310 



THE POPULAE EDUCATOE. 



15. Eeduce . , , .and to fractions having a com- 



2ab' 3bc 4cd 5de 6ef 



man denominator. 



132. To reduce an improper fraction to a whole or mixed 

 quantity. 



Divide the numerator by the denominator, the quotient with the 

 remainder in a fractional form is tlie answer. [See Art. 106.] 



133. To reduce a mixed quantity to an improper fraction. 

 Multiply the integer by the given denominator, and add the 



given numerator to the product. [See Art. 122.] The sum will 

 be the required numerator ; and this placed over the given deno- 

 minator will form tlie improper fraction required. 



If the sign before the dividing line is , all the signs in the 

 numerator must be changed. [See Art. 124.] 



EXERCISE 16. 



1. Eeduce to a whole or mixed quantity. 



b 



_ , am a + ady Jir. , . 



2. Eeduce to a whole or mixed quantity. 



3. Eeduce a + - to an improper fraction. 



4. Eeduce o - to an improper fraction. 



c 



a c 



5. Eeduce ob 



6. Eeduce m + d 



to an improper fraction, 

 to an improper fraction. 



h-d 



7. Eeduce a; to an improper fraction. 



c 



8. Eeduce ax + * to an improper fraction. 



d 



9. Eeduce b to an improper fraction. 



d y 



10. Eeduce & + ax + a 3 + to an improper fraction. 



x a 



11. Eeduce Zx 4a -t- 



to an improper fraction. 



x + 2a 



12. Eeduce 3o - 4z + * ~ to an improper fraction. 



4a 3x 



13. Eeduce 1 * - to an improper fraction. 



tc + a 



134. To reduce a compound fraction to a simple one. 

 Multiply all the numerators together for a new numerator, and 

 all the denominators for a new denominator. 



1. Eeduce ? of ,-=_ 



7 b + 2 



EXERCISE 17. 

 to a simple fraction. 



2. Eeduce - of - of to a simple fraction. 



3 5 2a m 



3. Eeduce of of to a simple fraction. 



o a 3 f 



4. Eeduce of of ^ to a simple fraction. 



b c* d 3 



_ _, x y ax + as . x + a 



5. Eeduce ~- of 



- - ---- 



x l + ax + a* x a 



6. Eeduce 



of 



a; -4 



to a simple fraction. 

 to a simple fraction. 



7. Eeduce - of - of - to a simple fraction. 



8. What is the value of 



2aay 



9. What is the value of 



abed/ 



10. What is the value of x 4 ? 



a 



11. What is the value of 16a ^ -^ 4x ? 



a 



12. What is the value of ^? when the denominator is multiplied 

 by 4? 



13. What is the value of _.! when the denominator is divided by 6ax? 



14. What is the value of 'Ll. when both numerator and denomi- 

 nator are x 2d ? 



6abc + 12abi 



15. Eeduce 



2ab 



to a whole or mixed number. 



16. Eeduce 



itr T> A 



17. Eeduce 



_ 



I2x 



to a whole number. 



, . . 



to a whole or mixed number. 



18. Eeduce the four next examples to their lowest terms : 



(1.) abc . (2.) 

 aac 



(3.) 



bx + 



ab + bn 



aaxy aab 

 ac + abc 



19. Eeduce and to a common denominator 



V d 



20. Eeduce - _, -L, and X - to a common denominator. 



bag y 



21. Eeduce a to an improper fraction. 







22. Eeduce a + b to an improper fraction. 



4m 



23. Eeduce - of - of - of - to a simple fraction. 



3 b d y 



24. Eeduce f of *? of i of 2c of 4d * of abc to a simple fraction. 



7 4b 2 4* 2a 2d 



KEY TO EXEECISES IN LESSONS IN ALGEBBA. 

 EXERCISE 13. 



fianb mtt intent SSatet cr einer Sittc, tit 

 Mal-vee'-na shtant mit ee'-rem fah'-ter fore i'-ner lee'-ll-ai, dee 



untet etncm 5Rofcnjhait<$ btu^ete JBlenbenb tuet^, Wte etn 



CSn'-ter i'-nem ro'-zen-shtrouc^ bltt"-hai-tai. Blen'-dent vice, vee ine 



erljcb kte ft^one Stume t^ren offnen buften 

 Kyt'-shtralil, err-hope' dee sho'-nai bloo'-mai ee'-ren 8f-nen d55f-ten- 



ben Sttlfy. llefccr t^r !)tng etne oott. aufgebtu^te fraftige 

 den kely. ii"-ber eer hink i'-nai f51 ouf'-gai-blii"-tai kref-tt-gai 



Slofe, unb toarf etiten rot^tic^en Dimmer auf bte jarten 

 ro'-zai, 55nt varrf i'-nen ro't'-Hy-yen Bhim'-mer ouf dee tsahr'-ten 



@t(6erbtfttter ber Sitte, unb fo flop au^> beiber S8(umeH 

 zil"-ber-blet'-ter dair lee'-H-ai, Cont zo floss oudj) bi'-der bloo'-men- 



buft in etnanbcr. 

 d65ft in ine-an'-der. 



O, toelcj) ein fc^oner 23unb ! rtef CKaliotna, unb ntigte 

 Oh, vely ine sho'-ner b3i5nt ! reef Mal-vee'-na, 55nt ni'y'-tai 



^r -fcaupt u ben Stumen ^ 



ley'-yelnt eer houpt tsoo dain bloo'-men hln-ap'. 



@ tfl bee 33unb ber Unfcf^utb unb SteBe! erttrieberte 

 Ess ist dair bfiont dair oon'-sh5051t 55nt lee'-bai ! err-vee'-der-tai 



bet SSater. @o flanben fte fcfjtoetgenb we ben Stumen. 

 dair fah'-ter. Zo shtan'-den zee shvi'-ghent fore dain bloo'-men. 



3nbe trat Ofar in ben arten, 9WaMna'3 fitlter 

 In-dess' traht Oss'-karr in dain garr'-ten, Mal-vee'-nass shtil'-ler 



etiebter. a flog ctn rotfitidber aitcf) uber 3KaMna'8 

 gai-leep'-ter. Dah floss ine ro't'-lliy-yer houc^ ii"-ber Mal-vee'-nass 



aBangen, tote bet JKofe @tan$ ubet bte Stlte. 

 vang'-eu, vee dair ro'-zai giants u ! '-ber dee lee'-ll-ai. 



2)a fa^ bet 3Satet fte an unb fttac^ : Sfttcpt toa^t, 

 Dah zah dair fah'-ter zee an S6nt shprahd) : Nlyt vahr, Mal-vee'-na, 



bte aSttimen ^a6en etne @racfye ttnb etn 



dee bloo'-men hah'- ben i'-nai shprah'-cfjai Cont ine ant'-lits? 



gut bte Wnftyulb unb Stebe ! fe^te DSfot Ijtnju. f 

 Fu'r dee 56n'-shoolt oont lee'-bai! zets'-tai Oss'-kaxr hm-tsoo. 



