRACTICAL QUESTIONS.

1. What part of a shilling is 1 penny? 2 pence? 3 pence? 4 pence? 5 pence? 6 pence? 7 pence ?

2. What part of a penny are 2 farthings? 3 farthings? 4 farthings? 5 farthings? 6 farthings? 8 farthings?

3. What part of a shilling is 1 farthing? 2 farthings? 3 farthings?

What part of a shilling is 1 penny and 1 farthing? 1 penny, 2 farthings? 3d 3 qrs.? 4d 2 qrs.? 6d 1 qr.? 9d 2 qrs.? 4. What part of a pound is 1 shilling? 2 s.? 3 s.? 5 s.? 18. 1d.? 2s. 1d.? 4s. 3d.? 5s. 6d.? 7s. 9d.? 3s. 8d.? 5. What part of a pound is 1 farthing? 2 qrs.? 3 qrs.? 2d. 3 qrs. ? 5d. 2 qrs.? 1s. 1d. 1 qr.? 6s. 7d. 3 qrs. ? 6. What part of a pound avoirdupois is 2 oz.? 8 oz.? 4 oz.? 5 oz.? 6 oz.? 7 oz.? 8 oz.? 9 oz.? 10 oz.?