29. 3289943865410010432, 30. 784363254871-99834369 After the pupil has gone through all the foregoing examples in Long Division, he should be taught the ITALIAN METHOD, as it is usually called, of which the following is an example, worked at length; and as the Italian method is so much neater, and with practice full as easy, and taking up only half the space, it is recommended that the learner should repeat the former examples by this mode of operation. Ex. Divide 6452800 by 765. 765)6452800(8435 the number of digits in the quotient will be equal to the number of dots in the dividend. 2. Every remainder must be less than the divisor; for, if it be equal to or greater than the divisor, the quotient figure which has produced the result is too small. 3. When the product of the divisor by the quotient figure is greater than the particular part of the dividend used, the quotient figure is too large. ILLUSTRATION.-I find the divisor is contained 8 times in the first four figures, I accordingly put 8 in the quotient, and multiply in this manner, 8 times 5 are 40, and put down as a remainder 2, the difference between 0 the units place of the number gained by multiplication, and the figure in dividend, from which it was to be subtracted, carry 4: 8 times 6 are 48 and 4 are 52, the difference now between the 5 in the dividend and the 2 in 52 is 3, which I put down and carry 5: 8 times 7 are 56 and 5 are 61, between which and 64 is 3, which I put down. Bring down the next figure 8, and proceed: the divisor will now go 4 times, I put 4 in the quotient, and say 4 times 5 are 20, put down 8 and carry 2 4 times 6 are 24 and 2 are 26, the difference between 26 and 32 are 6, which I put down and carry 3: 4 times are 28 and 3 are 31, which taken from 33 leaves 2. Bring down 0, the divisor now is contained three times in the dividend, the 3 I put in the quotient, and say 3 times 5 are 15, this taken from 20 leaves 5, which put down and carry 2: 3 times & are 18 and 2 are 20, which taken from 28 leaves 8, carry 2: 3 times 7 are 21 and 2 are 23, which taken from 26 leaves 3. Bring down the other 0, the quotient is now 5, with which proceed as before. 2. MISCELLANEOUS EXAMPLES. Ex. 1. Divide fifty millions by four thousand and seventy-nine. The planet Mercury goes round the sun in 88 days, which is the length of her year, how many years of Mercury would make 50 of our years, supposing each year contained exactly 365 days? 3. It is estimated that there are a thousand millions of inhabitants in the known world: if one thirty-third of this number die annually, how many deaths are there in a year? 4. The national debt, at present, cannot be less than five hundred millions sterling: how long would that be in paying off, at the rate of two millions and twenty-five pounds per annum? 5. The taxes annually collected amount to full thirty-three millions of pounds: how many poor families of six persons each would that sum support, supposing the annual expenses of the father and mother to be 20l., and of each child 7 l. ? 6. My friend is to set sail to Jamaica on the first of March, 1812; the distance is reckoned to be 3984 miles: at what rate will he go, supposing he reaches the Island on the 10th day of April, that is, in 41 days? 7. What is the difference between the 12th part of 20,100, and the 5th part of 9110? 8. The prize of 30,000l. of the last Lottery became the property of 15 persons: how much was each person's share, after they had allowed 750l. to the office-keeper for prompt payment? 9. The sum of two numbers is 1440, the lesser is 48: what is their difference, product, and quotient? 10. The crew of a ship, amounting to 124 men, have to receive, as prize-money, 1890l. ; but as they are to be paid off, they determine to make their commander and boatswain a present, the one of a piece of plate, value 251.; the other of a whistle, which is to cost 51.: how much will each receive after these deductions are made? 11. In all parts of the world a cubical foot of water weighs 1000 ounces how many pounds are there, supposing 16 ounces make a pound? 12. A cubical foot of air weighs one ounce and a quarter, how many pounds avoirdupois of air does a room contain, which' is 10 feet high, 14 feet wide, and 16 feet long?* 13. Hydrogen gas, or, as it was formerly called, inflammable air, that is, the gas with which balloons are filled, is full nine times lighter than the common air which we breathe: how much less would a balloon, containing 27,000 cubical feet, weigh if filled with hydrogen gas, than if filled with common air?+ 14. At what rate per hour and per minute does a place on the equator move, supposing the great circle of the earth to be 25,000 miles, and the earth to turn on its axis exactly in 24 hours? NOTES. * The number of cubical feet is obtained by multiplying the height, width, and length together, or 10 X 14 X 16, and the product mul tiplied by 1 gives the number of ounces, which, divided by 16, is the answer, viz. 175 pounds avoirdupois. This circumstance cannot fail of exciting surprize, to conceive that in a moderate sized room, the air, which is invisible and scarcely observed to exist, should be known to weigh more than three half-hundred weights. The answer will be found to be 1875 lb.; of course the balloon would ascend, with several persons in its boat; because it will ascend, when the balloon and persons are together, lighter than an equal bulk. of common air, REDUCTION. REDUCTION is the method of converting numbers from one name, or denomination, to another of the same value; and it is divided into Reduction descending, and Reduction ascending. When numbers of a higher denomination are to be brought to a lower, it is called Reduction descending, and it is performed by Multiplication. When numbers of a lower denomination are to be brought to a higher denomination, it is called Reduction ascending, and is performed by Division. REDUCTION DESCENDING, OR CONVERTING GREAT INTO SMALL. RULE. Multiply the given number by as many of the lower denomination as make one of the higher. Thus, in reducing 557. into shillings, I multiply the 55 by 20, and the answer is 1100 shillings; in both cases the value is the same, that is, 557. is equal to 1100 shillings. REDUCTION ASCENDING, OR CONVERTING SMALL INTO GREAT. RULE. Divide by as many of the lower denomination as make one of the next higher. Thus, in bringing 890 pence into shillings, I divide the number by 12, and the answer is 74 shillings and two pence over.* NOTE. The remainders, if any occur, are always of the same denomina... tion as the respective dividends. denotes a farthing: two farthings, or a halfpenny: and three farthings. 20 683 into farthings. 586 shillings 12 I multiply the 29 by 20, and take in the 6, thus I find 586 shillings are equal to 291. 6s. To reduce the shillings to pence, I multiply the 586 by 12, and take in the 8, which give 7040, the number of pence equal to 291.6s. 8d. I next multiply 7040 by 4, and take in the 2, and find the Answer 28163 farthings. answer is 28163 farthings, equal to the given sum 291. 6s. 83d. 7040 pence Ex. 2. In 28163 farthings how many pounds sterling? 4)28163 2,0)58,6 8d. Answer £.29 683 I divide the 28163 farthings by 4, because 4 farthings make a penny; the answer is 7040, and 3 over, which are farthings, because the remainder is always of the same denomination as the dividend. I next divide the 7040 by 12, and the answer is 586 and 8 over, which are pence; and now 586 divided by 20, gives 29 and 6 over, which are shillings; the true answer is therefore 291. 6s. 83d.† Ex. 3. Reduce 28 shillings to pence. 4. Bring 56 pounds into shillings. 5. Reduce 672 pence into farthings. NOTES. * £. s. d. and q. are the initials of the Latin words Libræ, Solidi, Denarii, and Quadrantes; signifying pounds, shillings, pence, and farthings. + Hence it is evident, that the best method of proving the truth of examples in Reduction is, to reverse the operation. It will also be the best method of teaching the pupil to depend on his own exertions, by making him prove each example, and not permitting him to begin the second till he has made the first come right. |