long division for short division, which you should always do whenever you can. * Reduce 3275 lb. avoirdupois to cwts. By the table we see that 112 lb. make one cwt.; it is also easy to see that 112 will divide by 2; the quotient is 56, which is 7 × 8; therefore, 112 = 2 × 7 × 8; so that we may here use short division, employing these three factors for divisors, as in the margin. Instead of the divisors 2, 7, 8, we might have used 4, 4, 7, as is obvious.t 2)3275 7)1637 1lb. 8) 233 12 lb. 29 14 lb. 29 cwt. 27 lb. NOTE. You may sometimes have to divide by a number having a fraction joined to it, as, for instance, by 5, in order to reduce yards to perches. In this case, the best way is to divide twice the dividend by twice the divisor, that is, by 11. If the fraction in the divisor be a quarter or three quarters, instead of a half, then you should reduce both dividend and divisor to quarters before you begin, multiplying the divisor, without the fraction, by 4, and taking in the odd quarter or quarters. You must observe, however, that the remainder you get must be divided by 2, if you are dealing with halves, and by 4, if you are dealing with quarters, in order that you may obtain the proper remainder, in the same denomination as the dividend. Thus, suppose it were required to divide 37810 by 5, and by 304, respectively, you should work as follows: Quotient 6874...3, half the remainder. * A table of factors, suitable for short division, of all numbers up to 10000, is given at the end of the book; the arithmetician will find it very useful on many occasions. †The learner will readily see that the above method of getting the final remainder corresponds with what has already been explained at page 35. The only difference is, that there the remainder is expressed in the final denomination, while in Reduction it is made to preserve the original denomination. If in the above example the final result were required in cwts. and fractional parts of a cwt. without lbs., the result of the first division would have been written 1637, the result of the second division 23314, and the result of the third 29.27; that is, 2927 cwt., which is 29 cwt. and the 112th part of 27 cwt.; this 112th part being, of course, 27 lb. Quotient 1249...27, one-fourth of the rem. In the first of these operations, 37810 yards are reduced to poles or perches; in the second, 37810 square yards are reduced to square poles or square perches. The first result is 6874 per. 3 yds.; the second is 1249 sq. per. 27 sq. yds. You divide the remainder by 2, in the first case, because that remainder is halves, and by 4, in the second case, because it is quarters or fourths. Exercises. 1. Reduce 26493 farthings to pounds. 2. Reduce 397024 yards to miles, furlongs, perches, and yards. 3. How many hours are there in 28635 seconds? 4. How many pounds of silver are there in 12875 grains? 5. Reduce 176432 lb. to tons. 6. How many yards are there in 24631 nails? 7. Reduce 42657 square poles to acres. 8. How many square yards are there in 27568 square inches? 9. Reduce 100000 pints to gallons. 10. How many degrees and minutes of a circular arc are there in 132530 seconds? 11. How many cubic yards are there in 100000 cubic inches? 12. It is related by Josephus, that the battering-ram employed by Titus against the walls of Jerusalem weighed 100000 lb. how many tons did it weigh? 13. If an omnibus carry, on the average, 1000 persons weekly at the rate of 3d. each, what are the gross receipts for a year? 14. If all the letters which passed through the Post Office during the week ending Feb. 21, 1851, had only penny stamps on them, what was the cost of the stamps? (See Ex. 16, p. 19.) 15. If a steam-vessel sail across the Atlantic Ocean, a distance of about 3000 miles, at the average rate of 9 miles an hour, in how many days will she perform the voyage? 16. The distance of Plymouth from Adelaide in Australia is estimated at 9080 miles: in how many days would the voyage be performed in a steamer sailing at the average rate of 94 miles an hour? 17. The iron railing round St. Paul's Cathedral weighs 448081 lb. how many tons does it weigh? 18. The duty paid on advertisements in English and Scotch newspapers is 1s. 6d. each: how much was paid for advertisement-duty on all the English newspapers in the year ending Jan. 5, 1851? (See Ex. 14, page 9.) 19. In quick marching, soldiers take 108 steps a minute, each step being about 2 feet 8 inches: at this rate, how long would a regiment be in marching from London to Richmond, a distance of 10 miles? 20. An imperial gallon of distilled water weighs 10 lb. avoirdupois: how many tons of water would the great vat mentioned in Ex. 18, p. 47, hold? (See foot-note, p. 42.) 21. A cubic foot of water weighs very nearly 1000 ounces avoirdupois how many cubic yards are there in the vessel referred to in the last Exercise? 22. In how many days could the above-mentioned vessel be emptied by a tap which discharges half a gallon in a second? 23. What is called a ship-load of coals weighs 949760 lb.; a sack weighs 2 cwt.: how many sacks are there in a ship-load? 24. The number of Electric Telegraph stations now open (Jan. 1, 1852) is 226; of these, constant attendance, day and night, is given at about 70: all are in connection with the central station at Lothbury, in London. The length of lines of communication already completed is upwards of 2500 miles: you can send a short message, for any distance not above 100 miles, for 2s. 6d., which message will be forwarded by other means to the house of the person you send to: what would it cost, for the use of the telegraph, to send a message from one end of England to the other, a distance of about 642400 yards,* and to receive an *The railway distance would, of course, be more than this, as the lines do not run directly north and south. It may interest the learner to be here informed, that the electric wire answer back, which you might do in a few minutes: you will observe, that 2s. 6d. is to be paid for any distance not exceeding 100 miles ?* (38.) ADDITION OF COMPOUND QUANTITIES. RULE 1. Place the quantities to be added together under one another, so as that all in the same column may be of the same denomination. £. s. d. 31 17 4 27 13 2. Add up the first column on the right, that is, the column in which the quantities of lowest denomination are placed; find how many quantities of the next denomination are contained in the sum: put what is over under the colunin, and carry the quotient to the next column. Proceed in this way, from column to column, till all have been added up. 1. Suppose, for example, you had to find the total amount of the following bills: namely, Baker's bill, £31 178. 44d.; Butcher's bill, £27 13s. 8d.; Grocer's bill, £19 0s. 62d.; Tailor's bill, £21 78.; Shoemaker's bill, £11 2s. 9d.; Washerwoman's bill, £8 16s. 3d.; Bookseller's bill, £7 13s. 8d.; and Stationer's bill, 17s. 6d. Then, arranging these sums, as in the margin, putting pounds under pounds, shillings under shillings, pence under pence, and farthings under farthings, you would begin with the column of farthings, and say, 2 and 19 0 21 7 8 63 4 0 11 2 9 8 16 3 £128 8 101 has been extended under the sea, from Dover to Calais; it is embedded in a thick cable, and sunk across the Channel. Occurrences that take place at Paris, 160 miles from Calais, at 7 or 8 o'clock in the evening, are now fully described in print in the London newspapers by 7 o'clock the following morning. Electricity brings the news to London, delivers it in symbols, which require to be translated into common words; the translation is carried in the ordinary way to the printing-office, the compositors set up the type, the pressmen work off the printed sheets, and have thousands of them ready for the public by 7 o'clock in the morning! We owe this wonderful facility to the genius and industry of Professor Wheatstone, of London. The velocity of the electric current is calculated by this gentleman to be at the rate of at least 288000 miles a second; so that it would travel completely round the world in about the twelfth of a second! * " The most wonderful application of electricity to the purposes of life, is the facility it affords to persons separated by hundreds of miles to 2 are 4, and 3 are 7, and 2 are 9; 9 farthings contain 2 pence, and 1 farthing over; therefore, you put down the 1 farthing, and carry the 2 pence to the column of pence. 2 and 6 are 8, and 8 are 16, and 3 are 19, and 9 are 28, and 6 are 34, and 8 are 42, and 4 are 46; and since 40 pence make 3s. and 4d., 46 pence are 3s. and 10d.: 10 and carry 3. 3 and 7 are 10, and 3 are 13, and 6 are 19, and 2 are 21, and 7 are 28, and 3 are 31, and 7 are 38; then, proceeding downwards, you point to the several ones in the shillings column, on the left, each 1 standing for ten, and say, 48, 58, 68, 78, 88; so that this column amounts to 88 shillings; and since 80 shillings make £4, you put down the 8 shillings, and carry 4 to the column of pounds, the sum of which is 128; so that the total amount of all the bills is £128 8s. 101d. The work of examples in compound addition is all so similar to this, that you cannot require any further explanation to prepare you for the following Exercises, which are chiefly intended to give you practice in the tables. hold instant communication, by night or by day, giving them the power, as it were, to annihilate space, enabling them to consult, admonish, inform, condole with each other, as if they were in the same room; and, having ended their conversation, to turn aside, and one to find himself in London and the other in Edinburgh."-(Sir W. Snow Harris's "Rudimentary Electricity," page 191.) |