THE POPULAE EDUCATOR. 



LESSONS IN ARITHMETIC. XV. 



DECIMALS (continued). 



10. Division of Decimals. 



CASE 1. Divide 120-3033 by 3'27. 



120-3033 -i- 3-27 = ^ggg? -f- ?& - J?gip X t^Jo = ^ 

 3679 is the quotient arising from dividing the dividend by the 

 divisor as if thoy were whole numbers, and the denominator 100 

 shows that there must be two decimal places in the quotient. 

 These two decimal places arise, as will be seen by the fraction 

 Su8ub> from the . fact of there being two decimal places more in 

 the dividend than in the divisor. 



CASE 2. If the number of decimal places in the divisor and 

 dividend were the same, the result would be exactly the same 

 as if the divisor and dividend were whole numbers. Thus, 



1203-033 -4- -327 = - 3 -=- $Jb = ^$22 X jgg = 3679. 



CASE 3. Suppose that there are more decimal places in the 

 divisor than in the dividend. 

 ' Take, for example, 120303-3 -f- '327. 

 120303-3 -T- -327 = ^^ 4. gg. - isosp x ^ = 367900. 



The true* quotient in this example is an integer, but it will 

 not be so in all cases. . 



It will be better in practice, before commencing the operation, 

 to annex ciphers to the dividend sufficient to make the number 

 of decimal places equal to the number in the divisor, in which 

 case the result will be exactly the same as if the division had 

 been in whole numbers. 



ADDITIONAL EXAMPLE OF CASE 2. Divide 411-95 by T25. 



1-25) 411-95,00 (329-56 

 375 



1195 

 1125 



700 

 625 



750 

 750 



Dividing as in whole numbers, we get a quotient 329, and a 

 remainder 70. Now annex ciphers to the dividend, which will 

 not alter its value, and continue the division. We now find 

 that, the true quotient is 329'56. 



ADDITIONAL EXAMPLE OF CASE 3. To divide 356-7 by 2-31. 

 Annexing a cipher to 356'7 before commencing the operation, 

 we have 



2 31 ) 336-70 ( 154 

 231 



1257 

 1155 



96 



The part of the true quotient already obtained is an integer, 

 the division being in fact the same as that of 2 Sp. If more 

 ciphers be annexed to the dividend, we shall get decimal places 

 in the quotient, and the more we obtain the nearer to the true 

 quotient shall we arrive. 



11. These examples will sufficiently illustrate and explain the 

 following 



Rule for the Division of Decimals. 



Divide as if the divisor and dividend were whole numbers. 



If the number of decimal places in the dividend exceed the 

 number in the divisor, cut off from the quotient as many 



* ^Ve shall use the expression true quotient to indicate the total 

 result obtained by the division of ono number by another, thus distin- 

 rniishins it from the quotient defined in Lesson V., Art. 1 (page 69), 

 wtiich is only the Integra', part arising from a division. 



decimal places as are equal in number to this excess, prefixing 

 ciphers if necessary. 



If the number of decimal places in the dividend and divisor 

 be equal, the division will be the same as in whole numbers. 



If the number of decimal places in tho dividend be less than 

 the number in tho divisor, annex as many ciphers to the dividend 

 as will make the number equal to the number in the divisor, 

 and then proceed as in whole numbers. 



12. We subjoin other examples of division of decimals. 



EXAMPLE. Divide 1 by 10-473, carrying the quotient to 5 

 places of decimals. 



We are at liberty to write 1 thus 1 -00000, putting as many 

 ciphers after the decimal point as may bo required. Since there 

 aro to be 5 decimal places in the quotient, and since there are 

 3 in the divisor, we must add 8 ciphers. 



10-473) 1-00000000(9518 

 94257 



57430 

 52365 



60650 

 41892 



87580 

 83781 



3796 



Hence tho required answer is '09548, prefixing a cipher In 

 order to get 5 decimal places in the quotient. 



13. EXAMPLE. Divide '8 by -00002. 



Annexing 4 ciphers to '8, since there arc 5 decimal places in 

 the divisor, we have 



00002) -80000(40000 



the division by tho rule being, in fact, the same as that of 

 80000 by 2. 



14. It will be observed that wo are not required in some 

 cases to find more than a certain number of figures of tho 

 quotient when it is a decimal. Sometimes, by continuing the 

 division far enough, we shall find that there is no remainder 

 i.e., that tho quotient can exactly be found in the form of a 

 decimal. But if by continually dividing we cannot arrive at a 

 stage where there is no remainder, then we can only get what is 

 termed an approximation to the result. Tho more figures of the 

 quotient we take, the nearer we shall be to the value of the 

 truo quotient. 



Thus, in the division above performed in Art. 12, if we 

 stopped at four decimal places in tho quotient, the result would 

 be "0954. Carrying on the operation one step further, we seo 

 that 8 is the next figure of the quotient, and therefore this 8 

 meaning ^^ we aro nearer to tho true quotient by usSjoa- 

 Where wo are required to find a quotient to a given number of 

 places, it is customary to carry on the division to one place 

 more than is actually required, in order to see whether tho next 

 igure is greater or less than 5. If it is greater than 5, then we 

 shall be nearer to the true result if we increase the last figure 

 of tho required number of places by unity. 



Thus, in the case above given, finding that the fifth decimal 

 place is 8, tlia quotient to four decimal places will be more, 

 accurately written '0955 than '0954, because -0955 or, what is 

 the same thing, '09550 is nearer to '09548 than -09540 is. 

 Now -09550 is ^^ more than -09548 5 whereas -09540 is j^g 

 less than -09548. 



The same method is applied whenever a limited number of 

 decimals is employed. We shall return to this subject hereafter, 



EXERCISE 33. 

 1. Find the quotients of the following examples in division of 



decimals : 



