CHAMBERS'S INFORMATION FOR THE PEOPLE. 



divide 7. In other words, the greatest 

 common measure of 35 and 14 is the same 7) I 4( 2 

 as that of 14 and 7 numbers less than H 

 the given numbers. It now remains to 

 find the greatest common measure of 14 and 7. 

 Dividing 14 by 7, we find an exact 

 quotient ; hence 7 is the number re- 14)35(2 

 quired. It is usual to connect these 28 

 separate processes, and form one . . 

 continuous operation, thus : 



As another example, let it be re- 



auired to find the greatest common 

 ivisor of 1155 and 351. We first begin by trying 

 the lowest number namely, 351. It is sufficient 

 to sec if this number 

 divides the greater, 351)1155(3 

 for we know that it 1053 



divides itself. On "TrTw,/- 



dividing 1155 by M )gS 



351, however, we find ^ 



a remainder, 102. 45)102(2 



This shews that 351 90 



is not the number . 



required, but, as in I2 )45(3 



the last case, this 



division shews that 9)12(1 



the number required o, 



is the greatest com- - 



mon divisor of 351 3)9(3 



and 1 02. Then, as be- 9_ 



fore, the lowest num- 

 ber namely, 102, must first be tried, to see if it 

 divides 351. It is found that there is a remainder 

 of 45, and therefore 102 is not the number re- 

 quired ; but, as before, we see that the greatest 

 common divisor of the given numbers is the same 

 as that of 102 and 45. Continuing this process, 

 we find that the greatest common divisor of the 

 given numbers is the same as that of 9 and 3, but 

 the greatest common divisor of these numbers is 

 3 ; hence 3 is the number required. From these 

 two examples, it will be seen how to find the 

 greatest common divisor of any two numbers. 

 Two numbers whose greatest common divisor is 

 unity, are said to be prime to each other. 



The least common multiple of two or more given 

 numbers is the least number which can be divided 

 exactly by each of the given numbers. 



Thus, for example, the least common multiple 

 of the two numbers 8 and 15 will be the least 

 number which can be divided both by 8 and by 15. 

 Now, it is evident that the product of these two 

 numbers that is, 120 can be divided both by 8 

 and by 15. Hence it is a common multiple ; and it 

 is the least, since no number less can fulfil these 

 two conditions. 



To find the least common multiple of any two 

 numbers for example, 12 and 15. Here the 

 greatest common divisor of 12 and 15 is 3; and 

 therefore we may express these numbers as 3 X 4 

 and 3X5. Now, 3 being the highest common 

 divisor, it is plain that 4 and 5 are prime to each 

 other. Then, whatever the least common mul- 

 tiple is, it is clear, since it is divisible by both 

 numbers, it must be divisible by 3, and also by 4, 

 and also by 5. Hence the least common mul- 

 tiple is 3 X 4 X 5. Now, this number is not 

 altered by multiplying by 3 and dividing by 3. 

 Hence the least common multiple is 3 X 4 X 3 X 

 5, divided by 3'; but 3 X 4 is the one given 

 number, and 3 X 5 is the other. Hence the least ' 



600 



common multiple is 12 X 15, divided by 3 ; or the 

 least common multiple of two numbers is found 

 by dividing the product of the two numbers by their 

 greatest common divisor. 



VULGAR FRACTIONS. 



A fraction, as the name implies, is a part broken 

 off from a whole. We have already treated of 

 a particular class of fractions namely, decimal 

 fractions. All other fractions are called vulgar 

 fractions. Such, for example, are 



* 



If, 



The upper figure (or figures) is called the numer- 

 ator; the under, the denominator. Fractions like 

 the above, where the numerator is less than the 

 denominator, are called proper fractions ; other- 

 wise, they are improper. The denominator of the 

 fraction shews us into what parts the whole 

 number has been divided ; and the numerator 

 shews us how many of these parts are repre- 

 sented. Before shewing how to perform the 

 fundamental operations on fractions, it is neces- 

 sary to prove (i) that a fraction is not altered by 

 multiplying or dividing numerator and denom- 

 inator by the same number. Thus, for example, if 

 we take the fraction , and multiply numerator 

 and denominator by 2, we have the fraction % ; 

 which we say is the same as the original fraction 

 I ; for doubling the numerator, and afterwards the 

 denominator, means that we first double the frac- 

 tion, and then halve the result, which leaves the 

 value of the fraction undisturbed. 



Again, taking the fraction |, and dividing 

 numerator and denominator by 2, we get for the 

 resulting fraction f , which is equivalent to f ; for 

 dividing numerator by 2 means that we are 

 halving the fraction, but multiplying denominator 

 by 2 means that we are doubling the result ; so 

 that by these two operations we have first halved 

 the fraction, and then dotibled the result, which, 

 of course, leaves the value of the original fraction 

 undisturbed. The application of this principle 

 enables us to simplify fractions. 



To compare two or more fractions. For example, 

 let it be required to compare the values of the 

 following fractions, $, f, |. We shall, in the first 

 place, transform these fractions into others which 

 have the same denominator ; and for this common 

 denominator we shall take the least common 

 multiple of the denominators : this is evidently 24. 

 Arranging the fractions as is here shewn, 

 we see that, taking the first fraction, in % = if 

 which the denominator is 3, we have to f- = i 

 change it into another whose denom- I = if 

 inator is 24. But this is done by multi- 

 plying the denominator by 8. Hence, to keep 

 the fraction of the same value, we must multiply 

 the numerator also by 8. This makes the first 

 fraction equivalent to if ; similarly, the second 

 fraction is if, and the third is if. Then it is 

 evident that if is the greatest, and if the least of 

 the fractions. 



Addition and Subtraction of Fractions. 



To add or subtract fractions, we must first trans- 

 form the fractions into others which shall have the 



