326 



THE POPULAR EDUCATOB. 



The Emperor of Germany, as president or head of the several 

 states or kingdoms previously enumerated, represents the em- 

 pire in all cases of international law, and in respect of declaring 

 war, making peace, treaties, &c. Notwithstanding, before any 

 declaration of war can be made by him, he must obtain the 

 consent of the Bundesraih, a council of delegates of all the 

 confederate governments. The Emperor is also the commander- 

 in-chief of the whole army and navy in time of war, as he is 

 also in time of peace, except in the cases of the kingdoms of 

 Bavaria and Wurtemburg. 



The Ottoman Empire had formerly very extensive territories 

 in Europe, but these have been largely reduced during the 

 present century, and especially since the Russo-Turkish War. 

 Amongst states which have won their independence from 

 Turkey are Montenegro a small principality of some 1,710 

 square miles, with a population of about 286,000 and 

 Roumania, with an area of 49,262 square miles, and a population 

 of about 5,110,000. Practically independent of Turkey is 

 Bulgaria a principality created by the Treaty of Berlin in 

 1878, with an area of 24,659 square miles, and with 1,859,000 

 inhabitants. Eastern Roumelia tributary to Turkey, and like 

 Bulgaria created by the Berlin Treaty has an area of 13,663 

 square miles, with 751,000 inhabitants. Under the provisions 

 of the same Treaty Greece has acquired a considerable acces- 

 sion of Turkish territory, an International Commission having 

 awarded to her parts of Thessaly and Epirus. 



LESSONS IN ARITHMETIC. XXXIII. 



EULE OF THREE-SINGLE AND DOUBLE (continued). 



8. IN Simple or Single Rule of Three, the method of performing 

 which was explained in the last lesson, it will be found that 

 questions of the following kind often occur : 



EXAMPLE 1. If 8 men can reap 32 acres in 6 days, how 

 many acres can 12 men reap in 15 days ? 



Such questions can always be solved in a manner similar to 

 the following : 



Since 8 men can reap 32 acres in 6 days, 



32 

 1 man acres in 6 days, 



o 



32 

 5-% acres m 1 day ; 



And 1 man 



Therefore, 12 men 



And 12 men 



12 x 



8x6 



12 x 15 x 



32 



32 

 8x6 



acres in 1 day, 

 acres in 15 days. 



And 12 x 15 x -^ = 120 days, the answer, 

 o x o 



EXAMPLE 2. If the carriage of 6 cwt. 3 qrs. for 124 miles 

 eosts 3 4s. 8d., what weight would be carried 93 miles for 

 .1 4s. 3d. ? 



Since 6 cwt. 3 qrs. is carried 124 miles for 3 4s. 8d., or 8/5, 

 Therefore, 6| cwt. 1 mile for 



And 1 cwt. 

 Therefore, 1 cwt. 



124 

 1 mile for 



93 miles for 93 x 



i.e., for 



124 x ,6f ' 



1 4s 3d 

 Hence ^-^ '- cwts. will be carried 93 miles for 1 4e. 3d. 



The answer therefore is 3| cwt., or 3 cwt. 1 qr. 14 Ibs. 



9. Questions of this kind can always be solved by the method 

 given above i.e., by finding what quantity of one kind corre- 

 sponds to one unit of each of the other kinds. Thus we have 

 found, in the first example, how many acres can be reaped by 

 one man in one day. In the second example we have found 

 what is the cost of carrying one cwt. one mile. After this has 

 been done, the process is easy. 



The result, can, however, be always arrived at more simply 

 by means of the following rule, which depends, however, upon 

 an algebraical principle which we cannot explain here. 



10. Double Rule of Three. 



There are five quantities given to find a sixth. Call this 

 sixth quantity x. These six quantities will consist of 3 kinds 

 in pairs. Observe which kind increases with the increase of 



both the other kinds. Then the ratio of the two quantities of 

 this kind will be equal to the ratio (See Lesson XX., Art. 2, 

 Vol. I., page 342) compounded of the ratios of the others. 



11. We will work out the previous examples by this rule. 



EXAMPLE 1. Here the acres increase if the men increase, 

 and if the days increase. 



Hence, the sixth quantity, x, being days, we have - 



32 

 Therefore x 32 x 



x _ 12 x 15 



8x6 ' 

 12 x 15 



= 120 days. 



8x6 



EXAMPLE 2. Here the price increases if the weight increases^ 

 and if the distance increases. 



Hence, the sixth quantity, x, being weight, we have 

 1^5= x j 

 3s 7 o ~6f 124 ; 

 _ 3 x 

 Or = -. 

 8 9 



Therefore x = V cwt. = 3 cwt. 1 qr. 14 Ibs. 

 EXAMPLE 3. If 27 men can do a piece of work in 14 days 

 of 10 hours each, how many hours a day must 24 boys work, in 

 order to complete the same in 45 days, the work of a boy being 

 half that of a man ? 



do the work in 



Therefore 



And therefore 



27 men 

 1 man 



24 boys or 12 men 

 27 x 140 



140 hours ; 



27 x 140 hours ; 



27 x 140 , 



1 2~ tours; 



is the number of hours in the days, which 

 27 x 140 . 



12 x 45 



are such that 45 contain 



And 



27 x 140 



12 



hours, 



hours = 7 hours, the answer. 



12 x 45 



EXAMPLE 4. How long will 20 men take to build a wall 

 feet high, if 11 men require 17 days to build one of the same 

 length, but only 7| feet high ? 



This we will work by the rule. 



Here the amount of wall built increases if the number 0f men 

 is increased, and if the time they work is increased. 



If x be the time required, we have therefore 



10 _ x_ 20 



7-1 "" 17 11 



Therefore x = 



10 x 2 



11 x 17 187 ... , 



= == = 12ft days. 



15 20 15 



EXERCIBE 52. EXAMPLES IN DOUBLE RULE OF THREE. 



1. If 12 horses can plough 11 acres in 5 days, how many horses can 

 plough 33 acres in 10 days ? 



2. If 40 gallons of water last 20 persons 5 days, how many gallons 

 will 9 persons drink in a year ? 



3. If 16 labourers earn 15 12s. in 18 days, how many labourers 

 will aarn 35 2s. in 24 days ? 



4. If 24 men can saw 90 loads of wood in 6 days of 9 hours each, 

 how many loads can 8 men saw in 36 days of 12 hours each ? 



5. If 6 men can make 120 pairs of boots in 20 days of 8 hours each, 

 how many days will it take 12 men to make 360 pairs, working 10 

 hours a day ? 



6. If 12 men can build a wall 30 feet long, 6 feet high, and 4 feet 

 thick in 18 days, how long will it take 36 men to build a wall 360 feet 

 long, 8 feet high, and 6 feet thick? 



7. If 250 gain 30 in 2 years, how much will 750 gain in 5 years ? 



8. What will 500 gain in 4 years, if 600 gain 42 in 1 year ? 



9. If 8 persons spend 200 in 9 months, how much will 18 persons 

 spend in 12 months ? 



10. If 15 men working 12 hours a day can hoe 60 acres in 20 days, 

 how long will it take 30 boys working 10 hours a day to hoe 96 acres, 

 3 men being equivalent to 5 boys ? 



11. If the 8d. loaf weighs 48 oz. when wheat is 54s. a quarter, what 

 is the price of wheat when the 6d. loaf weighs 32 oz. 8 dwt. ? 



12. If 35 barrels of water last 950 men 7 months, how many men 

 would 1464 barrels last for 1 month ? 



13. If 13908 men consume 732 barrels of flour in 2 months, in how 

 long will 425 men consume 175 barrels ? 



14. If 3 men with 4 boys earn 5 16s. in 8 days, and 2 men with 

 3 boys earn 4 in the same time, in what time will 6 men and 7 boys 

 earn 20 guineas ? 



15. If 5 men with 7 women earn 7 13s. in 6 days, and 2 men with 

 3 women earn 3 3s. in the same time, in what time will 6 men with 

 12 women earn 60 ? 



16. If the penny loaf weigh 6 oz. when wheat is 5s. a' bushel, what 

 should be the weight of the shilling loaf when wheat is 7s. 6d. a bushel? 



17. If 20 men can perform a piece of work in 12 days, how many 

 Ten will perform a piece of work half as large again in a fifth part of 



