49. When the multiplier does not exceed 12. 1. Let it be required to multiply 236 by 4.

ANALYSIS.-It is required to take 236 4 tines. If the entire number is taken 4 times, each order of units must be taken 4 times: hence, the product must contain 24 units, 12 tens, and 8 hundreds; therefore, the product is 944.

It is seen, from the preceding analysis, that,

OPERATION.

236

4

24 units.

12

tens.

hundreds.

8

944 Product.

1. If units be multiplied by units, the unit of the product will be 1.

2. If tens be multiplied by units, the unit of the product will be 1 ten.

3. If hundreds be multiplied by units, the unit of the product will be 1 hundred; and so on:

And since the product of the factors is the same whichever is taken for the multiplier (Art. 48), it follows that,

4. If units of the first order be multiplied by units of a higher order, the units of the product will be the same as that of the higher order.

The operation in the last example may be performed in another way, thus:

ANALYSIS.--Say 4 times 6 are 24: set down the 4, and then say, 4 times 3 are 12, and 2 to carry are 14; set down the 4, and then say, 4 times 2 are 8, and 1 to carry are 9. Set down the 9, and the product is 944 as before.

OPERATION.

The method of carrying is the same as in addition.

236

4

944

9. A merchant sold 104 yards of cotton sheeting at 9 cents a yard: what did he receive for it?

10. A farmer sold 309 sheep at four dollars apiece: how much did he receive?

11. Mrs. Simpkins purchased 149 yards of table linen a two dollars a yard: how much did she pay for it?

12. What is the cost of 2974 pine-apples at 12 cents apiece?

13. What is the cost of 4073 yards of cloth at 7 dollars a yard?

14. What is the cost of a drove of 598 hogs at 11 dollars apiece?

READING RESULTS.

50. Spelling, in multiplication, is naming the two factors which produce the product, as well as the words which indicate the operation; whilst the reading consists in naming only the word which expresses the final result.

ANALYSIS.-In multiplying 8325 by 6, we say,

6 times 5 are 30; then, 6 times 2 are 12 and 3 to carry are 15; 6 times 3 are 18 and 1 to carry are 19; 6 times 8 are 48 and 1 to carry are 49.

OPERATION. 8325

6

49950

This is the spelling. The reading consists in pronouncing only each final word which denotes the result of an operation, thus thirty, fifteen, nineteen, forty-nine.

With a little practice, the pupils will perform the operations mentally, and read with great facility, either separately or in concert in classes.

51. When the multiplier exceeds 12.

1. Multiply 8204 by 603.

49. Explain the multiplication of 236 by 4? What principles are established by this operation?

50. Explain the manner of reading the results in the operations of multiplication?

51. Give the rule for multiplication.

ANALYSIS.-The multiplicand is to be taken 603 times. Taking it 3 times we obtain 24612.

When we come to take it 6 hundreds times, the lowest order of units will be hundreds: hence, 4, the first figure of the product, must be written in the third place.

OPERATION 8204

603

24612

49224

4947012

The

NOTE. The product obtained by multiplying by a single figure of the multiplier, is called a partial product. In the above example there are two partial products, 24612 and 49224. sum of the partial products is equal to the result or product sought: hence, the following

RULE-I. Write the multiplier under the multiplicand, placing units of the same order in the same column.

II. Beginning with the units' figure, multiply the entire multiplicand by each figure of the multiplier, observing to write the first figure of each partial product directly under its multiplier.

III. Add the partial products and their sum will be the product sought.

PROOF.

52. Write the multiplicand in the place of the multiplier and find the product as before. If the two products are the same, the work is supposed to be right.

NOTE. This proof depends on the principle that the product of two numbers is the same whichever is taken for the multiplicand (Art. 48); and also on the fact, that the same error would not be likely to occur in both operations.

2. Multiply 365 by 84; also 37864 by 209.

18. Multiply five thousand nine hundred and sixty-five, by six thousand and nine.

19. Multiply eight hundred and seventy thousand six hundred and fifty-one, by three hundred and seven thousand and four.

20. Multiply four hundred and sixty-two thousand six hundred and nine, by itself.

21. Multiply eight hundred and forty-nine million, six hundred and seven thousand, three hundred and six, by nine hundred thousand, two hundred and four.

22. Multiply 679534 by 9185. | 26. Multiply 50406 by 8050. 23 Multiply 86972 by 1208. 27. Multiply 523972 by 1527. 24. Multiply 1055054 by 570. 28. Multiply 760184 by 1615. 25. Multiply 538362 by 9258. | 29. Multiply 105070 by 3145

CONTRACTIONS IN MULTIPLICATION.

53. Contractions in multiplication are short methods of finding the product when the multiplier is a composite number.

53. What are contractions in multiplication?

CASE I.

Of Components or Factors.

54. A composite number is one that may be produced by the multiplication of two or more numbers, which are called components or factors.

Thus, 2x3=6. Here, 6 is the composite number, and 2 and 3 are its components or factors.

The number, 16=8×2: here 16 is a composite number, and 8 and 2 are the factors. But since 4x4-16, we may also regard 4 and 4 as factors of 16.

Again, 16 8×2, and 8=4x2=2×2×2: hence, 16=2×2×2×2: therefore, 16 has also four equal factors. 1. What are the factors of 8? of 9? of 10? of 12? of 14? of 18? of 24 ?

2. What are the factors of 20 of 21? of 22? of 26 ? of 25? of 30?

3. What are the factors of 36? of 42? of 44 of 49? of 56 of 64? of 72 ?

4. Let it be required to multiply 8 by the composite number 6, of which the factors are 2 and 3.

If we write 6 horizontal lines with 8 units in each, it is evident that the product of 8×6=48 will express the num

ber of units in all the lines.

Let us first connect the lines in sets of two each, as at the right; the number of units in each set will then be expressed by 8x2=16. But there are 3 sets; hence, the number of units in all the sets is 16x3=48.

54. What is a composite number? Is 6 a composite number? What are its components or factors? What are the factors of the composite number 16? What are the factors of the composite number 12? How do you multiply when the multiplier is a composite number?