142 



THE POPULAR EDUCATOR. 



April 26th. 



Keceived of Thomas Jones, of Liverpool, 

 The following Beruittanees in Bills : 



No. 4, drawn on Parker and Co., due *Iay llth 

 No. 5, ,, Baring and Co., r une 3rd . 



27th. 



Sold at Liverpool, per Thomas Jones, 

 24 bales Madras Cotton (on credit), 



Net 8068 Ibs., at 6^d. per Ib 



His Commission and other expenses . . . 



190 10 6 

 36 10 



218 10 2 

 592 



213 1 



LESSONS IN ARITHMETIC. XXVIII. 



FRACTIONS IN CONNECTION WITH COMPOUND 



QUANTITIES. 

 1. To find the Value of any Fraction of a Compound Quantity. 



It is evident that we have only to divide the given compound 

 quantity by the denominator, and then multiply by the nume- 

 rator. The first part of the process determines the magnitude 

 of the equal parts into which the denominator indicates that the 

 quantity is to be divided, and the latter takes as many of those 

 parts as are indicated by the numerator. 



Thus, ! of 1 = 3 x Vs. = 15s. 

 Again, T \ of 2 = A of 40s. =- 4 x 8 . = 4 * 8 B = Vs. ; 



Vs. =10|s. 3 



Again, ^f of a shilling is I of 12 pence, or 2 x Yd. = gd. 

 Therefore, -,V of 2 = 10s. 8d. 



2. Find | of 3 days 4 hours 25 minutes, 

 days. hrs. min. 



3 4 25 

 24 



72 + 4 = 76 

 60 



hra. 



4560 + 25 = 4585 minutes. 



Therefore, f of 3 days 4 hours 25 min. = 1 of 4585 minutes 

 = 2 x 917 = 1834 minutes. 



Beducing 1834 minutes to higher denominations, 

 6,0 ) 183,4 



24) 30 



34 minutes. 



1 ... 6 hours. 

 Therefore, f of 3 days 4 hrs. 25 min. is 1 day 6 hrs. 34 min. 



3. To reduce one Compound Quantity to a Fraction of any 

 other. 



Finding what fractional part of one compound quantity 

 another given compound quantity is, is called reducing the latter 

 quantity to the fraction of the first. 



Thus, finding what fraction of one pound 6s. is, is reducing 

 6s. to the fraction of a pound. 



This is, in fact, only another name for performing the opera- 

 tion of dividing one compound quantity by another, or of finding 

 the ratio of two compound quantities (see Art. 11, Lesson 

 XXVII., page 101). 



4. EXAMPLE. What fraction of .1 7s. 6d. is 3s. 6d.? 



1 7s. 6d. = 330 pence. 

 3s. 6d. = 42 pence. 

 Hence, if 1 7s. 6d. be divided into 330 equal parts, 3s. 6d. 



contains 42 of them ; 



Therefore, 3s. 6d. is s Vo of 1 7s. 6d. ; and ,*& = /r when 

 reduced to its lowest terms. 



This might have been more shortly performed as follows : 

 1 7s. 6d. = 55 sixpences. 

 3s. 6d. = 7 sixpences. 

 Therefore, 3s. 6d. is -ft of 1 7s. 6d. 



5. Hence the following 



Rule for reducing one Compound Quantity to the Fraction of 

 another. 



Having reduced them both to any the same denomination, 

 take the first quantity for a numerator, and the latter for a 

 denominator. 



The denomination to which the quantities are to be reduced 

 is a question of choice. Sometimes we can much simplify our 

 alculation by choosing one rather than another. 



6. EXAMPLE. Reduce 2 6s. 8d. to the fraction of 3 5s. 



6s. 

 5s. 



8d. = 2| pounds. 

 Od. 3J pounds. 



Therefore the required fraction is r, which is x ^, i.e., }$. 



O j 



7. The question of Art. 2 might also have been solved as 

 follows by the aid of the present method : 



25 



25 minutes = JJ of au hour = of a day. 



60 x 24 

 4 hours = j\ of a day = J of a day. 



Therefore, 3 days 4 hours 25 minutes are 3 + J + 



25 

 60 x 24 



days, 



or 3,V,- of a day, or ?J of a day 

 Therefore, ? of 3 days 4 hours 25 minutes = f x JJJ of a day. 



= I U = 1J-SJ of a day. 



$?,'; of a day = fJJ x 24 hours = \'J hours = 6JJ- hours. 

 sJ of an hour = iJ x 60 minutes = 34 minutes. 



Hence the required answer is 1 day 6 hours 34 minutes. 



EXEP.CISE 47. 



Examples in finding a Fractional Part of a Compound Quantity, 

 and in reducing One Quantity to the Fraction of another oj 

 the same kind. 

 Find the value, expressed in successive denominations, of- 



1. * of 1; Jof 1; f of Is. 



2. j of 1; f of Is. ; fi of 3s. 2d. 



3. f of 1 Ib. avoir. ; f of 1 oz. Troy ; ^ of 1 cwt. 



4. | of 1 ton ; $ of 1 yard ; f of 1 rod. 



5. | of 1 mile ; f of 1 gallon ; |j of 1 peck. 



6. f of 1 hour; f of 1 minute ; f of 1 degree. 



7. 3f of f of a mile ; of ^ of a week. 



8. 



24 of a guinea; "j- of a crown. 



9. If If of a sum of money = | 

 10. Find the value of i* + | 

 of i" of 2s. 



of 5s. 10d., find it. 



fl of 2 - 



6fd. + J of 3 2s. 



15. Find the value of 



J-S of 1. 



"" 



11. Find the sum 



of 5 7s. 



12. And of of 16~s. 6Jd. + J of 12s. 10-Jd. + \ of 2 4s. 



13. Find the value of ^ of 7 4s. Id. -fa of 4 Os. Id. 



14. Find the value of -fa of 4 Os. Id. -f s of 3 10s. Id. 



"LzifL* 



f of f + of 

 Reduce to the fraction of a pound 



16. 4|s. ; 4s. 7d. ; 9s. 2Jd. 



17. 13s. 03-d. ; 3id. ; 3 15s. 9d. 



18. What part of 1 is of a penny ? 



19. What part of 1 Ib Troy is 7 oz. ? 



20. Eeduce % of a quart to the fraction of a gallon. 



21. Eeduce * of 1 secou to the fraction of a week. 



22. Reduce 3 17s. ll'd. to the fraction of 7 3s. 



23. Reduce 3 pecks 2 gallons to the fraction of 2 bushels. 



24. Reduce 15 cwt. 65 Ibs. to the fraction of 2 tons 3 cwt. 



25. Reduce 1 15' 10" to the fraction ol a right angle. 



26. Eeduce 1} acre to the fraction of 5 acres 2 r. 40 p. 



27. Reduce ,fr of 1 to the fraction of a penny. 



28. Reduce iV of a week to the fraction of a minute. 



29. Reduce f of | of 2 8s. 9d. to the fraction of 1 Is. 8d. 



30. Express as a fraction of 10 tho difference betwen 



and | of 8. 



31. Find the value of 



32. Find the value of 



136 gals. 2 qts. 

 178 gals. 3 qts. 



33. Find the value of 



2 Ib. 7 oz. 10 dwts. 

 77 dys. 4 hrs. 30 

 6 dys. 



4 dw *?: of 15 guineas. 



1 IT 

 17' 15" 



8f 



of 



of 517 square feet 72 inches. 

 3 18s. 8d. 



2s. 9d. 



of 104 yards 9 inches. 



KEY TO EXERCISES 44, 45, 46, LESSONS XXVI., XXVII. 

 (Vol. II. pages 78, 102.) 



