Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[blocks in formation]

Ex. 49. In this example the multiplier consists of 11, 4, and 12. I multiply first by 12, putting the right hand figure 8 of the product under the units 2. I next multiply by 4, putting the right hand figure 6 of the product under the said multiplier 4. Lastly, I multiply by 11, putting the right hand figure 4 of the product under the right hand 1 of the 11. And note, that always in multiplying by a double figure, the units place of the product must stand under the right hand figure of such multiplier.

Ex. 50. Here the multipliers are 12, 12, and 11. The truth of these operations may be proved by multiplying by every figure singly (Art. 31.), or by changing the places of the factors, viz. multiplying the multiplier by the multiplicand.

35. When the multiplier is a composite number, that is, the product of two or more numbers in the table.

RULE. Multiply by one of the component parts, and multiply the product by another, and this last product by another, and so on, when there are several component parts; but this rule is seldom applied when the multiplier is found to consist of more than two parts, or three at most P.

The operations may be proved by Art. 31.

• A number which is the product of two or more numbers (each greater than unity) is called a composite number; and the numbers of which it is the product are called its component parts. Thus 45 is a composite number, the component parts of which are 9 and 5; for 9 × 5 = 45. The terms composite and component are derived from the Latin con, with, and pono, to place.

P Suppose it were required to multiply 234 by 14: by Art. 28, if I add 234 taken successively 7 times, the sum will be the same as 234 multiplied by 7. Now if I double this sum, or multiply it by 2, the result will be the same as though I had taken down 234 14 times, and added the whole together; that is, 234 multiplied by 14 is the same as if it were multiplied by 7 and the product multiplied by 2. Again, if 234 be multiplied by 2, and the product be taken 7 times, the result will evidently be the same as it would had I taken the given number 14 times; and the same thing may be shewn in every similar case. The truth of the rule may likewise be proved by Art. 31.

51. Multiply 63958 by 24.

Here we have a multiplier 24, which may be divided into two component parts a variety of ways; thus 3 X 8, or 4 × 6, or 2× 12; and these may be taken in a different order, thus 8 X 3, 6 × 4, and 12 × 2; wherefore the operation may be performed, according to this rule, six different ways: thus,

[blocks in formation]

52. Multiply 52794 by 21; viz. 3×7 and 7×3. Product 1108674.

53. Multiply 28537 by 35; viz. 5x7 and 7x5. Product 998795.

54. Multiply 81764 by 48; viz. 6×9...8×6... 4 × 12, and 12 x 4. Product 3924672.

55. Multiply 39468 by 132; viz. 11 x 12 and 12 × 11. Product 5209776.

56. Multiply 13896 by 1728; viz. 12 x 12 x 12. Product 2401228S.

36. When the multiplier is any number between 12 and 20, the operation may be performed in one line.

RULE. Multiply by the right hand figure, and (besides what you carry) to each particular product add the figure which stands next on the right (in the multiplicand) of that which you multiplied; if the last product be a single figure, the left hand

figure of the multiplicand must be placed on its left; but if it be a double figure, the left hand figure of the multiplicand must be added to its last, or left hand figure. The operations to be proved by Art. 31.

57.

Multiply 351482
By
Prod. 4569266

Proof.

13 5 4
2

Explanation.

I say 3 times 2 are 6, and put it down; then 3 times 8 are 24 and 2 (the right hand figure) 26, put down 6 and carry 2; then 3 times 4 are 12 and 2 carried 14 and 8 (the right hand figure) 22; put down 2-and carry 2; 3 times 1 are 3 and 2 carried 5 and 4 (the right hand figure) 9, put it down; 3 times 5 are 15 and 1 (the right hand figure) 16, put down 6 and carry 1; lastly, 3 times 3 are 9 and 1 carried 10 and 5 (the right hand figure) 15; put down 5, and to the I carried add the last figure 3; this makes 4, which I put down in the last place.

[blocks in formation]

PROMISCUOUS EXAMPLES FOR PRACTICE.

66. Multiply 371948 by 28 three ways. (Art. 35, 31.) Product 10414544.

67. Multiply 185974 by 14 four ways. (Art. 31, 35, 30.) Product 2603636.

68. Multiply 671612 by 114 two ways. (Art. 31, 34.) Product 76563768.

69. Multiply 230605 by 819000. Product 188865495000. 70. Multiply 128121 by 72001. Product 9224840121. 71. Multiply 241643 by 1212 two ways. Product 292871316.

In this rule the operation of adding continually the back figure amounts exactly to the same as though the top line were taken down one place to the left and added; which is the process directed in Art. 33, by that article it is recommended that each of these examples should be proved.

72. Multiply five million forty-six thousand and one, by four thousand and eight. Product 20224372008.

73. Multiply eight hundred and seventy thousand four hundred and ten, by two thousand and fifty. Product 1784340500. 74. Multiply nine thousand eight hundred and seven, by nine hundred and eight. Product 8904756.

75. Find 42378 × 100, and 4237800 × 30000.

76. Find 385746 × 463, and 178600398 × 1200.

77. Find 3526 × 2534, and 8250704 × 301.

78. Find 10802 × 10203, and 12345 × 6789.

79. A beggar collects 478 halfpence in a week; how many halfpence is that a year? Ans. 24856.

SO. There are 12 signs in the ecliptic, and every sign contains 30 degrees; how many degrees does the ecliptic measure ? Ans. 360.

81. A clerk calls at 14 places, and receives at each 1201.; how many pounds does he carry home? Ans. 1680.

82. The workmen on an estate eat 2167 penny cakes in a day; will be sufficient to serve them 128 days? Answer

how many 277376.

DIVISION.

37. Simple Division teaches to find how often one whole number, called the divisor, is contained in another whole number, called the dividend, or to divide a whole number into any proposed number of equal parts; and is a short method of performing continual subtraction'.

The number arising from the operation is called the quotient; it shews how often the divisor is contained in the dividend, that is, into how many equal parts the dividend is divided.

* The term Division comes from the Latin divido to divide, distribute, or part ; and quotient from quoties, how many times; also remainder from remaneo, to tarry behind.

This rule, like all the former, is derivable immediately from Notation. Thus, to divide 12 by 5, resolve the 12 into its constituent units; then count off as many fives from these as you can, putting a stroke between the divisions; there will then be as many fives in 12 as there are divisions; thus, 1.1.1.1.1. | 1.1.1.1.1.1.1. here are two complete divisions, and 2 units over, therefore there are 2 fives in 12 and 2 remainder; and the like may be shewn in every instance under this rule.

If any thing be over after the division is performed, it is called the remainder.

The mark for division is; it is named by, and shews that the number standing before the sign is to be divided by the number which follows it.

37. B. When the divisor does not exceed 12.

RULE I. Write down the dividend with a small curve line at each end, draw a line below, and place the divisor on the left hand of the dividend.

II. Find how often the divisor is contained in the first or left hand figure of the dividend, or (if it be not contained in that) in the two or (if necessary) three first figures.

III. Set the quotient, or number denoting how many times, below the number divided.

IV. If there be any remainder (it will be less than the divisor,) carry as many tens to the next figure as there are ones in the remainder.

V. Divide the sum as before, set down the quotient underneath, carry tens for the units in the remainder to the next figure, divide, and so on until the work is finished, and if there be a remainder at last it may be placed to the right over the divisor, with a small line between them'.

"

* We here suppose the dividend resolved into parts or denominations; then, beginning at the superior denomination, we find by trials how many times the divisor is contained in it; if there be a remainder, we increase the next inferior denomination by it, observing (according to the established principles of Notation) that every unit of the superior becomes ten of the next inferior; in this manner we proceed through all the parts or denominations in the dividend. Thus let 963 be divided by 3; here the dividend resolved into its constituent denominations is 900+60 +3; now there are 300 threes in 900, 20 threes in 60, and 1 three in 3; therefore the quotient will be 300 +20+1, or 321. Let 573 be divided by 4; this number resolved into parts, of which each of the superior denominations is a multiple of 4, will be 400 +160 + 13, each of which divided by 4, we obtain 100, 40, and 3 for the quotients, with 1 remaining from the last division; therefore 100+40 +3 or 143 is the quotient, and 1 the remainder. We can easily prove that these are respectively the true quotients; for (since multiplication and division are converse rules, it follows that) the quotient multiplied by the divisor, with the remainder added in, will give the dividend; thus, 321 × 3=963, and 143 X 4+1=573, which are the proposed: dividends; and since this is the mode of operation prescribed in the rule, it is shewn to be right.

[ocr errors]
« ΠροηγούμενηΣυνέχεια »