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THE POPULAR EDUCATOE. 



LESSON'S IN ARITHMETIC. XII. 



FE ACTIONS (continued). 



15. Multiplication of Fractions. 



To multiply | by |. 



This means to take four-fifths of the fraction | ; that is, it is 

 the same thing as finding- the value of the complex fraction | 

 of . 



Now, if | be divided into five equal parts, i.e., if f be divided 

 by 5, we get T 2 ? ; because, to divide a fraction by a whole num- 

 ber, we multiply the denominator by that number (Art. 5) ; and 

 taking four of these fifth parts of f viz., four times f s we get 

 as the required result . 



This result is plainly got by multiplying the numerators 

 together and the denominators together of and |, to form a 

 numerator and denominator respectively. The same method 

 would evidently apply to any other two or more fractions. Hence 

 the following 



Rule for the Multiplication of Fractions. 



Multiply together all the numerators for a numerator, and all 

 the denominators for a denominator. 



06s. -In multiplying fractions we can often simplify the 

 operation by striking out or cancelling factors (as we arc at 

 liberty to do, Art. 6) which are common to the numerator and 

 denominator of the fraction formed by multiplying the numera- 

 tors and denominators together. 



EXAMPLE. Multiply together f, f, , s . Their product is 

 equal to 



2 x 5 x 6 x 55 J x 5 x --^Y.^ 

 3x8 x~ll x 108 "^xI^px^Fxlf 



11x5 



x 108. 



And 2 occurs twice in both numerator and denominator, 

 11 once s , 



3 once 



~ 



Therefore the product = - 



16. Division of Fractions. 

 To divide by f . 



Dividing by a whole number is finding how many times the 

 divisor is contained in the dividend. Now. a seventh is con- 

 tained in unity 7 times, and therefore a seventh is contained in 

 ?;, | X 7 times ; 5 sevenths will be contained therefore in | one- 

 iifth of this number of times, and therefore the quotient of by 

 f- is | X f, that is, f, and the same method will be true for 

 any other two fractions. Hence the following 



Rule for the Division of Fractions. 



Invert the divisor, and then proceed as in multiplication, i.e., 

 multiply the numerators together for a numerator, and the 

 denominators for a denominator. 



Obs. In performing the process, the Ols. of Art. 15, with 

 reference to cancelling factors which are common to both nume- 

 rator and denominator, must be attended to. 



17. By this and the foregoing rules we are able to simplify 

 complex fractions. 



EXAMPLE. To MJZjfe, add j B I5 , and multiply the result 

 bv 8?' 



In a case like this it will be better to simplify each portion 

 separately before performing the operation indicated. Now 



J_ 



226 



_1 339 - 226 



339 ~ 226 x 339 = 220 x 339 



113 



2 x 



55 - 21 



J54_ 



120 



(120 being the L.C.M. of 24 and 40). 



rp, ., 226 339 

 Therefore -tr = 



"24" ~ 40" 



1 



2 x 339 



~60~ H3 



TT _JIO _6_ _ __10 6 x 17 112 



Ce 113 x 17 + 113 - 113 x 17 + 113 x 17 = IIsT 



_ 

 17 



17 



And therefore tL.e required result will be given by the following- 



7 



ri iia is- 7 



Sr X iS x ~===L Answer. 

 Utrx^f -it> 144 144 



EXERCISE 28. 



EXAMPLES IK MULTIPLICATION AND DIVISION OF 

 FRACTIONS, ETC. 



1. Find the following products : 



3. Divide ?- of f by 21. 



4. Divide i + 11 + by -J- + =* 



5. Divide | + 2i + i + by | + & 

 C. Reduce to their simplest forms 21 - 



i , ; 



and 3T- 



7. Divide 21 + 11 + 31 by i of of V- 



8. Find the difference of 2f X | and X jj. 



9. Simplify 6| X If of (1& |). 



10. What is that number | of which is 27? 



11. If a certain number be multiplied by 2? tho result 

 What is the number ? 



12. By what must 29 bo multiplied to obtain 67? 



13. Express the difference of the first two of the followini 

 tioiis as a fraction of the difference of the last two : ^, |W|}, 



14. Perform the same operation on 3 B g , f? l} r *^. 



5 + f 



is 52. 

 rfrao- 



15. Of what quantity is 



~ 



seven-tenths ? 



{ 

 16. Find the products of the following fractions : 



1. ^.7 ^ 1- X "^ > 



3 -^ I1 -X 49 x ' 



4. 5 x -5| x -55. x VA x 5 



5- I X i X A X A X f X -f,. 



6. -J of J of x I of -*- of i 9 ff 



17. Find the products indicated by the following expressions :- 



1. 79 x |. 



2. 86 x T j. 



3. 1423 x T ? 



4. / ff x 112." 



5. J T x 476. 



6. /,-V x 2. 



18. Find the products indicated in the following expressions :- 



1. 4J x 



2. 7| x 



3. 49 f V x 

 4*225*1 



5. 1000*. X 



6. 47685* x 376|. 



19. Divide th'e following fractions by each other, according to 

 the indicated expressions : 



4. |j * II I 



20. Find tho quotients indicated by the following expressions : 



1. * -f- 9. 



2. A -7- 12. 



3. Jf r ~ 11. 



4. if -f- 52. 



400. 



21. Find the quotients indicated by the following expressions : 



1. 12 H- f. 



2. 172 -T- *-. 



3. 5 -H _j_ 



4. 1000 -T- 



5. 1 * T J-5. 



6. 75 -T- J. 



22. Find the quotients indicated by the following expressions : 



1. 112 -4- 7*. I 3. 1000 -f- 1*. I 5. 1 -7- If. 



2. 160 -f- 9|. I 4 800 -i- 800J. | 6. 2 -f- 2000*. 



23. Findthe quotients indicated by the following expressions: 



1. 17-| -4-7. [ 3. 1400SJ -T- 9. I. 5. lOOOf ^ 18. 



2. 100 -T- 12. I 4. 4783965^- -T- 112. | 6. 1J- -=- 800. 



24. Find the quotients indicated by the following expressions : 



1. 7* -f- 5i. I 3. 407| -T- 55i. I 5. 1000| -T- 10|." 



2. 93^ -T- 17^. I 4. 1423? -r J. 1 6 If H- 21J. 



