198 



THE POPULAR EDUCATOB. 



jcncr tft fieufc fef;r (abm. 10. SBent fdjtcfen ie ben fdjencn 3ling ? 11 

 3fy fcfjicfe t^n bent SJZanne, n>eld;en ie fo fetyr getobt I;a0cn. 12. -fcakn 

 @ic bie grcunbe meineS JBrubev! getofct ? 13. 3a, id; fyabc fie getofct. 14 

 abcn ie btcfclfcen nid;t gclicbt ? 15. 3d? fyafce eine Heine @d>n>efter 

 tt>e(d;c id; Itcfee; lie&en @ie bicfelfee ? 16. 2)er %im liefct (einen Sleffen 

 abet berfet&e tft untanfbar. 17. 3)er SJater Uef>t feinen Heinen o!)n, line 

 betfetbe gut tft. 18. SBarum fmb fo ie(e tuwm in bcr Statt? 19 

 SQ3eU fie "S bem JTricge gefommen ftnb. 20. SBarum liefccn uitS unfen 

 <ltcrn ? 21. 2Bei( n)ir ifjre Jlinfcer fmb. 22. 3u n>em ge(;cn ic ? 23 

 3d; gefye $u meinem SSetter. 24. 2)lit ioem gef;cn ie ? 25. 3d; gcfje mit 

 mcinem SSkuber. 



EXERCISE 36. 



1. Is your brother at homo ? 2. Yes, but he is ill. 3. Where 

 have you bought this watch ? 4. I bought [gefaitft] it of the 

 watchmaker. 5. These rings are beautiful, will you give me one 

 of thorn ? 6. The troops which went to Leipsic returned yes- 

 terday. 7. The teacher loves the boy, because ho writes beaiiti- 

 fully. 8. Do you go to your parents ? 9. I go with my brother 

 10. These children love their teacher, because he is good to 

 them. 11. Do you require my books any longer ? 12. I wili 

 give you them back [juntcf] to-morrow. 



LESSONS 



ARITHMETIC. XIII. 



DECIMALS (continued). 



6. IT is evident, also, from the explanations given in Lesson 

 XII., that to multiply a decimal by any power of 10, we need 

 only move the decimal point as many places to the right as 

 there are ciphers in the multiplier. For example : 



345S7 x 100 is 34'567. 



Tor -34567 x 100 - iVJaVo x 100 = Vol" = 34'567. 



Similarly, to divide a decimal by any power of 10, we must 



move the decimal point as many places to the left as there are 



ciphers iu the divisor. If there are more ciphers in the divi.sor 



than there are places in the decimal, we must prefix a sufficient 



number of ciphers (Art. 5). For example : 



456-329 + 100 is 4'56329. 



For * 58 ^? 29 -- rz- = isow? 4-56329 



*X \<M 1000 x 10 y loiwuu 



329 * 100 is -00329. 



399 



For ^ 2 = - - 32 J = -00329 

 * or 100 1000 x 100 10UUUO 



Here, in order to move the decimal point two places to the left, 

 we must place two ciphers before 3, the first significant digit of 

 the dividend. 



EXHECISE 29. 



1. Express as decimals 



I- A, TSo, fS- 



2. 25/ ff , Via, 9ro53o- 



3. 7^0, 43iAA^, 3 wjiiVa. 9fJ-^;ii, 7 ;a. 



2. Express as fractions, or mixed numbers 



1. '32, -246, -3624. 



2. -03687, "00046. 



3. 42-068, -007006, 1-100492, 'OOOW08. 



3. Multiply and also divide each of the decimals in the pre- 

 ceding examples by 100 and by 1 0000. 



4. Divide '1 and 40'0059 by 10000 and also by 10000000. 



5. Express as fractions or mixed numbers the following 

 decimals : 



82344 

 13236 

 46274 

 00368 

 00009 



17-401 



23-07 



81-4339 



90-0104 



12-683 



20-064 



35-0072 



67-4008 . 



9-0007 



6-00754 



3-0463 



2-306843 

 1-710236 

 2-463126 

 6-004534 

 9-000028 

 8-001249 

 0-100010 

 4-306702 



1-13004 



9-203167 



9-2000076 



8-0403842 



4-3008004 



7-4627350 



1-0006003 



5-8493001 



6. Write the fractional part of the following mixed numbers 

 in decimals : 



1. 30^ 1 3. Sj'Jjj I 5. 41 T = ,^, 5 



2. 72^% 4. ISyiftky 6. B^,*^ 



7. Addition of Decimals. 



To add together 28'35, 345-3294, -0018, and 6'4. 



Write the units under units, tenths under tenths, ete.; or, 



what is the same thing, write the decimal points under one 

 another, and then proceed to add thus : 8 ten thousandths and 

 4 ten thousandths are 12 ten thousandths, i.e., 1 

 thousandth and 2 ten thousandths ; write down 2 

 under ihz ten thousandths' place, and carry the 1 to 

 the ness cciimiu of figures, as in simple addition. 

 The same method will evidently apply for all the 

 columns, since the value of each place of figures in- 

 creases tenfold from loft to right. The decimal point in the 

 answer mil clearly fall under the column of decimal points. 

 We oay also exhibit the process thus : 



28-Sfs = Iff, 345-3294 = si^, -0018 = I J^ 5 , 6-4 = %. 



And -bnerefcre reducing all these fractions to a common denomi- 

 nator, IOOO&, cad adding them, we get for their sum- 



28-35 

 345-3294 

 0018 

 6-4 



380-0812 



283500 + 3453294 + 18 +64000 

 10000 



= 380-0812. 



Hence we get the following 



Rule for tJie Addition of Decimals. 



Write the decimals under one another, so that the decimal 

 points may fall under each other. Begin at the right hand, or 

 column of the lowest order, and add as in simple addition, 

 placing the decimal point in the row of figures so obtained 

 under the other decimal points. 



EXERCISE 30. 



1. Find the sum of the following decimals : 



1. 25-7, 8-339, 23'OoG, and 57'145. 



2. -001G2, -1701, 325, 27031, and 3'000701. 



3. 1-03041, 6-578031, 2-4178, and 472103. 



4. 467-3004, 28-78249, 1-29468, and 3782il. 



5. 293-0072, 89-00301, 29-84567, 924'00369, and 72'39602. 



6. 394-01, 81-928, 3624 '8103, 640'203, and 51216291-30003. 



7. 36-258, 2-0675, 382'45, and 7'3984. 



8. 327o4, 5-78, 16'0037, and 49 '3046. 



9. 4-25, ti 293, 4'612, 3S'07, 2'056, 3'248, and 1'62. 



10. 35-7603, 47-0070, 129'03, 100'007, and 20'32. 



11. 24-6434, SOO'7, 29'461, 1-7506, and 3'45. 



12. 45-001, 163-4231, 20-3045, 634-2104, and 234'90213. 



13. 1-721311, 8-620047, 51720345, 2'684, and 62-304607. 



14. 1-293062, 3-OOOi2, 97003146, 3'600426, 7-0040031, and 87200489. 



2. Add together the following, after writing them as decimals: 



1. 45 thousandths, 6 millionths, 9 tenths, and 11 ten millionths. 



2. 25 huudredtlis, 8 tenths, 65 thousandths, 16 hundredths, 142 

 thousandths, and 39 huiidredths. 



3. 9 tenths, 92 hundredths, 1C2 thousandths, 489 thousandths, and 

 92 millionths. 



4. 29 hundredths, 7 millionths, 62 thousandths, and 12567 ten 

 millionths. 



5. 95 thousandths, 61 millionths, 6 tenths, 11 hundredths, and 265 

 hundred thousandths. 



6. 1 tenth, 2 hundredths, 16 thousandths, 7 millionths, 26 thou- 

 sandths, 95 ten milKonths, and 7 ten thousandths. 



7. 96 hundred thousandths, 92 millionths, 25 hundredths, 45 thou- 

 sandths, and 7 tenths. 



8. Subtraction of Decimals. 



It is evident, from the remarks we have made with respect to 

 the addition of decimals, that the process of subtraction will be 

 performed in exactly the same way as in simple subtraction. 



Thus, to subtract 3'275 from G'14, we write the decimal 

 points under each other, as in the margin, adding a cipher 

 to 6" 14 for convenience, to make the number of decimal 

 places correspond with that of the number to be sub- 

 tracted. We then say borrowing 1 (really ^ or jJJk) 

 Torn the next highest order of figures, as in simple addi- 

 tion 5 from 10 leaves 5, then 8 from 14 leaves 6, and so on, 

 ;he decimal point in the row of figures obtained falling under 

 ho other decimal points. 



We may also exhibit the process as follows : 



6-140 

 3-275 



2-865 



6-14 - 



3.97C 

 A'd 10 



Therefore 6'14 3-275 = TJOO- = fSSo = 



06s. The methods of simple addition and subtraction apply 

 :o decimals, because the only condition upon which their truth 



depends is, that the places of figures should increase in value 

 n a tenfold ratio from right to left, which is the case with 



decimals. 



