490 



MATHEMATICSALOEBRA. [gonmoaa TO THB 



ft* 



* x 

 Multiplying by 4, Sz + y + j - *, or 3* + 2x - 4, 



thfttu,6x-4,.-.x- g', 



hence, the time is jjh. - 48 minutes. 



10. After A ha. been working 4 day. at a iob, which 



bare done the whole ? 

 Suppose B can finish it in x days ; then he can do i- 



of it in 1 day, so that in tho 2 days he does - of it. Now, 

 as A does f of it in a day, in the 4 days, working alone, 



be has done ^or | of it ; hence, when B commences 



3 223 



there is only g of it to be done, so that -+ fo = 6 or 



2,1 322 11 

 i^S'S' 'x = 6' ' x = 6' 



Multiplying by z, and then by 5, we have 5 

 hence, B can finish tho work alone in 5 days. 



17. Divide 143 among A, B, and C, so that A may 

 receive twice as much as B, and B three times as much 

 asC. 



Suppose C's share to be x pounds 

 then B's ,, 3x 

 and A's ,, 6x ,, 



. . the sum of the shares is Wx= 143, 

 



.- . C's share = 14 6s. ; B's, 42 18s. ; A's, 85 16s. ; 

 anil the sum of which is the whole = 143. 



18. A person has 40 quarts of superior wiue worth is. 

 a quart ; he wishes, however, so to reduce its quality as 

 that he may sell it at 4s. 6d. a quart ; how much water 

 must he add 1 



Suppose the water to be x quarts, then the entire 

 number of quarts in the mixture will be 40 + z, and by 

 the question the worth of the pure wine is 280s., and 

 that of the reduced wine is 41(40 + x) shillings. As the 

 worth is to remain the same," we have the equation 

 4J(40 + x) = 280, or 180 + 4}* = 280 

 100 200 



100 ; and .' . x = -j- 



= 22g; 

 2)62? 



hence, the quantity of water to bo added 

 is 22J quarts ; so that tho mixture will 

 make 62J quarts. The value of this at 4Js. 

 a quart is found, as in the margin, to be 

 280., which is the value of the unreduced 

 40 quarts. 



19. Divide 90 into four parts, such, that 

 if the first be increased by 2, the si-coinl 

 (liminuthcd by 2, tho third multiplied by 

 2, and tho fourth divided by 2, tho results may all be 



M8I 



_S1* 



280s. 



increased by 1, the second diminished by 2, the third 

 multiplied by 3, and the fourth divided by 4, the results 

 may all bo equal. 



Let the first part be x 1 : this increased by 1 is*. 

 Then the second will be x + 2 : this diminished by 2 is x. 



third g : this multiplied by 3 is x. 

 fourth **J__ ti" 8 divided by 4 is x. 



x 



The sum of those is 6z+ ^ + 1 = 39, .'.6* + jj = 38. 



Multiplying by 3, 18* -f x = 114, . . 19* = 114, . '. 

 x = 6 ; hence, tho required parts are 5, 8, 2, and 24, the 

 sum of which is 39. 



From tho last two examples you will perceive, that 

 although, in general, the unknown quantity sought is best 

 represented by a single symbol x, yet tho conditions of 

 the question may bo such as to suggest a more conve- 

 nient form for it : a judicious form of representation at 

 the outset will often save several steps of work in the so- 

 lution. When fractions are foreseen to enter the equa- 

 tion, when tho symbol for the unknown quantity is x, it 

 will always be better to use instead of x, such a multiple 

 of x as will preclude their entrance, as in Examples 7 and 

 8, just given. 



MULTIPLICATION. CASE I. PAGE 459. 



EXAMPLES FOE EXEKCISE. 



MULTIPLICATION. 



1. 



CASE II. ' PAGE 460. 

 2. 5aV 4* 



