198 



THE POPULAE EDUCATOE. 



and South America, which are connected with each other by the 

 Isthmus of Darienor Panama (pronounced pan-a-mar'). Between 

 these continents, on the eastern side, north of the equator and 

 within the torrid zone, are situated the West Indies, a range of 

 islands stretching in a curved line from the Gulf of Florida to 

 the mouth of the Orinoco. South of Asia, and east of the 

 Arabian Sea, consisting partly of the continent and partly of the 

 islands south of it, are situated the East Indies, lying almost 

 wholly within the torrid zone, and comprehending the penin- 

 sulas of India and Further India, Hindostan within and India 

 beyond the Ganges, with the island of Ceylon and the group of 

 islands denominated the East Indian Archipelago, the Asiatic 

 Archipelago, or Malaysia. Sumatra, Borneo, and Celebes, 

 the principal of these islands, are situated directly under 

 the equator. 



The relative position of the greater part of the places men- 

 tioned in this lesson may be ascertained from an inspection of 

 the figures in the preceding page, or the Map of the World in 

 page 144. Our readers will find it useful, when studying our 

 Lessons in Geography, to make a map of the world on a large 

 scale according to the directions given in the last lesson, and to 

 mark in the position and name of each place, as soon as it 

 occurs for the first time. 



LESSONS 



IN ARITHMETIC. - 



PEACTICE. 



-XXIX. 



8. Definition. Any fraction of a quantity the numerator of 

 which is unity, is called an aliquot part of that quantity. 



Thus 4s. and 6s. 8d. are each aliquot parts of a pound, being 

 respectively | and | of it. 



In finding the value of any given compound quantity from 

 the given value of any other given quantity of the same kind, 

 a convenient form of multiplication, called Practice, is often 

 employed. It depends, as will be seen, upon the principles of 

 fractions and the judicious choice of aliquot parts. 



9. EXAMPLE 1. Find the value of 3589 cwts. at <! 11s. 6|d. 

 per cwt. 



This might be effected in various ways. We might, for 

 instance, reduce the money to farthings, multiply by 3589, and 

 then reduce the result to pounds, shillings, and pence ; or we 

 might reduce the money to the fraction of a pound, and then, 

 multiplying by 3589, reduce the resulting fraction to pounds, 

 shillings, and pence. But we may also evidently obtain a correct 

 result if we divide the whole sum into portions, multiply each 

 of these portions separately by 3589, and then add the results 

 together. This we are able to do, simply by the aid of aliquot 

 parts, as follows : 



3589 cwts. at 1 per cwt. will cost .... 3589 



Since 10s. is , 3589 cwts. at 10s. each will cost -J of 



3589, or 1794 10 Q 



Since Is. is TS of 10s., 3589 cwts. at Is. each will 

 cost iV of the same number at 10s. each, or ^7 of 

 1794 10s., which is 179 9 o 



Since 6d. is i of Is., 3589 cwts. at 6d. each will cost -J- of 



the same number at Is. each, or -J of 179 9s., or 89 14 6 



Since |d. is T \ of 6d., 3589 cwts. at -Jd. each will cost 

 & of the same number at 6d. each, or t^ of 

 89 14s. 6d., which is .... 7 9 'gl 



Since id. is of -Jd., 3589 cwts. at -Jd. each will cost i 

 of the same number at Jd. each, or -J- of 7 9s. 6|d., 

 which is 3 14 9 



Hence 3589 at 1 + 3589 at 10s. + 3589 at Is. + 3589 



at 6d. + 3589 at id. + 3589 at d. will cost . . 5663 17 9| 



The above is the explanation of the process, which may be 

 arranged as follows : 



3589 cwts. would cost at 1 per cwt 

 10s. which is I of 1 



Is. & f 10s. 



6d. I of Is. 



Id. /jof 6d. 



id. i of -Jd. 



8. 



3589 



1794 10 



179 9 



89 14 



7 9 



3 14 



5663 17 



10. If the quantity whose value is to be found, and also the 

 p rice given, be each expressed in various denominations, then a 

 somewhat different method must be adopted. 



For instance, suppose it to be required to find the value of 

 375 cwt. 3 qrs. 21 Ibs. at 4> 14s. 6d. per cwt. 



First find the value of 375 cwt. at ,4 14s. Gd. per cwt. by 

 the previous method. This will be done as follows : 



s. d. 

 375 cwt. would cost at 1 per cwt. . . 375 



Therefore 375 cwt. 3 qrs. 21 Ibs. would cost . 1776 6 1 i 



The fraction being | of a farthing. If, however, the fractional 

 part of the farthing were put in terms of a fraction of a penny, 

 the result would be written .1776 6s. lid. 



11. Sometimes, by inspection, we can see that one or both 

 of the compound quantities which are expressed in different 

 denominations can be simply expressed as a fraction of one of 

 the denominations. This will much simplify the operation. 



EXAMPLE. Find the value of 24 cwt. 1 qr. 9 Ibs. 5^ oz. at 

 .2 5s. 6d. per cwt. 



Here it is readily seen that 1 qr. 9 Ibs. 5| oz. is | of a cwt. 



Hence the question is reduced to finding the value of 24| cwt. 



s. d. 

 24 cwt. would cost at 1 per cwt. 24 



,, ,, ,. 2 

 5s. which is i of 1 

 6d. iV of 5s. 



At 2 5s. 6d. 

 I cwt. would cost 



48 

 600 

 12 



54 12 

 15 



55 7 2 Ans 



12. If both commodity and price are easily expressible by 

 fractions, it will generally be found most convenient to treat 

 the question as in the following 



EXAMPLE. Find the value of 15 cwt. 2 qrs. 7 Ibs. at .1 6s. 8d, 

 per cwt. 



15 cwt. 2 qrs. 7 Ibs. = 15 + + ,V = 15^ cwt. 



1 6s. 8d. = 1|. 

 Hence the required value will be IS^- x 1-J- pounds, 



83 



Or, H x |> = ^| = 20f = 20 15s. 



4 



13. In employing the method of practice, a good deal must be 

 left to the student's judgment as to dividing the compound 

 quantity into separate portions, so that the aliquot parts shall 

 be the most convenient. 



Tables of aliquot parts of 1, of a hundredweight, an acre, 

 etc., are drawn up for the convenience of persons much engaged 

 in calculations ; but the learner had better trust to his memory 

 and knowledge of fractions in solving any question of the kind 

 with which he may be concerned. 



EXERCISE 48. EXAMPLES IN PRACTICE. 

 Find the cost of 









1. 1625 yards at 



2. 1429 yards at 1 



3. 749 yards at 



4. 1689 yards at 



5. 2476 yards at 



6. S13 cwt. at 



7. 9999 tons at 7 



8. 5926 articles at 



9. 1000 articles at 5 



10. 2010 articles at 6 



11. 89 articles at 3 



12. 535 articles at 1 



13. 112 cwt. 1 qr. 17 Ibs. at 8 



s. d. 

 2 8g per yard. 

 2 9 per yard. 

 5 8 per yard. 

 4 10^ per yard. 

 18 6 per yard. 



per cwt. 



per ton. 



each. 



each. 



each. 



each. 



8 8 

 17 6 

 11 8 

 6 



7 



8 10 

 11 5 



5 10 each. 



11 4 per cwt.. 



