1 7. 14 cwt. 1 qr. 9 lb. = 18 9. 128° 45' 52" ; 125 8. 823 m. 7 fur. 21 po. +11 10. 315 d. 17 h. 38 m.: 112 11. 1784 ac. 3 roo. 32 per. • 105. 12. £15 16s. 9d. ; £2 13s. 5d. 13. £89 11s. 7£d. = £7 38. 4.d. 14. £126 78. • £34 18s. 1d. 15. £321 178. 3 d. • £47 68. 9d. 16. 73 cwt. 3 qr. 13 lb. = 5 cwt. 1 qr. 14 lb. 17. 78 d. 18 h. 49 m. ; 5 d. 1 h. 2 m. 18. 79° 13' 46" ; 13° 5' 18" 19. 2 tons 13 cwt. 5 lb. = 3 qr. 17 lb. 20. The mint price of standard gold is £3 178. 10 d. an ounce: what is the value of 1 lb. ? 21.* The amount of money expended for the maintenance of the poor among the 607 Unions of England and Wales for the year ending at Michaelmas, 1851, was £3288192: how much, on the average, was expended by each Union ? 22. A fruiterer offers a market-woman 120 oranges at 3 a penny, and 120 of a better sort at 2 a penny, and refuses to take any less; the woman offers to purchase the whole at 5 for 2d. ; and the man, thinking that this is the same thing, lets her have them: how much did the woman save by this arrangement ? 23. The building of the new Royal Exchange in London cost £400000; and after it was opened by the Queen on Monday, the 28th of October, 1844, the public were admitted to it for three days, when a subscription was made for the widows of four men killed during the progress of the works: the money received was as follows : 4 sovereigns, 1 half-sovereign, 1 crown piece, 88 half-crowns, 992 shillings, 842 sixpences, 142 fourpenny pieces, 5 threepenny pieces, 665 pence, 667 halfpence, and 25 farthings: what was the total amount, and what was each widow's share? 24. If a dozen teaspoons weigh 9 oz. 18 dwt. 20 gr., what is the weight of each spoon? 25. Although a sovereign, when quite new, weighs very * These questions are of a miscellaneous kind; some of them require only the rules for reduction. This mixed character is intentionally given to them, that the learner may be accustomed to work examples without requiring to know what rule they come under. ounce. nearly 1234 grains, yet there are only 113 grains of pure gold in it, the rest is called alloy, and is either of pure copper, or a compound of silver and copper :* for , how many sovereigns was there gold sufficient in the 312500 ounces supposed to have been collected in California during the year 1850 ? 26. Find the value of the mass of gold in last Example, at the rate of only £3 178. 101d. an NOTE. 312500 = 10 x 10 x 5 x 5 x 5 x 5 x 5: see table of factors at the end. 27. If 64 lb. of tea, at 48. 8d. a pound, be mixed with 42 lb. at 4s. 4d. a pound, what per lb. will be the price of the mixture? 28. How much would the money which the Royal Exchange cost weigh in sovereigns, at 123 grains each ;t and how many times as high as the Monument (202 feet) would they reach, if piled one on another, allowing a pile of 16 to reach an inch, which is about the case ? I 29. A bankrupt owes his creditors £2831, and proposes to pay them 138. 2żd. in the pound: how much money must he have to do this? 30. What is the difference in the weight of 100000 sove reigns and 100000 guineas; the weight of a sovereign being 1234 grains, and the weight of a guinea 1297 grains ? 31. From the 31st of December, 1829, to the 15th of Fe bruary, 1831, there were coined at the Mint 2387881 sovereigns: what was the weight of these, and what weight of pure gold was used in the coinage ? $ (See Ex. 25.) * The pound troy, of sterling gold, contains 11 oz. of pure gold, and loz, of alloy; it is coined into 464 sovereigns. + That is, supposing each sovereign had lost, on the average, a quarter of a grain by use. It may be here mentioned, that gold coins are allowed to pass under the mint weight, in consideration of the effects of wear. A sovereign weighing 1224 grains, is considered a legal tender ; but not if it be below this weight. I Of sovereigns much worn by use, about 17 would be required to make an inch : of new sovereigns, 16 would do. § In expressing the weight of gold, it is not usual to employ higher denominations than pounds; when cwts. and qrs. are mentioned in troy weight, cwt. means 100 lb., and qr. means 25 lb. 67 FRACTIONS. (45.) I am now about to explain to you one of the most important parts of arithmetic,—the arithmetic of fractions. Learners generally consider it to be the most difficult part; but I am sure that if you carefully attend to the explanations to be given, you will find the arithmetic of fractions to be quite as easy as the arithmetic of integers. I have been obliged to mention the term fractions already, and to say a little about them; for you see they will force themselves upon our notice at a very early stage of arithmetic. I am now to speak of them more at length, and must begin by defining a fraction; that is, by telling you what a fraction really is. A fraction, strictly speaking, means a part of unit, or 1; thus, one-half, two-thirds, three-fourths, &c., are fractions ; they are represented by the figures of arithmetic, in the following way, namely, i, j, , &c. These are so many examples of the notation of fractions; the number below the short line is always called the denominator, and the number above it the numerator ; because the lower number always makes known to us the denomination of the parts, as to whether they are halves, or thirds, or fourths, &c., and the upper number tells us how many of those parts are meant; that is, it enumerates them. If therefore a fraction such as ☆ were presented to you, you would at once know what was meant by it. Looking at the numerator, you would see that 5 parts were represented, and looking at the denominator to learn what those parts were, you would see that they were sevenths ; you would thus know that the fraction, translated into words, is five-sevenths ; that is, if unit, or 1, were cut up into seven equal parts, five of those parts are represented by You must see that it is very convenient to have a notation for parts of a unit as well as for whole units or integers ; and from what has already been said, you also see that, in strictness, a fraction is always something less than 1. But this strictness is departed from ; the term fraction is extended to quantities greater than 1: thus, }, , , &c., are all called fractions, though each is greater than 1. The first stands for seven-fifths, the second for eight-thirds, the third for elevenfourths, and so on. The meaning of the first is, that if i be cut up into five equal parts, seven of such parts, that is seven times one of those parts, are to be taken ; the meaning of the second is, that if i be divided into three equal parts, eight times one of those parts is to be taken ; and the third fraction ineans, that if i be divided into four equal parts, eleven times one of them is to be taken. Fractions such as these, where each denotes a quantity greater than units, are called improper fractions, the prefix improper reminding us that the strict meaning of the word fraction is departed from, a proper fraction always having the numerator less than the denominator. Even when the numerator is equal to the denominator, the thing is still called a fraction,-an improper fraction, of course; thus, š, , , &c., are all called fractions, though in fact each is only a peculiar manner.of writing down unit, or 1, since three-thirds, four-fourths, &c., each make one whole. There is thus no restriction upon fractional notation ; you may write you please for numerator, and any number you please for denominator, and what you put down will be entitled to be called a fraction. (46.) I think from what has now been said, you will see that fractions are a good deal like those quantities with which you have been occupied in the preceding articles, where both the number and the denomination of the things dealt with are to be considered, the chief difference being merely in the notation. When the things dealt with were pounds, in money, the denomination was expressed by the mark, or symbol, £, written against the number of pounds; when the denomination was ounces, the symbol oz. was employed in the same way; and so on. In like manner in fractions, both number and denomination have to be expressed; but here a different notation is used, the number of the things being written above, and their denomination below a short line of separation ; so that when you are familiar with the notation, you ought to find no more difficulty in the arithmetic of fractions than in the arithmetic of whole quantities of different denominations. (47.) You have already been told (p. 28), that this notation for fractions is also the notation for division ; so that sought to mean not only two-thirds of unit, or 1, but also 2 divided by 3; and that should express, indifferently, either threefourths, or 3 divided by 4; and so on, And it is pretty plain that such is really the case ; for two-thirds, or the third part of 1 taken twice, is obviously the same as the third part of 2 taken once ; that is, 2 divided by 3: also, that threefourths, or the fourth part of 1 taken 3 times, is the same as the fourth part of 3 taken once; that is, 3 divided by 4, and so on. Consequently, whenever you see a fraction, as , , &c., you may read it either five-sevenths, nine-fourths, &c., or 5 divided by 7, 9 divided by 4, &c. Suppose, for instance, you had (that is, three-fourths) of 1 shilling, or 12 pence; then, since one fourth is 3d., three fourths would be 9d., which you see is one fourth of 38., or of 36 pence; that is, 36 pence divided by 4 gives 9d. Again, 1 (that is, ninefourths) of a shilling is 9 times one-fourth, or 9 times 3d., which is 27d., and 98., or 108 pence, divided by 4, is also 27d.; and similarly in all cases: and it is of importance that you keep this fact always in remembrance. (48.) Before proceeding to the arithmetic of fractions, I have only further to add, that 1. A whole number, that is an integer, may be written in the form of a fraction, by merely putting under it 1 for denominator ; thus, 3 may be written i ; 7 may be written 1, and so on. And that any may be written for denominator, provided only the product of that number and the proposed one be written for numerator; thus, if we wish to write 3 in a fractional form, with 5 for denominator, we must write 3 x5, or 15, for numerator, the fraction beings, which is, of course, the same as 3. In like manner, 7=, or =%, or = &c. The numerator and denominator of any fraction are called the terms of the fraction. 2. A number consisting of two parts, one whole and the other fractional, is called a mixed number: thus, 2, 31, 22 11, &c. are all mixed numbers. Such mixed numbers may always be reduced to improper fractions; and, on the other hand, an improper fraction may always be reduced to a mixed number. It will be as well to commence the subject by showing how these reductions are to be made. (49.) To reduce a Mixed Number to an Improper Fraction. RULE. Multiply the whole number by the denominator of the fraction connected with it. Add the product to the numerator, and write the denominator of the fraction underneath, with the short line of separation between, and you will have the improper fraction required. *Thus, if 3be the mixed number, consisting of the integer |