25. What will 175 barrels of flour cost, at 6 dollars per barrel?

26. A man bought 460 pair of boots, at 5 dollars a pair: how much did he pay for the whole?

27. What cost 196 acres of land, at 7 dollars per acre? 28. What cost 340 ploughs, at 8 dollars apiece? 29. What cost 691 hats, at 7 dollars apiece?

30. What cost 865 heifers, at 9 dollars per head? 31. What cost 968 cheeses, at 8 dollars apiece? 32. What cost 1260 sheep, at 7 dollars per head? 33. What cost 9 farms, at 2365 dollars apiece? 41. When the multiplier contains more than ONE figure. 34. A man sold 23 sleighs, at 54 dollars apiece: how much did he receive for them all?

Suggestion.-Reasoning as before, if 1 sleigh costs 54 dollars, 23 sleighs will cost 23 times as much.

figure 3, by which we are multiplying, and carry the 1 as above. 3 times 5 are 15, and 1 (to carry) makes 16. Next, we multiply by the 2 tens thus: 20 times 4 units are 80 units or 8 tens; or we may simply say 2 times are 8. Set the 8 under the figure 2 by which we are multiplying, that is, in tens' place, because it is tens. 2 times 5 are 10. Finally, adding these two products together as they stand, units to units, tens to tens, &c., we have 1242 dollars, which is the whole product required.

Note. When the multiplier contains more than one figure, the several products of the multiplicand into the separate figures of the multiplier, are called partial products.

35. Multiply 45 by 36, and prove the operation.

36. What is the product of 234 multiplied by 165?

Suggestion.-Proceed in the same man

ner as when the multiplier contains but two figures, remembering to place the right hand figure of each partial product directly under the figure by which you multiply.

Operation.

234

165

1170

1404

234

38610 Ans

37. What is the product of 326 multiplied by 205 ? Operation.

Suggestion.-Since multiplying by a cipher produces nothing, in the operation we omit the 0 in the multiplier. Thus, having multiplied by the 5 units, we next multiply by the 2 hundreds, and place. the first figure of this partial product under the figure by which we are multiplying.

326

205 1630

652

66830 Ans.

42. From the preceding illustrations and principles we derive the following

GENERAL RULE FOR MULTIPLICATION.

I. Write the multiplier under the multiplicand, units under units, tens under tens, &c.

II. When the multiplier contains but ONE figure, begin with the units, and multiply each figure of the multiplicand by the multiplier, setting down the result and carrying as in addition. (Art. 23.)

III. If the multiplier contains MORE than one figure, multiply each figure of the multiplicand by each figure of the multiplier separately, and write the first figure of each partial product under the figure by which you are multiplying.

Finally, add the several partial products together, and the sum will be the whole product, or answer required.

43. PROOF.-Multiply the multiplier by the multiplicand, and if the second result is the same as the first, the work is right.

OBS. 1. It is immaterial as to the result which of the factors is taken for the multiplier. (Art. 38.) But it is more convenient and therefore customary to place the larger number for the multiplicand and the smaller for the multiplier. Thus, it is easier to multiply 254672381 by 7, than it is to multiply 7 by 254672381, but the product will be the same.

2. Multiplication may also be proved by division, and by casting out the nines; but neither of these methods can be explained here without anticipating principles belonging to division, with which the learner is supposed as yet to be unacquainted.

QUEST.-42. How do you write numbers for multiplication? When the multiplier contains but one figure, how do you proceed? When the multiplier contains more than one figure, how proceed? 41. Note. What is meant by partial products? What is to be done with the partial products? 43 How is multiplication proved ?

EXAMPLES FOR PRACTICE.

1. Multiply 63 by 4. 2. Multiply 78 by 5. 3. Multiply 81 by 7. 4. Multiply 97 by 6. 5. Multiply 120 by 7. 6. Multiply 231 by 5. 7. Multiply 446 by 8. 8. Multiply 307 by 9.

9. Multiply 560 by 7.

10. Multiply 46 by 10. 11. Multiply 52 by 11. 12. Multiply 68 by 12. 13. Multiply 84 by 13. 14. Multiply 78 by 15. 15. Multiply 95 by 23. 16. Multiply 129 by 35. 17. Multiply 293 by 42. 18. Multiply 461 by 55.

19. If 1 barrel of flour costs 9 dollars, how much will 38 barrels cost?

20. If 1 apple-tree bears 14 bushels of apples, how many bushels will 24 trees bear?

21. In 1 foot there are 12 inches: how many inches are there in 28 feet?

22. In 1 pound there are 20 shillings: how many shillings are there in 31 pounds?

23. What will 17 cows cost, at 23 dollars apiece? 24. What will 25 tons of hay cost, at 19 dollars per ton? 25. What will 37 sleighs cost, at 43 dollars apiece? 26. What will a drove of 150 sheep come to, at 13 shillings per head?

27. What cost 105 acres of land, at 15 dollars per acre? 28. How much will 135 yards of cloth come to, at 18 shillings per yard?

29. In 1 pound there are 16 ounces: how many ounces are there in 246 pounds?

30. A drover sold 283 oxen, at 38 dollars per head: how much did he receive for them?

31. If you walk 22 miles per day, how far will you walk in 305 days?

32. In one day there are 24 hours: how many hours are there in 365 days?

34. Multiply 6207 by 235. 36. Mult. 25271 by 2579.

33. Multiply 2345 by 175. 35. Mult. 10645 by 1262. 37. Mult. 162537 by 21268. 38. Mult. 425231 by 30765. 27.* What will 15 caps cost at 18 shillings a piece?

Analysis. The multiplier 15 is a composite number, the factors of which are 5 and 3. That is, 15=5 × 3. We first multiply the multiplicand by the factor 5, and this product by 3. The last product is the answer. Hence,

Operation.

18

5

90

3

270 shill.

44. To multiply by a composite number. Multiply the multiplicand by one of the factors of the multiplier, and this product by another; and so on till you have multiplied by all the factors. The last product will be the answer.

OBS. 1.-A composite number is one which is produced by multiplying two or more factors together. Thus, 14=7 × 2; 55=11 × 5. 2. Some numbers may be resolved into more than two factors; and also into different sets of factors. Thus, 8=2×2×2=4× 2. 28. What are the factors of 4, 6, 9, 10, 21, 35, 77 ? 29. Name the different sets of factors of 12, 16, 18, 20, 27, 32, 63.

30. Name the different sets of factors of 24, 30, 36, 48,

Ans. 8680.

60, 72, 100.

31. Mult. 248 by 35 using the factors of 35.

QUEST.-44. How do you multiply by a composite number? is a composite number? 45. How multiply by 10, 100, 1000, &c.

Obs. What