160 



EbtjiCATOii 



One of these parts, or a collection containing any number of 

 them, ia called a. fraction of the original number or thing. 



Thus, if a straight line be divided into seven parts, each part 

 is one-seventh of the line, and any number of the parts as, 

 for instance, five of them, i.e., five-sevenths of the whole is a 

 fraction of the whole line. 



The number of parts into which the unit or whole is divided 

 is called the denominator, because it indicates or denominates 

 the number of parts into which the whole is divided. 



The particular number of these parts taken to form any frac- 

 tion of the whole is called the numerator, because it expresses 

 the numbei of parts taken. 



Thus in the case given above, 7 is the denominator, because 

 the line is divided into seven parts, and 5 is the numerator of 

 the fraction. 



Fractions are expressed by writing the numerator above the 

 denominator, and drawing a line between them. Thus the 

 above fraction would be written f, one half would be written , 

 eight-ninths, g, and so on. 



The word fraction, which means a part or portion broken from 

 any integer or whole, is derived from fractus> broken, a part of 

 the Latin verb frangere, to break. The word integer is simply 

 a Latin adjective meaning fKeeh, entire, or unbroken, which has 

 been adopted into the English language. 



2. A proper fraction is one whose numerator is less than its 

 denominator, as ^, f, . 



An improper fraction is one whose numerator is not less than 

 its denominator, as , f, etc. 



A mixed number consists of a whole number and a fraction 

 expressed together; for example, 3 and f. This is generally 

 written thus, 3 ; similarly, 4jj, 7J, eto. 



Fractions in which the denominators are 10, or any power 

 of 10 (Lesson VI., Art. 5), are called Decimal Fractions, or 

 Decimals. All other fractions are called Vulgar Fractions. 



A compound fraction is a fraction of a fraction, as i of |, f of 

 | of J ; for any fractional part of a unit may be regarded as a 

 new unit. This fractional part may itself be divided into any 

 number of equal parts, and a certain number of them may be 

 taken. 



A complex, or mixed fraction, is one which has a fraction in 

 its numerator or denominator, or in both ; as, for instance 



|' 5* 3| SA -Jof9' 



Every whole number may be looked upon as a fraction, of which 

 the denominator is unity ; thus 5 is f . 



3. Fractions, it will readily be seen, are expressions of un- 

 executed division, the numerator being the dividend, and the 

 denominator the divisor. Take, for example, |. We are sup- 

 posed to have one unit or thing to be divided, and dividing it 

 into 9 equal parts, to take 4 of them. But it will be the same 

 thing if we take 4 such units, and dividing this collection into 

 9 equal parts, take one of these parts. This gives the same 

 fraction of the original unit as before ; but, looked at in this 

 lightj it expresses the quotient which results from dividing the 

 numerator by the denominator. 



4. To multiply a fraction by a whole number. 



Multiply the numerator by the whole number. For instance, 

 to multiply | by 4. Here the unit is divided into 9 parts, two 

 of which are taken ; four times as many of these parts will give 

 eight parts, or ; therefore 4 x = jj. 



5. To divide a fraction by a whole number. 



Either divide the numerator or multiply the denominator by 

 the whole number. Thus, f 2 = J ; for, the unit being 

 divided into 7 parts, 6 are taken, halving which gives 3 parts, 



or f . Again, f -r- 2 = -- = -. In {j the unit is divided into 



7 parts, 5 of which are taken. In the unit is divided into 

 14 parts, 5 of which are taken. But each of these latter 14th 

 parts is equal to each one of the former 7th parts divided by 

 2, and therefore ve of the latter will be equal to five of the 

 former divided by 2, or = -i- 2. 



}. From the above reasoning we see that it produces exactly 

 the same result whether we divide the numerator or multiply 

 the denominator by any number. Hence, if we multiply both 

 numerator and denominator by the same quantity, the value of 

 the fraction is unaltered. Multiplying the denominator divides 

 the unit into so many more parts, and multiplying the nume- 



rator takes exactly so many times more of them. Similarly it 

 follows, that if we divide both the numerator and the denomi- 

 nator by the same quantity, the value of the fraction remains 

 unaltered. 



7. To reduce a fraction to its lowest terms. 



The numerator and denominator of a fraction are sometimes 

 called the terms of the fraction. If both the numerator and 

 denominator of a fraction can be divided by the same number 

 (an operation which we have just seen does not alter its value), 

 it is said not to be in its lowest terms. A fraction, then, may 

 be defined to be in its lowest terms when the numerator and 

 denominator have no common factors. Hence to reduce a frac- 

 tion to its lowest terms, we must first find the greatest common 

 measure of the numerator and denominator, and then divide 

 them both by it. 



EXAMPLE. Eeduce g to its lowest terms. 

 The greatest common measure of 27 and 36 is 9, and there- 

 fore dividing numerator and denominator by 9 we get the frac- 

 o x g tion expressed in its lowest terms. It is not 



U = j-^-g = j necessary always to find the G. C. M. of the 

 numerator and denominator, but it is often 

 more convenient in practice to divide the numerator and deno- 

 minator by numbers which are seen to be factors common to 

 both until we arrive at the lowest terms. Thus 



10 x 231 



3 x 77 _ 7x11 _ 



~ 5t ~ 7 x 13 ~ "' 



57Si 10 x 273 2 3 x 91 

 the fraction in its lowest terms. 



EXERCISE 22. 



1. Eeduce the following fractions to their lowest terms : 



1- ft- ' 9. ti. 17- HJS- 



2. f f . *0. tf$3. 18. fHj. 



3. is. 11. H. 19. *m. 



4. . 12. m- 20. H-JiJ. 



5. ^r- 13. Ml. 21. igttiSS* 



6. ??;, 14. T VA- 22. ttttJA. 



7r, 22 It 379* oo Apeoe 



io*. lij . flSl' * TIsT' 



8. J5-S-. 16. !-;-;. 24. $**. 



2. In a joint-stock company which was divided into 10,800 

 shares, what part of the whole concern belongs to the individual 

 who holds 4,050 shares ? 



3. A ship is worth .21,600; what fraction of the ship belongs 

 to him who contributed to this sum no less than .12,960 ? 



8. To reduce an improper fraction to a whole or mixed number. 

 Divide the numerator by the denominator. If there is no 



remainder, the quotient will be the equivalent whole number. 

 If there is a remainder, the improper fraction is equivalent to a 

 mixed number, of which the quotient is the whole number (or, 

 as it is called, the integral part), and the remainder the nume- 

 rator of the fractional part, which will evidently have the same 

 denominator as the original improper fraction. Thus, ^ = 3, a 

 whole number ; and S T 3 = 3f . Since 7 sevenths make one whole 

 unit, 23 sevenths will make as many whole units as 7 is con- 

 tained in 23, i.e., 3 whole units, and 2 sevenths over. Hence 



13 Q2 



1 6 1- 



9. To convert a mixed number into an improper fraction. 

 Multiply the integral part by the denominator of the frac- 



tional part, to which product add the numerator of the fractional 

 part. This sum will be the numerator, and the denominator of 

 the fractional part will be the required denominator. Thus, 



+ f = y + f , and 28 



4 = Lli- 6 = s ? * ; f or 4f = 4 + ? = 



sevenths and 6 sevenths make 34 sevenths, or y. 



EXERCISE 23. 



1. Reduce the following improper fractions to whole or mixed 

 numbers : 



1. H- 



2. -?: 



!! 



7. 



3. V- I 



4. Vo'. I 6. V- I & i Hs 9 - I 10. i-H 



2. Eeduce the following mixed numbers to improper fractious j 

 in their lowest terms : 



1. 17J. I 3. 115.V \ 5. 25|. I 7. 4725|. I 9. 62'!. 



2. 48?. I 4. 1304*. I 6. 856JJJ. | 8. 525&. I 10. 891 T \. 



3. Reduce 445 to tenths, fourteenths, seventeenths and thirds. 



4. Eeduce 672 to eighths, twelfths, eighteenths and fourths. 



5. Eeduce 3830 to hundredths, fifteenths ami jfchirty-fiftha. 



