MATHEMATICS. ALGEBRA. 



[SIMPLE EQUATIONS. 



But the part filled by the first tap in 1 hour is the j 

 part, and by the seoond tap the J part ; 



.1,1 1 



Multiplying by 15, 8-15 



SB 



by x, 8 z = 15, .*. x - 1 J 



.-. tho time is one hour and seven-eighths. 

 These specimens of the solution of questions by simple 

 equations must suffice for tho present. It will be seen, 

 from an examination of them, that the only thing of any 

 difficulty, in thus applying the first principles of Algebra, 

 is the translation of the conditions of tho question into 

 algebraical language, under the form of an equation ; tho 

 process by which this equation is to be solved, and the 

 unknown quantity in it discovered, is in general suffi- 

 ciently suggested by the appearance of the equation 

 itself. As tho aim is to isolate tho x, so that it alone 

 may occupy one side of the equation, while known num- 

 bers occupy the other, every step in the solution is made 

 to contribute to this end. By transposition, clearing 

 fractions, collecting like terms, <tc. , the equation is made 

 to pass through one change after another, till at length 

 a single unknown term appears on one side, and a known 

 number, which is its interpretation, on the other. It 

 then only remains to divide each side by the coefficient 

 of z, if it have a coefficient other than unity, and x itself 

 becomes known. 



QUESTIONS FOB EXERCISE. 



1. There are two numbers, of which the difference is 

 9, and the sum 43 : what are the numbers ? 



2. From two places, 108 miles apart, two persons, A 

 and B, set out at tho same time to meet each other. A 

 travels 17 miles a-day, and B travels 18 : in how many 

 days will they meet ? 



3. Find two numbers of which the difference is 13, 

 and which are such that if 17 be added to their sum, the 

 whole will amount to 62. 



4. There are two numbers, of which the difference is 

 15, and which are such that if 7 times the less be sub- 

 tracted from 5 times the greater, the difference is 19 : 

 what are the numbers ? 



6. A person starts from a certain place, and travels at 

 the rate of 4 miles an hour. After he has been gone 10 

 hours, a horseman, riding 9 miles an hour, is dispatched 

 after him : how many hours must the horseman ride to 

 overtake him ? 



0. A person has 264 coins sovereigns and florins ; ho 

 has 4} times as many florins as sovereigns : how many of 

 each coin has he ? 



7. A person spends Jth of his yearly income in board 

 and lodging, f th in clothes and other expenses, and he 

 lays by 85 a-year : what is his income ? 



8. What number is that whose third part exceeds its 

 fifth part by 72 ? 



9. 1 have a certain number In my thoughts. I mul- 

 tiply it by 7, add 3 to the product, and divide the sum 

 by 2. I then find that if I subtract 4 from the quotient, 

 I get 15 : what number am I thinking of ? 



10. A man 40 years old has a son 9 years old ; the 

 father is therefore more than 4 times as old as his son : 

 in how many years will the father be only twice as old as 

 his son ? 



11. Two persons, A and B, 120 miles apart, set out at 

 the same tiino to meet each other. A goes 3 miles an 

 hour, and B 5 miles : what distance will each have 

 travelled when they meet ? 



12. Divide 250 among A, B. and C, so that B may 

 have 23 more than A, and C 105 more than B. 



13. A can execute a piece of work in 3 days which 

 takes B 7 day* to perform : in how many days can it bo 

 done if A and B work together f 



14. A cistern can bo filled by three pipes ; by the first 

 in 2 hours, by the second in 3, and by tho third in 4 : in 

 what time can it be filled by all the pipes running 

 t. :.. r I 



15. Solve the preceding question when the first pipe 

 fills the cistern in 1 hour 20 minutes ; the second in 3 

 hours 20 minutes ; and the third in 5 hours. 



10. After A has been working 4 days at a job which ho 

 can finish in 10 days, B is sent to help him ; they finish 

 it together in 2 days : in what time could B alone have 

 done the whole ? 



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

 ceive twice as much as B, and B three times as much as 0. 



18. A person has 40 quarts of superior wine, worth 7*. 

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

 as that he may sell it at 4s. Cd a quart : how much water 

 must he add ? 



19. Divide 90 into four parts, such that if tho first be 

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

 multiplied by 2, and the fourth divided by 2, the results 

 may all be equal 



20. Divide 39 into four parts, such that if the first be 

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

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

 may all be equal. 



The preceding examples may serve to show how the 

 first principles of Algebra may be applied to inquiries of 

 a practical nature. It is time that we now proceed to 

 the other two rules, multiplication and division ; for, as 

 in Arithmetic, the four rules comprehend all the opera- 

 tions of the science. But two or three particulars must 

 be previously defined. 



We know that when factors are multiplied together, 

 tho result is called a product ; if tho factors are all equal, 

 the product is called o power of the factor whose repeti- 

 tion in the multiplication has produced it : thus, in tho 

 following instances, namely 



5 + 5 = 25; 3 + 3+3 = 27; 2 + 2 + 2 + 2 = 16, <tc., 

 25 is the second power, or the square of 5 ; 27 is the third 

 power, or the cube of 3 ; 16 is the fourth power of 2 ; ',','2 

 is the fifth power of 2 ; 64 the sixth power, and so on. 

 The fourth power of a is aaaa ; the fifth power of x is 

 xxxxx, and so on. But as this repetition of the factors 

 is tedious and cumbersome, it is agreed to represent a 

 power by writing down the factor only once, and placing 

 over the right hand upper corner the number which de- 

 notes the repetitions; thus, the fourth power of a is 

 written a 4 , the fifth power of x is written x s , and so on. 

 Suppose, for example, that x stands for 3 ; then x=3, 

 x 2 =9, * 3 =27, x=81, x s =243, x=729, x T =2t87, <fcc. 

 The small figures, thus used in the notation for powers, 

 are called exponents or indices; the exponent or index 

 for the cube, or tirird power, is 3, that for the fourth 

 power 4, and so on. 



The number or quantity, which thus produces a pototr, 

 is called a root of that power ; thus, 3 is the square rout 

 of 9, the cube root of 27, the fourth root of 81, and so 

 on. There is a convenient notation for roots as well as 

 for powers. 



The sign N /, for a root, is called the radical sign ; it is 

 prefixed to the quantity whose root is meant, and a small 

 Sgure. denoting what root is to be understood, is con- 

 nected with it ; thus, !^/4 means the second, or square 

 root of 4, that is, 2 ; because the second power, or square 

 of 2, is 4 ; in like manner, ^/8 means the third, or cube 

 root of 8, namely, 2 ; because tho third power, or cube of 

 2, is 8 ; V/x moans the fourth root of y ; that is, it is a 

 number such that the fourth power of that number is the 

 number that x stands for. It must be observed, how- 

 ever, that in tho case of the second or square root, tho 

 ittle 2 is always omitted ; so that when there is no itulex- 

 igure connected with the radical sign J, the square root 

 s always to be understood.' You will now easily make 

 out the following statements or equations : 



^9=3, -V125-5, V<*- 1 , V* 5 =* *'*'- 

 But besides the radical sign, there is another contri- 

 vance for indicating roots a contrivance like that f i T 

 lenoting powers, namely, the attaching an exponent to 

 he quantity whose root is meant ; thus, the equations 

 above, expressed in this other form of notation, are as 

 ollows : 



9^-3, 125&-6, (a*)*"-o, (*)*- x, (*)*-*. 



