LESSON'S IN ALUKBKA. 



LKSSONS IN ALGEBRA. XXIV. 



following problems, the student may now employ two, 

 three, or more unknown quantities in their solution, just M the 

 nature of oooh may require ; or ho may still limit the number 



unknown quaiititii-.-i, by first supposing one unknown 



y, and then finding from the conditions of the question 

 -ions for the other unknown quantities in terms of that 



which has been assumed. 



EXERCISE 41. ALGEBRAICAL PROBLEMS. 



I. Find two numbers such that their sum shall be a, and 



1100 b. 



-. Divide the number 20 into such parts, that three times the 

 one added to five times the other will make 76. 



.:. Two gamesters, A and B, sat down to play. A had 80 

 guineas, and B had GO. After a certain number of games were 

 won and lost between them, it was found that A had three times 

 as many guineas as B. How many guineas did A win of B ? 



4. Find two numbers suoh that half the first and a third part 

 of the second shall make 9 ; and that a fourth part of the first 

 with a fifth part of the second shall make 5. 



5. Divide the number 2 into two such parts that a third of 

 the one added to a fifth of the other shall make $. 



G. Find three numbers such that the sum of the first and 

 second shall be 7, tho sum of the first and third 8, and the sum 

 of the second and third 9 ; and give a general solution, by sup- 

 posing these three sums to bo a, 6, and c respectively. 



7. The sum of the three digits in a certain number is 16 ; the 

 sum of the hundreds' digit and the tens' digit is to the sum of 

 the tens' digit and the units' digit, as 4J is to 5i ; and if 198 be 

 added to the number, the hundreds' digit and the units' digit 

 will change places. What is the number ? 



8. Divide 72 into four such parts, that the first increased by 

 5, the second diminished by 5, the third multiplied by 5, and the 

 fourth divided by 5, the sum, difference, product, and quotient, 

 shall all bo equal to one another. 



9. A farmer hired 4 men and 8 boys for a week, and paid 

 them in all .8 ; the next week he paid 7 men and 6 boys at the 

 same rate each, and paid in all 10. How much did ho pay 

 each man and each boy by tho week ? 



10. A father bequeathed .2,800 to his daughter and son, in 

 euch a manner that for every half-crown tho daughter had, the 

 eon should have a shilling. What were their shares ? 



II. A bill of .100 was paid in half -guineas and crowns ; and 

 202 pieces of money were employed in the payment. How many 

 pieces were there of each kind ? 



12. Find four numbers such that the sum of the first, second, 

 and third, shall be 13 ; the sum of tho first, second, and fourth, 

 15 ; the sum of tho first, third, and fourth, 18 ; and the sum of 

 the second, third, and fourth, 20. 



13. Two numbers are to each other as 20 to 30; but if 6 bo 

 added to each, then the sums are to each other as 40 to 50. 

 What are the numbers ? 



14. There are two numbers such that the greater is to the 

 less as their sum is to 20, or as their difference is to 10. What 

 are the numbers ? 



15. Three boys were playing at marbles. In tho first game, 

 A loses to B and C as many as each of these two had when they 

 hegan ; in the second game, B loses to A and C as many as each 

 of these two had at the end of the first game ; in the third game, 

 C loses to A and B as many as each of these two had at the end 

 of the second game. Each has now 16 marbles ; how many had 



' p.t first ? 



lo. A person goes to a coffee-house with a certain quantity of 

 money in his pocket, where he spends 2 shillings ; he then 

 borrows as much money as he had left, and going to another 

 3offee-house, ho there spends 2 shillings also. Then, borrowing 

 again as much money as was left, he went to a third coffee- 

 house, where likewise he spent 2 shillings ; and thus repeating 

 the same at a fourth coffee-house, he then had nothing remain- 

 ing. What sum had he at first, and what was he in debt ? 



17. A man with his wife and child dine together at an inn. 

 Tho landlord charges 1 shilling for the child ; for the woman, as 

 much as for the child and a quarter as much as for the man ; 

 and for tho man, as much as for the woman and child together. 

 How much was that for each ? 



18. A cask which held GO gallons was filled with a mixture 



of brandy, wine, and aider, so that the cider WM 6 gallons more 

 than the brandy, and the wine WM M much as the cider and 

 of the brandy. How much WM there of each ? 



19. Says A to B, " If you give me 10 guinea* of 700* money, I 

 hall then have twice M much M you will hare left ; " hot says B 

 to A, " Qire me 10 of your guinea*, and then I shall hare three 

 times M many M you." How many had each ? 



20. Three persons, A, B, and C. make a Joint contribution, 

 which in the whole amounts to 400; of which sum B con- 

 tributes twice M much M A, and 20 more ; and C M much 

 as A and B together. What um did each contribute ? 



21. The stock of three traders amounted to 760. The- 

 shares of the first and second exceeded that of the third by 

 240, and the sum of the second and third exceeded the first 

 by 360. What WM the share of each ? 



22. What two numbers are those which, being in the ratio of 

 3 to 4, their product is equal to 12 times their sum P 



23. A certain company at an inn, when they came to settle 

 their reckoning, found that had there been 4 more in company, 

 they might each have paid a shilling less than they did ; but that 

 if there had been 3 fewer in company, they must each have paid 

 a shilling more than thej did. What, then, was the number of 

 persons in the company, what did each pay, and what WM tho 

 whole reckoning ? 



24. A farmer has two horses, and also two saddles, the ono 

 valued at 18, the other at 3. Now when he sets the better 

 saddle on the first horse, and the worse on the second, it makes 

 the first horse worth double the second ; but when he places tho 

 better saddle on the second horse, and the worse on the first, it 

 makes the second horse worth three times the first. What 

 were the values of the two horses ? 



25. It is required to divide the number 24 into two such parts, 

 that the quotient of the greater part divided by the less, may be 

 to the quotient of the less part divided by the greater, as 4 

 to 1. 



26. A cistern is to be filled with water from three different 

 stop-cocks. From tho first it can be filled in 8 hours, from the 

 second in 10, and from the third in 14. How soon would they 

 altogether fill it ? 



27. A labourer engages to work for 3s. 6d. a day and hid 

 board, but to allow 9d. for his board each day that he is un- 

 employed. At the ead of 24 days he has to receive 3 2s. 9d. 

 How many days did he work ? 



28. Three workmen are employed to dig a ditch of 191 yards 

 in length. If A can dig 27 yards in 4 days, B 35 yards in 6 

 days, and C 40 yards in 12 days, in what time could they do it 

 if they worked simultaneously ? 



29. A farmer wishes to mix 28 bushels of barley at 2s. 4d. a 

 bushel, with rye at 3s. a bushel, and wheat at 4s. a bushel, so 

 that the whole may consist of 100 bushels at 3s. 4d. a bushel 

 How much rye and wheat must he use for this purpose ? 



30. A sum of money was divided equally amongst a certain 

 number of persons. Had there been three persons more, each 

 would have received 1 shilling less ; and had there been two 

 persons fewer, each would have received 1 shilling more. Be- 

 quired the number of persons, and what each received. 



31. How may a bill of 7 4s. be paid with half-guineas and 

 crowns, so that twice the number of crowns may be equal to 

 three times the number of half-guineas ? 



32. A person rows a distance of 20 miles and back in 10 hours, 

 tho stream flowing uniformly in the same direction all the time. 

 He finds that, with tho stream, he can row three miles in the 

 same time that it takes him to row 2 miles against it. How 

 long was he going with the stream, and how long against it ? 



KEY TO EXERCISES IN LESSONS IN ALGEBRA. 

 EXERCISE 39. 



(a b 



1. z 6, v = 4, and i = 



2. - $ (a + b c), f 



+ c),andi=j( 



3. A's money = 64 dollars, B's = 



72, and C'a = 84. 



EXERCISE 40. 



4. A' distance la 46 miles, Fs - 



9, and C's = 7. 



5. 24, y - 80, and * - 130. 

 . * = 30, y = 20, and * - 10. 



1. A. 



2. 18, 22, 10, and 40. 



3. 50, 65, and 75, 



4. 10 and 2. 



5. The port 3 guineas per dozen. 



the sherry guineas. 



6. 78 of brand/ and W of 



7. - 



