In the annexed example, 6 and 8 are 5 and 5 are 897. 6 480179 617373 548776 .... 61531509.... 3 14, which exceeds 9, by 5; 10, which exceeds 9, by 1; 8, which excess is set opposite to the multiplicand. In the multiplier, in like manner, 8 and 7 are 15; which exceeds 9, by 6, which excess is set opposite to the multiplier. In the product, 6 and 1 are 7 and 5 are 12, which exceeds 9, by 3; 3 and 3 are 6 and 1 are 7 and 5 are 12, which gives an excess of 3, to be set opposite to the product. Then, the product of the two excesses 6 and 8 is 48, the sum of the digits in which is 12, which gives an excess of 3, the same as the excess of the product; and hence we judge the work to be correct. multiplied by 5×9+3; which, by multiplying the several terms, becomes 12X9X5×9+5×9×4+12×9×3+4×3. In this product, the first three terms are evidently divisible by 9, without remainder: consequently, the remainder obtained by dividing the whole product by 9, will be the same as the remainder obtained by dividing 4×3, the product of the excesses of the factors, by 9; and thus the reason of this method of proof is evident. The same property belongs to the digit 32 but it is more convenient in practice to use 9. : 37. Multiply fifty-six millions seven thousand eight hundred and fifty-four by eight millions six hundred thousand nine hundred and seventy-six. Ans. 4514287696065504. 38. Multiply eighty millions seven thousand six hundred by eight millions seven hundred and sixty. Ans. 640121605776000. 39. How many yards of linen are in 759 pieces, each containing 25 yards? Ans. 18975. 40. Sound is known to move about 1130 feet per second; how many feet will it move in 69 seconds? Ans. 77970. 41. During the 50 years between 1700 and 1750, the quantity of linen exported from Ireland, each year, at an average, was four millions of yards; during the next six years, 11796361 yards in each; during the next seven, 17776862 yards per year; and during the succeeding seven years, ending 1770, the average quantity was 20252239 yards annually. Required the whole quantity exported from 1700 till 1770. Ans. 638565684. 42. A boy can point sixteen thousand pins in an hour: how many will he do in six days, supposing he works eleven clear hours in a day? Ans. 1056000. 43. How many miles will a person walk in fifty-five years, supposing he travels, one day with another, six miles, and there are 365 days in a year? Ans. 120450. 44. A man spends 99 cents a day: how much does he spend in 49 years, each consisting of 365 days? Ans. 1770615. 45. Supposing that one acre of land produces 30 bushels of wheat; how many bushels will thirty thousand three hundred and sixty-five produce? Ans. 910950 Questions. What is multiplication? Is the product of any two numbers the same, whichever of them be made the multiplier ? How is the product of any number multiplied by 10, 100, 1000, &c. obtained? Repeat the rule for performing the operation of multiplication. If the multiplier be the product of any two known fac tors, how is the operation performed? DIVISION OF WHOLE NUMBERS. 27. Division, in the primary view of it, is but an abridged method of subtraction. Here we inquire how often one number, called the divisor, may be subtracted from another number, called the dividend. The quotient expresses the number of times that the divisor may be subtracted from the dividend, or is contained in it. Thus, when we divide 96 by 12, the quotient is 8: for we may subtract the divisor 12 from the dividend 96 just 8 times. This might be ascertained by performing the successive subtractions, and reckoning the number of them: but is at once discovered by the multiplication table, which informs us that 96 is equal 8 times 12, and therefore contains 12 in it exactly 8 times. If we divide 103 by 12, it is plain that after subtracting 12 from 103 eight times, there will remain 7; so that the quotient is still 8, but with 7 for a remainder. 28. When one number is contained in another a certain number of times exactly, without leaving any remainder, the former number is said to measure the latter. Thus, 12 measures 96, but does not measure 103. The numbers 8 and 12 measure 24; 8 being contained in it exactly 3 times, and 12 exactly twice. We often express division by writing the dividend above the divisor, with a line interposed between them. Thus, expresses the division of 84 by 7: and the following symbols, 412, express therefore that the quotient of 84 divided by 7 is equal to 12. The symbol also is sometimes employed to express division, the dividend standing on the left hand of it, and the divisor on the right. Thus, 42÷6 is another way of expressing the division of 42 by 6, as well as 43. We shall have such frequent occasion for the signs +, →, ×,÷, and the terms plus and minus, that the young arithmetician cannot too soon become familiar with them. A little patient explanation and illustration will soon make a child as familiar with them as with the Arabic characters; and it is ridiculous to think how many have been deterred from attempting the study of algebra, by the mere formida ble appearance of its outworks, a number of strange symbols and terms which they do not understand. But every thing the most simple is obscure till it is understood; and every term is alike unintelligible, till its meaning is explained. 29. If any quotient be made the divisor of the same dividend, the former divisor will be the new quotient, and the same remainder (if any) as before, Thus, dividing 103 by 12, the quotient is 8, with the remainder 7. Now, if we divide 103 by 8, the quotient must be 12, leaving the same remainder. For the first division shows that the divisor contains 12 eight times, and 7 over; 8 times 12 and 12 times 8 being equal. And thus also it is manifest that if any product be divided by either of the factors, the quotient must be the other factor; and that any dividend may be considered as the product of the divisor and quotient, with the remainder (if any) added: In the view of division which has hitherto been proposed, the divisor must be conceived not greater than the dividend; else it would be absurd to inquire how often it is contained in the dividend. But there is another view of division, closely connected with the former, in which we may easily conceive the division of a smaller number by a greater. When we are called to divide 96, for instance, by 12, we may consider ourselves called to divide 96 into twelve equa parts, and to ascertain the amount of each. The quotient, found as before, is a number of that amount, or the twelfth part of 96. For, since 96 contains in it just 8 twelves, it must contain just 12 eights; and therefore the quotient 8 is the twelfth part of 96. And thus universally, the quotient may be considered as that part, or sub-multiple, of the dividend which is denominated by the divisor, as the divisor may be considered that part, or sub-multiple, of the dividend which is denominated by the quotient. (Hitherto, the divisor is supposed to measure the dividend.) Thus, dividing 64 by 4, the quotient is 16; for, subtracting 16 fours from 64, there is no remainder; therefore, 4 is the sixteenth part of 64; and 16 is the fourth part of 64. Now, though it would be absurd to inquire how often 12 may be subtracted from 7, and therefore any division of 7 by 12 is inconceivable, according to that view; yet it is not absurd to inquire what is the twelfth of 7, or of dividing 7 by 12 according to the latter view. For instance, we might have occasion to divide 7 shillings among 12 persons equally, or into 12 equal shares; and then. it is plain that each person must get the twelfth part of 7 shillings. The quotient or twelfth part of 7, as has been already observed, may be represented by the notation; and the child ought to be familiarized to this notation, previous to his entrance on the doctrine of fractions. Let us now revert to the example of division at the close of 27, the division of 103 by 12. The quotient being 8, but leaving a remainder of 7. Therefore, 8 is not exactly the twelfth part of 103: for if we were dividing 103 dollars equally among 12 persons, after giving each of them 8 dollars, there would be 7 dollars; which 7 dollars we would proceed to divide equally among them; that is, we should give each of them the twelfth part of 7 dollars in addition to the 8 dollars he had received, in order to make the division accurate therefore, the twelfth part of 103 is exactly 8+; or 8 and the twelfth part of 7. And so, whenever there is a remainder on a division, the student should be taught to correct the quotient by annexing to it that remainder divided by the divisor. 7 As to the practical method of performing division, the grounds of it are obvious from § 27. Let us first suppose that the divisor does not exceed 12: for instance, let it be required to divide 5112 by 8. We immediately know from the multiplication table, that 8 may be subtracted 600 times from the dividend, but not 700 times; since 600 times E |