362 



THE POPULAR EDUCATOR. 



LESSONS IN ARITHMETIC. XXXIV. 



PER CENTAGE PKOFIT AND LOSS. 



1. Per cent, is an abbreviation for per centum. So much per 

 cent., therefore, is so much a hundred. 



Thus, 7 per cent, upon 400 is 28 ; 3 per cent, upon 345 is 

 fis x 31, for 100 is contained f*jj times in 345. 



The quantity or number per cent, is called the rate per cent. 



The number or quantity which is produced by reckoning a 

 certain rate per cent, upon any given number or quantity is 

 called the per centage. 



Thus, at 1 per cent. .28 is the per centage on .400. Again, 

 if we have to find how much per cent, one given number is of or on 

 another, it is the same thing as dividing the first number into as 

 many equal parts as there are hundreds in the second number. 

 One of these equal parts in the first will then correspond to 

 each hundred of the second number, which is the same thing as 

 saying that the first number is so much per cent, on the other. 



Thus, if we want to find how much per cent. 27 is of 900, we 

 must divide 27 into 9 parts, which gives 3, and hence we see 

 that for every 100 of the 900 there corresponds 3 in the 27. 

 Hence 27 is 3 per cent, on 900. 



Again, to find how much per cent. 311 is of 73 : 



100 ia contained in 73, ^ss times. 

 The result therefore required is j, or V ? x 31 J or ff. 



73 ) 3150 ( 43-15 

 292 



230 

 219 



110 

 73 



370 

 365 



Hence, correctly to 2 places of decimals, 31J is 43'15 per cent, on 73. 



2. The above remarks will be sufficient to explain the fol- 

 lowing rules : 



(1.) To find what a given rate per cent, upon any number 

 amounts to. 



Divide the number by 100, and multiply by the rate. 

 (2.) To find how much per cent, one given number is of another. 

 . Multiply the former number by 100, and divide by the latter. 



PROFIT AND LOSS. 



3. Questions relating to the amount gained or lost in com- 

 mercial transactions upon sales or purchases effected, can be 

 performed by the aid of the Eule of Three and the above remarks 

 upon per centage. 



EXAMPLE 1. If I buy 1 cwt. of coffee for 8 guineas, what 

 do I gain by selling it at l|d. per oz. ? Find also what I gain 

 per cent. 



112 Ibs. cost 168s. 

 Therefore 1 Ib. costs, lijs., or IJs. 



Now 1 Ib. is sold for 16 x lid., or 2s. Therefore the gain on 

 1 Ib. is 6d., and on 112 Ib. is '2 16s. 



Next, 6d. being gained on Is. 6d., we have to find what would 

 be gained on .100. 



Then, by the Eule of Three, 



As 3 sixpences : 1 sixpence : : 100 : required rate per cent. 

 K^nce 1 s, or 33J per cent, is the required answer. 



EXAMPLE 2. What is the prime cost of an article per yard 

 upon which a gain of 7 per cent, is obtained by selling it at 

 10s. 9d. a yard ? 



By the question, a quantity of the article which cost .100 

 would be sold for .107. 



Hence, by Rule of Three, we have 



As 107} : 100 : : 10s. 9d. : cost price per yard. 



Therefore cost price per yard = 



100 

 107* 

 200 

 215 



x 129d. 



< 129d. 

 120d. = 10 shillings. 



If the article had been sold at a loss of 7J per cent., the 

 question would have been stated thus. Since a" quantity, value 

 100, is sold for 100 7 10s., w have- 



As 92J : 100 

 Therefore cost price = Wj 



: 10s. 9d. : cost price. 

 129d. 

 = ?|| x 129d. 

 = & x 129d. = 4? 2 d. = W shillings = 11s. 7Hd. 



EXAMPLE 3. Having bought a quantity of tea, I find that 

 I lose 7 per cent, by selling it at 3s. a pound. What ought I 

 to sell it at, so as to gain 10 per cent. ? 



We have 



As 93 : 100 : : 3s. : prime cost per pound. 

 Therefore the prime cost per pound = Vs Q shillings = *? shillings. 



Now if I am to gain 10 per cent., we have the proportion 



As 100 : 110 : : Ws. : selling price required. 

 Therefore the selling price required = Vi* = 3s. 6Jfd. 



EXERCISE 53. EXAMPLES OF PER CENTAGE, PROFIT AND 

 Loss, ETC. 



1. How much is 4 per cent, upon 375 ? 



2. How much is 3| per cent, upon 373, to 4 places of decimals P 



3. How much is 3i per cent, upon 27| ? 



4. The population of London in 1861 being 2,803,034, and that of 

 the whole of England 29,307,276, how many per cent, of the population 

 lived in London ? 



5. If the population of a town in a certain time increases by 8 per 

 cent., and the actual population at the beginning of the time was 

 20,000, what was it at the end of the time ? 



6. If at the same rate of increase, it is found at the end of the time 

 to be 20,000, what was it at the beginning ? 



7. How much per cent, on 353 is 29 to 2 places of decimals ? 



8. How much per cent, on 534 is 27J to 2 places of decimals ? 



9. If standard silver can be bought for 5s. an ounce, what profit 

 does the Mint make per cent, by coining a pound Troy into 66 shillings ? 



10. A mixture professing to be coffee contains 24 per cent, chicory. 

 How much chicory would there be in one pound of coffee ? 



11. In the last question, if the cost price of pure coffee be Is. a 

 pound, and of chicory 4d. a pound, what does the grocer gain per cent, 

 by selling the mixture at Is. 4d. a pound? 



12. The population of a town increases in 10 years from 26,485 to 

 28,351. What is the rate of increase per cent. ? 



13. A grocer mixes 9 Ibs. of coffee at 2s. 3d. a pound with 6 pounds 

 of chicory at 7-Jd. a pound. At what price must he sell the mixture 

 to gain 25 per cent. ? 



14. If 3| per cent, is lost by selling steel pens at 3s. 6d. a gross, 

 how much would be gained or lost per cent, by selling them at 2s. 6|d. 

 a hundred P 



15. At what rate must skerry be sold which costs 40s. a doz., if on 

 every 100 of outlay the selling price of 5 doz. is gained? What is 

 the gain per cent. ? 



16. What sum must A bequeath to B, so that B may receive 1,000 

 after a legacy-duty of 10 per cent, has been paid ? 



17. What must be the gross rental of an estate, so that after de- 

 ducting 7d. in the pound for income-tax, and 4J- per cent, upon the 

 remainder for expenses of collection, there may be left a nett rental 

 of 1,000 ? 



18. By the sale of goods which cost me 3 19s. 2d., I lost a sum 

 equal to 5| per cent, of the proceeds ; and by the sale of another 

 quantity which cost me 5, I gained a sum equal to 3 If of the pro- 

 ceeds. What did I gain per cent, on the whole outlay ? 



19. A man bought a house which cost him 4 per cent, upon the 

 outlay to put into repair; it then stood empty for a year, during 

 which time he reckoned he was losing 5 per cent, upon his total outlay. 

 He then sold it for 1,192, by which means he gained 10 per cent, upon 

 the original purchase-money. What did he give for the house ? 



KEY TO EXERCISE 52, LESSON XXXIII. (Vol. II., page 326). 



13. 15}} months. 



14. 15 days. 



15. SSJdays. 



16. 48 oz. 



17. 100. 



18. 7aV days. 



LESSONS IN CHEMISTRY. XII; 



HYDRO-CARBONS, COAL GAS, FLAME, ETC. 

 THERE are many compounds of carbon and hydrogen. They 

 are all derived from the decomposition of organic bodies, ana 

 therefore properly belong to " organic chemistry." Only two 

 demand our attention at present 



Light carburetted hydrogen . . . CH 4 

 Heavy carburetted hydrogen . . . C a H 4 . 



Harsh Gas, or Light Carburetted Hydrogen (symbol, CH 4 ; com- 



