473 



MATHEMATICS. ALGEBRA. 



[SIMPLE EQVATIOXS. 



Let x bo tho numerator and y tho denominator ; then 

 the fraction u -; nd by the question 



6x-y 1, or y - 6x + 1- 



Hence, equating these two expressions for y, wo have 

 ox-r-l-Ar + 'J.-. 2* -.'. x-4, tho numerator. 



And y - 5x + 1 - 21, tho denominator, . . gj is the frac- 

 tion. 



4. A man and his wife could drink a barrel of beer in 

 15 days ; but after drinking together C days, the woman 

 alone drank the remainder in 30 days. In what time 

 could either alone drink tho whole barrel ? 



Suppose the man could drink it in x days, and the 



woman in y days ; then in one day tho man's share is - - 

 of the whole, and tho woman's - ; so that 15 times tho 

 sum of these shares is 1 whole barrel : that L 



Dut by the question 

 6 30 



So that dividing tho first equation by 15, and this last 

 by 6, wo have 



15 



By subtracting 



- 

 y 6 



: * y = 50. 

 15 



3 



"10 = 



= 150; .-.x- 



Consequently, the man alone could drink it in 213 

 days, and tho woman in 50 days. 



In this example, and _ are regarded as the unknown 

 quantities, as recommended at page 471, Example 9. 



1. Find a n timber such that if it be increased by one- 

 half, one-third, and ouo-fourth of itself, the sum shall 

 I " 



-. Tlicro is a fraction such that if -1 bo added to tho 

 denominator the value is a J- ; and if 3 l>e added to tho 

 numerator, the value is J : required the fraction. 



:: \Vh.it nuinlxjr is that such that if it bo increased 

 !;.- 7, tho square root of the sum shall be equal to tho 

 square root of tho numlxjr itself and 1 moro / 



I. Fifty labourers are engaged to remove an obstnic- 

 ti'in on a railway: some of thorn arc by agreement to 

 receive ninepcnco each, and the others fifteen pence. 

 Just 2 are paid to tlium : but, no memorandum having 

 been made, it is required to find how many worked for 

 9</., and how many for 15<t 



6. A person ordered a quantity of rum and brandy, 

 for which ho paid 19 4. : the brandy was 9*. a quart, 



ho rum G. He has, however, forgotten tho exact 

 quantity of each ho has to receive ; but ho remembers 

 that if his brandy had been mm, and his rum brandy, 

 his outlay would have been 1 13*. loss. How many 

 quart* of each did ho buy ? 



>. A person has spirits at 12. a gallon, and at 1 a 

 gallon ; how much of each sort must ho take to make a 

 gallon worth 1UJ 



7. A merchant has spirits at a shillings a gallon, and 

 at b shillings a gallon ; how much of each must ho take 



to make a mixture of d gallons worth c shillings a 

 gallon 1 



& In tho composition of a certain quantity of gun- 

 powder, two-thirds of tho whole + lOlbs. was nitro ; 

 one-sixth of tho whole 4 Jibs, was sulphur ; and tho 

 charcoal was one-seventh of tho nitre, all but 21bs. How 

 many Ibs. of gunpowder were there ? 



9. A fanner wishes to mix 23 bushels of barley at 

 2s. 4d. a bushel with ryo at 3i. a bushel, and wheat at 4. 

 a bushel, so that tho wholo may make 100 bushels worth 

 Si. 4ci. a bushel ; how much rye and wheat must ho 

 use? 



10. Two persons, A and B, are engaged on a work 

 which they can finish in 1C days ; but after woi 

 together 4 days, A is called off ; 'and B alone finishes it 

 in 36 days more. In how many days could each do it 

 separately ? 



11. A composition of copper and tin, containing 100 

 cubic inches, weighed 005 ounces ; how many oun< 

 each metal did it contain, supposing a ruliic inch of 

 copper to weigh 5J oz., and a cubic inch of tin to weigh 

 4}ozJ 



12. A cask is supplied by three spouts, which can fill it 

 in o minutes, b minutes, and c minutes respectively ; in 

 what time will it be filled if all flow together i 



Simple equations with three unknown quantities may- 

 be solved by imitating tho operations in equations with 

 two unknown ; that is, by first eliminating one of the un- 

 known from two of tho equations, and then eliminating 

 the same unknown from one of those two and the third ; 

 the results of the eliminations will be a pair of equations, 

 with only two unknown quantities: for example, 



A and 13 can perform a piece of work in 8 days ; A and 

 C in 9 days ; and 15 and C iu 10 days. In how many 

 days can each alone perform it ? 



Suppose A, B, and C can do the xth part, the yth part, 

 and tho Mi part respectively in one day : then by tho 



ion, 



8* + 8y = 1 (the w.V.V), 0,- + 9s = 1, 10;/ + 10; = 1. 

 To equalise the cocflieients of y, multiply the first by 

 5, and tho third by 4 ; and wo have 



4- 40i/ = 5 

 40x+40y-4 



.'. Subtracting, -\->JT 40s = 1. Mult, this by 9, and tho 

 secon 1 eqiia. !>. + 9s = 1 by 40 : 

 3GOx 360z = 9 

 3CO.E + 3GOs = 40 

 Adding and subtracting, 720.C = 49 and 720.- = 31, 



":-& ft 



.-. lOy + 10, = lOy + g - 1 A 10 - *i ' " = f 

 Hence A can do 



j' 1 of the whole in 1 day; B can do ** ; and C, 

 720 7-v 



i j-U 



/. A can do tho whole in 



* days, B in ^. days, and C iu ?*> daya . 

 I'.i 41 31 



That is, in 14JJ days, 17J? days, and 23,f r daya, re- 

 spectively. 



Equations which, liko those considered in the pre- 

 ; pages, imply distinct conditions, all existing 

 together, are called simultaneotu equations: tho con- 

 ditions of the foregoing problem aro implied in the 

 three simultaneous equations at the commencement of 

 the solution. 



Quadratic Equations with One Unknown Quau 

 A quadratic is an equation in which the square of tho 

 unknown quantity outers, and enters in such a way as 

 to require a peculiar process for its removal. It is pos- 

 sible that oven a simple equation may contain the square 

 of the unknown ; but then, in order that it may deserve 

 tho namo of a simple equation, this square must bo re- 

 movable by transposition or division, or by some other 

 of the operations common to simple equations in general ; 

 whereas a quadratic, properly so called, requires for its 



