Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/595

Rh ARITHMETIC 535 16) 12 oz. 28) 8-75 lb. 4) -3125 qrs. 2,0)1,3-078125 cwt. 65390625 tons. reduced form, without actually performing the reduction, and divide the numerator of the complex fraction so obtained by the denominator. Thus, to find the fraction of 46^1. that f- of 46f guineas is, both terms may be ex pressed as shillings, | x ^~ x ^, and 2 ^Q 1 x ~; whence, dividing the former by the latter, the required fraction is found to be ^. In reducing a compound quantity to a decimal of a higher denomination, it is generally best to proceed by duccessive divisions, beginning with the lowest denomination, and inserting the others as they occur, as in the accompanying example, where 13 cwt. 8 Ib 12 oz. is found to be -65390625 of a ton. It is to be noted that the integers here are inserted from the data, and the decimals obtained by division. The form, though somewhat incon gruous, is convenient. &amp;gt;actice. 41. Practice is the name given to a method of calculat ing prices from certain rates being aliquot parts (that is, exact measures) of other rates. The basis of this species of calculation is generally 1, and the given price is broken up into portions, the first of which is an aliquot part of a pound, and each of the others an aliquot part of some one before it. By the method adopted we virtually multiply the different aliquot parts in succession by the number of articles given. The sum of the result is the total price. ortion 42. Proportion. When the first of four quantities is the same multiple or the same fraction of the second that the third is of the fourth, the first is said to have the same ratio to the second that the third has to the fourth, and the four quantities are said to be proportionals, or in pro portion. Thus, since 30 is of 36, and 50 is of 60, the numbers 30, 36, 50, and 60 form a proportion. The proportion is written 30 : 36 = 50 : 60, or, more com monly, 30 : 36 :: 50 : 60. This is read 30 is to 36 as 50 is to 60. It may also be written ^ = |-. The ratio of two numbers is thus equivalent to the frac tion that the one is of the other; and a fraction may therefore be defined as the ratio of the numerator to the denominator. Ratio is a mere abstract relation between two numbers, or between two concrete quantities of the same kind. The ratio of 4s. to 6s., or of 48d. to 72d., is not shillings or pence, but the abstract fraction. If two quantities of one kind be proportional to two of another kind, when any three are given, the fourth can be found. If, for instance, we know the value of any quan tity of goods, we can determine the value of any other given quantity, or the quantity that has any assigned value, it being always understood that the rate is the same in both cases ; that is, that were the quantity doubled, trebled, halved, &c., so also would the value be, or, in other words, that the quantity is proportional to the value. Questions of this sort occur with very great frequency in practical arithmetic. The three quantities given are usually arranged as the first three terms of a proportion, whence the fourth term is found. The rule by which we proceed in such cases of Simple Proportion, as it is called, has often the name given to it of the Pule of Three. It is as follows: Of the three quantities given, set that down for the third term which is of the same kind as what is required. Consider whether the amount to be found will be greater or less than this third term ; if greater, make the greater of the two remain ing quantities the second term, and the other the first term ; but if less, put the less term second, and the greater first. Having thus arranged or &quot;stated&quot; the three terms of the proportion, multiply the second and third together, and divide the product by the first. The first and second terms must be reduced to the same denomination, and it will often be convenient to reduce the third term to the lowest denomination contained in it. Ex. 1. If 54 yards of cloth cost 63s., what will 30 yards cost at the same rate 1 Stating by the rule, we have 54 yards : 30 yards :: 63s., ,, n , 30x63 -_ whence the fourth term is = 3os. 54 This process is to be explained by the consideration that, since the rate of price is the same for both quantities, the one price must be the same fraction of the other that the one ... . ,., , ,, ,, ,, 54yds. 63s. quantity is of the other : that is, rr-^-j- = ^ 30 yds. the price required 30 the number of shillings required r 5l = 633. - Multiplying each of these equal fractions by 63, we have f x 63 = 35, tho number of shillings required, as above. The first and second terms are to be stated according to the rule, because their ratio is equal to that of the third and fourth, and must therefore be a proper or an improper fraction, according as the other is so. After the propor tion is stated, and the terms reduced, any common factor may be removed from the first and second terms, or from the first and third ; for, as will appear from the working above, this is virtually reducing a fractional expression to lower terms. Ex. 2. A bankrupt, whose debts amount to 1275, pays 14s. 6d. in the pound. What do his creditors lose 1 ? Since 14s. 6d. is paid, &quot;there is 5s. 6d. loss for every pound of debt, and the question is If 1 give 5s. 6d. loss, what will 1275 give 1 ? The &quot;stating,&quot; therefore, is 1 =1275 ::5s. 6d., and the result, 350, 12s. 6d. Par ticular care must be taken, when all the terms are money, as here, that the first and second be of the same kind. In this instance these terms are debt; and the third term, loss, corresponds to the term required. Ex. 3. If 91 men could perform a piece of work in 78 days, in what time could 21 men do it, working at the same rate 1 Here, if the number of men were doubled, trebled, halved, &c., the time required would be one-half, one-third, double, &c., the given time; or as the former is increased the latter is diminished in the same proportion, and vice versa. The time in this case is said to be inversely pro portional to the number of men. We have then as equal 21 men 78 days that is, the &quot;stating&quot; of 91 men days required the proportion is 21 men : 91 men :: 78 days, according to the rule, and the result 338 days. The process may also be explained thus: The work will be 91 x 78 times what 1 man can do in 1 day, or times what 21 men can 2i 1 91 x 78 do in 1 day, i.e., it will be days work for 21 men. 43. Compound Proportion. The ratio of two quantities Compound frequently depends on a combination of other ratios. If, r r P or tioa for instance, we have to compare the times required for building two walls, one twice the length, twice the height, and twice the thickness of the other, the men employed on the former being half the number employed on the other, and the day half the length, each of these separate conditions implies double the number of days. Each con dition gives the ratio 1 : 2, and the result must correspond to the product of all the ratios, that is, it is 1 : 32. This is an instance of what is called compound proportion. In such cases set down for the third term the quantity which is of the same kind as that required. State each proportion as though it alone had to be considered, writing