ONS IN 



-SONS IN Ai;rni.Mi:ri<'.--\ i. 



Aliki; 1IOD8OPM I > DIVISION. 



f multiplication and : i in t!m 



" . :MI:I.I: loaTUor 



himself perfectly familiar with tin m before pro- 



1-08808, however, in particular oases, can often bo 

 . various artifices. Some of those shorter 

 iilijuin, not only bocauHo they are useful in them- 

 selves, but booaiiMo they arc valuable as exorcises, in explaining 

 tho fundamental principles of arithmetic. 



J. Any number which is formed "by multiplying two or more 



numbers or factors together is called a composite number. It 



i>ody boon explained in a former lesson that tho same 



mimb'Ts multiplied together will give tho same product, in 



whatever in-ill"- tin! multiplication is effected. 



Hence, to multiply any number by one which is composite 

 i.e., which is composed of several factors we 

 have only to multiply the number first by one 

 factor, tho result by another factor, and so on. 



Thus, to multiply 352 by 28, since 28 = 7 x 4, 

 we can perform the operation as indicated in 

 the margin. 



3.VJ 

 7 



9856 - 28 x 353 



EXERCISE 10. 



(1.) Eesolvo tho following sets of numbers into their factors: 



1, 9, 10, It, H. 

 . 5t, 56, 63. 

 3, 45, 72, 64, 81, 96. 



4, 8, 16, 18, 20, 24. 



5, 27, 32, 36, 40, 48. 



6, 12, 28, 54, 72, 84. 



(2.) What will 2 1 horses cost at 74 crowns apiece. 



(3.) What will 45 hogsheads of tobacco cost, at 128 crowns 







(4.) What will 54 acres of land cost, at 150 crowns per acre ? 



(5.) At 118 shillings per week, how much will it cost a family 

 to board 49 weeks ? 



(6.) If a man travel at the rate of 72 miles a day, how far 

 will he travel in 64 days ? 



(7.) At 163 crowns per ton, how much will 72 tons of lead 

 cost? 



(8.) What will 81 pieces of broadcloth cost, at 245 shillings 

 apiece ? 



(9.) What will 84 carriages cost, at 384 crowns apiece ? 



(10.) What will a railway 132 miles in length cost, at the 

 rate of .1,960 a mile ? 



(11.) If I can walk a mile in 16 minutes, how long will it take 

 me to walk 374 miles ? 



3. Similarly, it will be seen that to divide by any composite 

 (7} 9856 number, we have only to divide by one 



23 ) ' L factor, then divide tho quotient by another 



\4/ 1408 factor, and so on. 



Thus, to divide 9856 by 28, arrange tho 

 352 Answer, process as indicated in tho margin. 



In this case there is no remainder. But suppose it bo re- 

 quired to divide 9873 by 28. 



?1 9873 Proceeding as before, wo get a remainder 3 



after tho division by 7, and a remainder 2 after 

 the division by 4. The first remainder 3 means 

 3 units ; and the 2 which remains after 

 dividing 1410 by 4, means two sevens of the 1410 

 sevens which are contained in 9873. Hence the whole remainder 

 will be 2 sevens + 3 units i.e., 17. 



The process may be exhibited analytically thus : 



9873 - 1410 x 7 + 3 



352x4x7+2x7 + 3 

 = 352 x 28 + 14 + 3 

 352 x 28 + 17 



Therefore 9873 divided by 28 has a quotient 352, and remainder 

 17. Hence, when there are two factors, to find the whole re- 

 mainder, multiply tho second remainder into the first divisor, 

 and add the first remainder. 



4. If there are more than two factors, similar considerations 

 will show that the following rule will give the whole remainder : 

 Multiply the last remainder into the continued product of all 

 the divisors but the last the lost but one remainder into the 



1410-3 



product of all tho divisors except the laat two, and no on. Add 

 all th0M result* and tho first remainder together ; the ram will 

 bo the whole rcmu 



EXAMPLE. To divide 17285 by 84. 



ftt-7 x 4x3 



7; 17286 



And tho whole remainder I* 2x1*7 + 1*7 + 2, that i, 

 56 + 7 + 2, or 5. . 



It does not follow that this process is in all cases oimpler 

 than tho method of Long Division ; sometimes, howev 

 more convenk-ut. 



EXERCISE 11. 

 (1.) Work tho following examples in division : 



1. ICU-s + u;. 4, 9ioeo + 72. 



2. 17220 + 84. 



3. 25760 + 56. 



5. 142857 + 112. 



6. 123456 + 168. 



(2.) How many acres of land, at 35 crowns an acre, can you 

 buy for 4650 crowns. 



(3.) A man divided 837 crowns equally among 27 persons, 

 who belonged to three families, each family containing nine per- 

 sons : how many crowns did each person receive ? 



(4.) A man bought a quantity of clover seed amounting to 

 507 pints, which ho wished to divide into parcels containing 64 

 pints each : how many parcels can ho inako ? 



5. Multiplying and dividing by powers of 10, and by numbers 

 ending in any number of cipJiers. 



The products of two tens, three tens, four tens, etc., are called 

 respectively the second, third, fourth, etc., powers of 10. They 

 are 100, 1000, 10000, etc. Thus, the second power is 1 followed 

 by two ciphers, the third 1 followed by three ciphers, and so 

 on ; the number of the ciphers in each case being tho same as 

 that of the power. 



It has been already explained that to multiply by 10, or any 

 power of 10, wo have only to annex to tho multiplicand the 

 number of ciphers corresponding to tho power. Thus, 345 

 multiplied by 1000 is 345000. 



If any number of the right-hand figures in the multiplier be 

 ciphers as, for instance, in 75000 then, as we have already 

 seen in Lesson IV., Art. 5, wo need only multiply the multiplicand 

 by 75, and annex to the product tho same number of ciphers, in 

 this case three. 



EXERCISE 12 



(1.) Work the following examples in multiplication : 



1. 153485 x 10000. 



2. 3120467 x 100000. 



3. 52690078 x 1000000. 



4. 689063157 x 10000000. 



5. 4946030506 x 100000000. 



6. 87831206507 x 1000000000. 



7. 67856005109 x 10000000000. 



8. 14376 x 25000. 



9. 350634 x 410000. 

 10. 4630425 x 6200000. 



11. 2370000 x 52. 



12. 48120000 x 48. 



13. S56300000 x 74. 



14. 1623000000 x 89. 



15. 540000 x TOO. 



16. 1563800 x 20000. 



17. 31230000 x 120000. 



18. 5310200 x 3400000. 



19. 82065000 x 8100000. 



20. 210909000 x 5100000. 



(2.) What will 10 boxes of lemons cost, at 63 shillings per 

 box? 



(3.) How many bushels of corn will 465 acres of land produce, 

 at 100 bushels per acre ? 



(4.) Allowing 365 days for a year, how many days are there 

 in 1000 years ? 



(5.) How much will 50 hogs weigh, at 375 pounds apiece ? 



(6.) If 1 barrel of flour weighs 192 pounds, how much will 

 500 barrels weigh ? 



LESSONS IN GEOMETRY. III. 



INSTRUMENTS USED IN PRACTICAL GEOMETRY. 

 IN the operations of practical geometry, a ease of mathematical 

 instruments must be considered as an essential requisite. 

 These instruments vary in number and quality, according to 

 their price. Some are made of wood, bone, and ivory as 

 rulers and scales ; others are made of brass and steel, German 



