o
00
co
O
>
LIBRARY
01 THK
UNIVERSITY OF CALIFORNIA.
( )K
Y
Received G2,c3f~ .
Accession No. 7^3 bO . Clots No.
ARITHMETIC
SHORT METHODS
* SWEET *
OF THE
UNIVERSITY
SWEET'S
Hand Book
OF
SHORT METHODS
Arithmetic
j. s. SWEET, A. M.,
'Principal of the Santa Rosa IJusiness College. Santa Ho.a, Cal.
formerly President of the Oix^oii State Xormal School.
Ashland. Or., Author of Sxveet ' Sv~um of Act
nal Hnsiness Practice, Element* of Geom
etrv, lousiness t"orm~. Etc.
SANTA ROSA, CALIFORNIA
Entered according to Act of Congress, in the yi;ir IS!)!!.
ByJ. S. SWEET,
In the Office of the Librarian of Congress, at Washington, D. C.
>.
J
PREFACE.
The principal object of this little work is to place in
the hands of the student, in compact form, many of the
briefer methods of rapid calculations. ''Time is money,"
and especially so to many of our young people who are
trying to obtain a business education in a brief time and
with limited means.
Hoping that many may profit by the suggestions here
in contained, I most respectfully dedicate this little volume
to the young business people of America.
Santa Rosa, Calif., 1893. J. S. SWEET.
6 XIIORT METHOD*
2. Slims Greater than 9.
54321543
56789678
6543 654
6789 789
765 76 87
789 89 89
8 9
9 9
.V. To Itetrtl fit Sight.
When a student sees the figures 1 and 3 written side by
side, he instantly recognizes "thirteen" or "thirtyone" ac
cording to their positions. The same facility may be ac
quired in regard to numbers in addition; thus, 4 over or
under 8, may be read "twelve" as readily as the figures 1
and 2 side by side. Ten minutes practice daily for one
month will accomplish the work.
4. Always add TWO or MORE figures at a time. Never
be guilty of adding single figures. Name the results of the
following as rapidly as possible :
24697563467489
35323678988723
38765725475399
48797999888789
IX AltlTHMKTIC.
56854322462537
49987873878689
73776298897778
84698549739894
J. Nine added to any number is always ONE LESS in its
unit's place than the number. Thus,
8 9  7 in its unit's place.
36 9 5
6*. Eight added to any number is TWO LESS in its unit's
place than the number. Thus,
7 8 == 15, 158 = 23.
T. To Add btf TenM.
A good method is to add by 10's, carrying the EXCESS
in the mind, as in the following :
8" 7 2
9 5
63 95
7 6
30 27
Here the 3 of the 13 is carried to the 7 of the 17 mak
ing three tens in all. Add in this manner the following:
3
9
6
5
9
8
8
8
8
8
7
5
5
7
9
9
9
5
9
6
4
3
4
4
5
6
4
9
8
6
8 *1IORT MKTHOIt*
8. When the Columns are Long.
When there are two or more columns of consider
able length, add each column separately as instructed, and
write the sum of each alone, then combine results into
one number, as follows :
32476
58976
76892
39428
73548
67943
28745
"^8
37
46
43
33
378008
This method is almost indispensable in bookkeeping, as
an error can be located much more readily than when the
separate results are not known.
9. To Add Two Columns at a Time.
To add two columns at a time practice on the fol
lowing, by adding the tens' column first, and by reading the
units' column, tell at a glance the number to carry :
23
36
72
49
35
44
66
27
38
79
38
44
59
71
88
64
39
89
88
26
86
49
94
87
75
89
85
94
f.\ ARITHMETIC. 9
10. Proof* of Addition.
In long columns the best proof is to add them again, up
or down, the opposite of your first addition. In short col
umns and several of them to add, you may prove the work
by casting out the 9's as shown below.
25189654  4
36972105  6
94375517  5
15155815  4
85310652  3
95315175 
352318918  4
Casting out the 9's of the first nnmber, we have an e.<
of 4 ; of the second, 6 ; of the third, 5 ; and so on, finally
casting out the 9's of these results which gives an excess of
4. Also by casting out the 9's of the sum, we have 4, we
therefore conclude that the work is correct.
XOTK. This is not always a sure test, the answer mi_<rht he wrong 1 and
vet prove bv this method.
SUBTRACTION
11. When the fortyfive combinations treated of in Ad
dition are thoroughly memorized, the process of subtraction
is a very simple one. This consists of being able to discern
at a glance the digit which will combine with one of those
given to produce the other. Thus,
8
3
are given, and the question is : what number combines with
3 to produce 8? The process is nearly the same as in ad
ding, the only difference is that we must furnish one of the
numbers to the combination, the result already being known.
Read the differences as rapidly as possible :
98767896757988
43234454326325
15 16 17 14 13 12 18
8986789
Daily drills in both addition and subtraction should not
be neglected. The process of this method is very simple
and is readily learned. Practice, only, will perfect it and
give value to it.
MULTIPLICATION
7V. With Multiplication we begin our Slinrt Methwls.
supposing the student to be sufficiently advanced to know
the multiplication table to the 12's. If not, he should learn
the following
MULTIPLICATION TABLE:
1
>
3
4
5
6
7
o
10
11
12
2
4
o
10
12
14
16
18
20
22
24
3
6
9
12
15
18
21
24
27
30
33
36
4
8
12
16
20
24
28
32
36
40
44
48
5
10
15
20
25
30
35
40
45
50
OD
60
6
12
18
24
30
42
:>4
60
66
72
,
14
21
28
42
4U
56
!63
70
77
84

16
Mi'
40
48
56
64
80
88
96
9
18
27
45
")1
68
72
81
90
99
108
10
20
40
:,d
C)()
70
80
90
100
110
120
11
22
33
44
55
66
. i
88
99
110
121
132
12
24
36
48
60
12
84
96
108
120
132
144
//>. The following squares of numbers should also be
memorized :
12 XI10RT METHODS
13 X 13 == 169 19 X 19 361
14 X 14 = 196 20 X 20 == 400
15 X 15 = 225 21 X 21 = 441
16 X 16 256 22 X 22 == 484
17 X 17 == 289 23 X 23 529
18 X 18 = 324 24 X 24 576
25 X 25 = 625
14. To multiply any number const stint/
of two digits by 11.
RULE. \Vrite the *inn of the digit* between
ilieiti, ilie number fltii* e.vpre^ned ?'s I lir product.
EXAMPLES. 11 times 24 = 264,
11 " 36 == 396,
11 " 57 == 627.
XOTK. \Yluii tinit tun is 10 or inure, carry one to tin liutulivd's diit.
EXERCISES.
15. 1. Multiply 45 by 11. 4. Multiply 75 by 11.
2. 38 by 11. 5. 96 by 11.
3. 92 by 11. 6. 88 by 11.
16. To multiply any number by 11.
RULE. Write the unit's iigurc; next, write the
sum of the ^lnits and tens, tJifit i lie ^nin of 1/ir fin*
and hundreds, etc., writing flic left IKI'IK/ figure
last, carrying i^lien uec:es?:r\'.
EXAMPLE. 11 times 12345 == 135795.
5
4 j 5 9
:i 4  7
2 3 5
12 3
1
7.Y ARITHMETIC. 13
EXERCISES.
/;. 7. Multiply 663 by 11. 4. 6731 by 11.
938 by 11. .7. 9884 by 11.
734 by 11. 6. 72596 by 11.
18. To multiply by 22, 33, etc.
Rl'LK. Mil hi ply h\> 11 //> <i}>,>r<>, ami rhru !>\' ?, .V,
<>r 4, etc.
EXAMPLE. 22 times 234 =. 2574 X 2 = 5148.
XOTK. The work should be done mentally, only results being; written.
EXERCISES.
W. 7. Multiply 64 by 22 4. 374 by 55.
65 by 33. .J. 874 by 66.
46 by 44. V. 336 by 77.
20. To multiply by ant/ tuttnbet* betirecn
12 and 2O.
RULE. Multiply by the unifs figure only, i.'rir
mg the r<\<it/r uJt>riln> innitlx>r <m<l <mr ~p}<ir t > r<>
tin' n'hr, thru ,i,J<l.
.KS. 13 times 24 24
312 An*.
14 times 175 175
700
2450 An.
EXERCISES.
21. 1. Multiply 262 by 13 .7. 9624 by 17
382 by 14. 6. 32694 by 18,
497 by 15. 7. 27314 by 19.
.;. 1824 by 16. 8. 98794 bv 12.
22. To multiple/ by 21, 31, 41, fil, etc.
RULE. Multiply by the tens only, i^n'ti'tig Hie
result render ilie ninuhe r <i n <1 our pluee 1o ilie left,
then add.
EXAMPLE. 31 times 24 24
72
744
EXERCISES.
23. 1. Multiply 35 by 31. k 728 by 51.
& 46 by 41. 5. 3824 by 61.
3. 245 by 21. 6. 8452 by 71.
24. To multiplt/ by lo.
RULE. Anne.\ one eipli.er /<> ////' number <nnl <i<I<t
its half.
EXAMPLES. 15 times 28 = 280
V 2 of 280 = 140
420
15 times 35 = 350
175
525
EXERCISES.
25. 1. Multiply 44 by 15. 4. 248 by 15.
87 by 15. 5. 7634 by 15.
3. 394 by 15. 6. 98768 by 15.
20. To multipJt/ by XI.
RULE. Take onehalf the nmnber <tnd write it
two places to the left <nul (J<!.
_
flTNlVERSITY )
IX ARITHMETIC. 15
KXAMPLKS. 51 times 72 = 72
V 2 of 72 36
4372
51 times 45 = 45
225
2295
EXERCISES.
1. Multiply 78 by 51. 4. 1384 by 51.
324 by 51. ,5. 4633 by 51.
723 by 51. 6. 78254 by 51.
28. To sqttftre ft n tun her tr/iose tntft fiy
ttre is ,7.
RULE. Multiply rite /;/>' <ligjr />v i
<t 11 (1 '
EXAMPLE. 25 times 25 = 625.
2 times 3 6, annex 25 625.
EXERCISES.
1. Multiply 35 by 35. 7. 75 by 75.
' 45 by 45. 6. 85 by 85.
55 by 55. ?. 95 'by 95.
4 65 by 65. 8. 105 by 105.
V0. To find the prodttct of two nttnthers
units' (I if/ its nre ,7's.
RULE. To rJir product of the tens add onehalf
tln'ir :uin mxl <nnn\\ 25 if
XOTK. Fractions of onehalf are dropped.
X 6 SHORT
EXAMPLES. 25 times 45 1125.
1/2 of (2 + 4) h 2_X 4 = 11, annex 25 = = 1125.
25 times 35 =875.
1/2 of ( 2 h 3) f 2 X 3 == 8, annex 75 875.
NOTK.  plus 1? is odd.
EXERCISES.
/>'/. J. Multiply 25 by 65. 4. 45 by 35.
2. 25 by 85. 5. 65 by 35.
3. 105 by 25. H. 75 by 65.
32. To find the product of two numbers
whose tens 9 digits are identical and the sum
of the units 9 digits is 10.
RULE. Multiply the tens' digit by OTIC greater
and annexe the product of the, unit*' <l,igi1*.
EXAMPLE. 43 times 47 = 2021.
4X5 and annex 7X3 == 2021.
EXERCISES.
33. 1. Multiply 29 by 21. 5. 38 by 32.
2. 28 by 22. n. 37 by 33.
27 by 23. 7. 49 by 41.
4. 39 by 31. 8. 48 by 42.
34. To find the product of two numbers
whose tens 9 digits are consecutive,, anil the
sum of the units 9 digits is 10.
Rl'LK. To (lie product of tin* /<>ss ten* and one
more than the grfutcr, <nni<'.\ tin complement of
the square of the greater number' * unit figure.
XOTK. Complement of ;i munher i TOO les^ the numlur.
EXAMPLE 87 times 73 6351.
7 < 9 63; complement of the square of 7 = 51;
annex it to 63 6351.
IX ARITHMETIC.
EXERCISES.
Multiply 47 by 33.
56 by 44.
64 by 56.
9 by 71.
(>. 84 bv 76.
#6*. To jitttt the product of tiro numbers
tcheti their fats' (lif/its are the sattte.
Rl'LK. Ta,he the prmluct <>f tin' unir*, ne.\~r rhe
print ncr ;/s riling the sum <>f tlie unit*, then
flu' pr>J i/ cr nf i/ir rcn*, always r,i ~rr\i m? rli*' r'7;s,
if itii v.
EXAMPLE.
73 times 75
5 3
8" 7
7 7
5475
15 write 5, carry 1
56 carry 5.
4V)
5475
EXERCISES.
/. Multiply 74 by 72.
85 by 83.
67 bv 65.
4. 97 by 94.
.5. 88 by 89.
'/. 79 bv 78.
V#. To /r tt ft the product of two tttrtttbet's
trhett the units 9 digits are iflettfieftt.
Rl'LE. Talif r/i.,' pnnlitcr of the nn.it ^
the sum <>f t lir ten* times the liuit*, mul the product
of the tens, currying i^'In'n necessary.
KXAMPLK. 44 times 74 3256.
EXERCISES.
/ Multiply 46 by 56.
54 by 34.
43 bv 53.
4 73 by 63.
6. 87 by 47.
n. 98 bv 28.
OF THE
i8 XIIORT METHODS
40. To find the product of fntt/ two num
bers consisting of two digit*.
RULE. Take the product of the units, the sum of
the products of each ten times the other unit, and
the product of the tens, carrying if necessary.
EXAMPLE. 47 times 36.
6 X 7 ~ 42
6X4 a <" 7 = 45
4X3= 12
1692
EXERCISES.
41. 1. Multiply 35 by 27. 4. 68 by 34.
2. 47 by 34. 5. 78 by 46.
52 by 46. 6. 39 by 35.
42. To find th e produ ct of numbers wh <> 1 t
one part of the multiplier is a factor of the
other.
RULE. Multiply by the factor, then tliit? product
by the quotient of the factor into tin oilier part,
<i tnl
EXAMPLE. 231
183
Multiply by 3 = 693
" this product by 6 = 4158
42273
423
126
Multiply by 6 = 2538
" this product by 2 = 5076
53298
EXERCISES.
43. L Multiply 1247 by 255.
2. 792 by 279.
3635 by 1089.
IX ARITHMETIC, 19
44. Jty tnnltijrtu ft// the factors of ft num
ber.
RULE. Multiply by one factor <iml /// /.
other.
EXAMPLE. 21 times 65 == 7 times 65 = 455
and 455 3 1365.
EXERCISES.
4X. 1. Multiply 73 by 42. 4. 97 by 14.
83 by 35. 5. 87 by 36.
123 by 27. 6. 79 by 49.
46. ToiHHltijrti/bt/ 1O, WO, 10OO, etc.
Annex as tna>ny cipln'r* <i* there ure in
1 lie in it It i pi ier.
EXAMPLES. 10 times 76  760.
100 times 125 12500.
4;. To nitdtiptt/ ft// ft nu multiple of 1O.
10O, WOO, etc.
RULJZ. Multiply by the digital number and tlieji
<t n n e.\ ciphers.
EXAMPLE. 400 times 123 49200.
2000 times 243 = 486000.
48. To inultinlt/ ft// .9, or fftit/ number
of ,9'x.
RTLK. Annex 11$ many ciphers as there &re 9* s
and finhtrarr thr nuniher multiplied.
EXAMPLES. 9 times 435 = 4350 435 = 3915.
99 X 267 = 26700 267 == 26433.
20 ^IIORT MKTIIOD*
EXERCISES.
49. I. Multiply 47 by 9. J t . 148 by 9. 
2 125 by 9. 5. 725 by 99.
238 by 9. ' 6'. 675 by 999.
50. To multiply btj any number endint/
in 9.
RULE. Multiply by //// n<'.\f greater number
and from the product subtract the number multi
plied.
EXAMPLE. 382 times 49 = 382 X 50 382.
382
50
"19100
382
18718
EXERCISES.
,7/. 1. Multiply 128 by 69. 3. 326 by 599.
2. 245 by 59. //. 262 by 499.
32. To multiply by ant/ number a little
less or a little greater than 100, 1000, etc.
HULK. JLnnex as many ciphers ^ liters a re fi^
nrc* in th.c. multiplier <nn1 mihtruci <>r <nld, the pr<>
<hn;t of tlic difference between 10O, 10()(),<>t<\, ami
(lie multiplier.
EXAMPLE. 423 times 996 423000 4 >< 423.
._ 1(51)2
421308
EXERCISES.
,T.V. L Multiply 993 by 624. J. 9994 by 425.
0. 997 by 529. <i. 9998 by 827.
992 by 895. 7. 99993 by 963.
.',. 326 hv 104. 8 1003 by 724.
":
IX ARITHMETIC. 21
34. To multiply by any multiple of 1),
t e.rceed i ny i)O.
RULE. ^Multiply b\' the multiple <>/ mi
tlnni. t he git 'ru ntu fri pi irr, and iii>tr<'icr ?'rs
netenth.
EXAM PI. K. 454 times 72
454
80
36320 product by 80
3632 " " 8
32688 " " 72
EXERCISES.
,7,7. /. Multiply 4(5 by 18. 5. 288 by 54.
75 by 27. r>. 384 by 63.
82 by 36. 7. 772 by 75.
4. 144bv 45. 8. 1244 bv 81.5
OI
.76*. To multiply by complements.
P'roiu eitliLT number subtract the comf>li'
of the Other, <tinl <nnn\\ t/n> pr<nlucr <tf t
com
XOTK. Tlie product should have ;i~ mail v figure as are in both nuiii
S: sup]>ly oi]ihirs To maki thtm the same.
p]>ly oi]i
EXAMPLES. 94 comp. 6 999 comp. 1
97 comp. 3 999 comp. 1
9118 998001
A'd
EXERCISES.
,7;. /. Multiply 92 by 87. 4. 996 by 995.
94 by 75. 5. 993 byi9\L.
99 by 93. n. 998 by 895.
22 SHORT METHODS
58. To find the product of two tt a in hers,
each of which is a little over 1OO.
RULE. From the sitin, of the numbers snl>rr<ict
100 (nid (.inncjc the product of tltc ejccesses.
EXAMPLE. 115 times 104 == 11960
115 f 104 100 == 119
To 119 annex 15 X 4 == 11960.
EXERCISES.
39. 1. Multiply 114 by 105. 4. 144 by 107.
2. 122 by 103. .7. 160 by 106.
135 by 102. 6. 138 by 108.
XOTH. Applv The same principle to the following:
1. Multiply 1008 by 1007. 8. 1250 by 1003.
2. 1125 by 1004. 4. 1475 by 1002.
60. To find the prod act of two n ambers
one of which is more ami the other Jess than
100.
RULE. From, tlic srrm of tlir nnn^hcr^ *ubtr<ict
100, aimex two <~ip//rr* <ni<l ^ul>fr<ict 1 lit product of
the excess and
EXAMPLE. 108
98
8 excess.
2 complement
10600
16
10584
EXERCISES.
61. 1. Multiply 102 by 94. 4. 125 by 92.
& 103 by 97. 5. 112 by 99.
115 by 96. 6. 116 by 95.
XOTK. Apply The same principle to the following;
1. Multiply 1004 by 92. S. 1015 by 92.
2. 1008 by 95. 4. 1025 by 96.
IX ARITHMETIC.
ALIQUOT PHRTS.
TABLE.
y 2 of 100 == so
Vs "  33^3
V4 " = 25
Vs " = 20
Ve "  16%
VT " 
Vs of 100 = 12%
1/9 " = 111/9
Vio "  10
Vl2
VlG
of 100 = 37 tf
= 62*4
 87y 2
 66^3
 83*3
6 " ^ 1834
s/ 16 of 100 31 H
7 /16 ' 4334
9 /i6 {>  56M
11/16 " = 6834
13 /16 " 81J4
9334
1>2. To multiply by an rtfiquot part of
100.
Anin\\ ti.'o ciphers, </
inn! multiply fry the ntnncrafor of t hr frac
tional f<n~t It /. of 100.
EXAMPLE. 50 times 12 = 7200 r 2 = 3600.
162 3 times 84 8400 * 6 == 1400.
EXERCISES.
1. Multiply 48 by 25
33% by 24.
35 by 20.
4. 63 by 14 1 
5. 184 by 12i o.
6. 960 by 8V 3 .
7. 3603 by
8. 2560 by
24 SHORT
1. Multiply 72 by 37V 2 . 4 423 by 66%.
2. 56 by 12V 2 . 5. 144 by 83V 3 .
96 by 87V 2 . tf. 216 by 18%.
fl/. To multiply by 10 times an aliquot
part of 100.
RULE. Annex three ciphers <m<l proered ^s In 
fore.
Ex. 166% times 84 = 84000 4 6 === 14000.
times 144 = 144000  12  12000.
EXERCISES.
6V>. J Multiply 125 by 48. 5. 112 by 62V 2 .
0. 1236 by 3331/3. 4 192 by 83V 3 .
66. To multiply by (t little more or a lit
tle Jess than an aliquot part.
RULE. Multiply by the nearest ali<im>t parr, </.
above, and add or subtract the difference 1 itm* the
mi i iiber.
EXAMPLE. 13 1X 2 times 64 864 or
12 1X 2 times 64 = 6400 s 8 = 800
1 times 64 64
864
EXERCISES.
6T. 1. Multiply 72 by 14 l  2 . 4. 78 hy
2. 84 by 152/ 7 . . 123 by
54 by 17%. n. 144 by 84 ' i
6*8. To mult inly by WO an ft an aliquot
^ art of 100.
Annex two ciphers ami <i <l <1 to the tntin
f ii indicated by llic <t I if/u of part.
IX AK1T1IMKT1C. 25
EXAMPLES. 125 times 128^ 12800  % of 12800
= 16000.
1331/3 times 36 3600  1200 = 4800.
EXERCISES.
0.9. 1. Multiply 96 by 116^3. 4 72 by 112 l _>.
120 by 137K. o. 84 by 1142/ 7 .
345 by 116%. 6. 106^4 by 144.
This same principle may be carried to more than 100
and an aliquot; to 200, 300, and even to thousands. The
student will find much in this field for original investigation.
DIVISION
70. To divide by 5.
RULE. Multiply by 2 and cut off one fig it re.
EXAMPLE. 125 divided by 5 == 125 X 2 = 25.0.
EXERCISES.
71. 1. Divide 135 by 5. 4. 265 by 5.
& 145 by 5. 6. 325 by 5.
3. 175 by 5. 6\ 875 by 5.
72. To divide by 25.
RULE. Multiply by 4 and cut off two figures.
EXAMPLE. 125 divided by 25 == 125 X 4 = 5.00.
EXERCISES.
73. 1. Divide 275 by 25. 4. 875 by 25.
. 325 by 25. 5. 925 by 25.
475 by 25. 6. 975 by 25.
74. To divide by 125.
R ULE. Multiply by 8 and cut off three figures.
Ex. 375 divided by 125 = 375 X 8 ^ 3.000.
L\ ARITHMETIC. 27
EXERCISES.
M. 1. Divide 500 by 125. S. 875 by 125.
625 by 125. 4. 1125 by 125.
;6*. To dh'ide ht/ an aliquot part of 10O.
Mn hi ply by the denominator of the frac
tion expressing the alf<{nt part, Divide In' tJie
numerator and a/r off'neo iigurcs.
EAMFLES. 240 r 5 = 240 X 20 48.00.
840 H 25 := 840 X 4 33.60.
1200 * 12^ == 1200 X 8 = 96.00.
1350 f 16^/3 == 1350 X 6 = 81.00.
EXERCISES.
;;. Divide 245 by 25. 820 by S
268 by 20. 725 by 83^.
475 by 33^3 446 by 125.
<H. To rtiriflr Inj 1O< 1OO, 1OOO, etc.
RULE. Cur off <i^ iiKiny figures as there are
in t lie <lirf*or.
EXAMPLE. 1240 divided by 100 12.40,
7.9. To wilnce tlte dirt'sor 1o sotue tttun
of tens, hnndwls, thousands, etc.
RULE. Multiply both divisor and dividend by
some nmnhcr that ?.'/'// make the divisor a, multiple
of tens, Jin ml reels, t it cnisa mis, etc., and divide as in
short division.
EXAMPLE. 15) 2365
2 ' 2
3.0)47370
157 and 10 rein.
XO'l'K. Divulo the remainder I'D hv '2 to find the true remainder.
EXERCISES.
80. 1. Divide 3845 by 35. 5. 8732 by 75.
*. 6492 by 45. .$. 6288 by 125.
28 SHOUT MKT110IIX
DIVISIBILITY OF .NUMBERS.
81. To tell when a number is fit risible bt/
2, 3, 4, 3, 6, 8, 9, 10, etc.
82. All numbers are divisible by 2 when they end in
0, 2, 4, 6, or 8.
83. By 3 when the sum of their digits is divisible by 3.
84. By 4 when the two right hand figures express a
number divisible by 4.
So. By 5 when they end in or 5.
86*. By 6 when divisible by 2 and 3.
87. By 8 when the three right hand figures express a
number which is divisible by 8.
88. By 9 when the sum of their digits is divisible by 9.
8,9. By 10 when they end in 0.
90. By 7 or 11 if they consist of four figures, the first
and fourth identical and the second and third ciphers.
91. By any composite number if divisible by all of its
prime factors.
CANCELLATION
92. Cancellation is a method of dividing by re
jecting equal factors.
RULE. Cancel any or 11 factors common to
both dividend and divisor. Ih'cide tlta product of
those remaining in the dividend by (lie product of
those remaining in the divisor.
EXAMPLES. 42 X 36 ^ 24 X 14 == ?
Arrange the numbers as follows :
03 ;5
At 30 9
U X U 2
'1
EXERCISES.
9:$. 1. Divide 84 times 72 by 36 times 21.
144 times 216 by 56 times 128.
3. 512 times 1728 by 144 times 216.
j FRACTIONS
. To a d ft ff(t ction* lt<iciny ft cot tun on
denominator.
RULE. Add their iiuimrator* ami write tin
.^11 It oi'fr tJir r<>in iitoii denoin iitatr.
EXAMPLE. V 7  2/7 + % = %.
EXERCISES.
,ai. /. Add 2 9 % T ( , .
f>6*. To rff7^ f*ro fractions
tton
RULK. ^Inltiplv tlir ^uin of the denominator s by
O'imnon ninii> : r<tror and ivripe the result over
of the denom imttinf.
Ex. i 2 i 3 == (2  3) > 1 over 2X3 =
% % (3 5) X 2 over 15 = 16 /i 5 
EXERCISES.
f>;. 1. Add 3 4 ^. 7 r> ii.
%  3 /7 ^ 6 /7 + 6 /H
5 f % & 10 ia  10 /7
30 SHORT METHOD*
98. To add fractions not having a com
mon numerator nor common denominator.
RULE. Multiply each numerator into all the
denominators except its own for new numerators,
and take the prod^lct of all the denominators for a
common denominator, then add.
EXAMPLES. % + % = ^jt" 19 /i5
12 r 1<> IS
EXERCISES.
.9.9. ^. Add % 4 y 7 . A V 2 % %
%f 6 /n. 4 %h 3 /7 r 8 /n.
NOTE. When .several fractions whose denominators are not prime to
each other are to be added, reduce them to their least common deuominaior
and add.
TOO. To add mixed numbers.
RULE. Add ivhole numbers and fractions sepa
rately and tlien unite results.
EXAMPLE. 8% f 12%.
8 r 12 20
% %=' 1 % 5  His
EXERCISES.
101. L Add 91/2+141/3 4 283/ 5 + 3 5 4/ 5 .
^. 18%J 252/7. 5. 431/5 72y 7 .
21%1275/y. ft 66% 231/4+ 17y 5 .
102. To subtract fractions h rinr/ a co tn
RULE. Take the difference of tli<>
<tinl write it over the common denominator.
EXAMPLE. % minus % = %  %.
IX ARITHMETIC. 31
EXERCISES.
1. Solve : % _ 44 .; 13/ 15 __ li/ 15 .
10/ 13 __ 5/ 13 . ,J. 42/_ 3 __ 27/_ 3 .
To subtract fractions liarina a com
mon
RULE. Multiply the (lijft'cn'iHf </' the <lcntnninu 
tors In' the common numerator <iiid write the re
*u It over the pr<l it ct of the >1 eu<n 7 7 ir/r ;..
2 / >
EXAMPLE. %  2/ 7 =
EXERCISES.
/OJ. /. Solve: % 3/ 7 . ,;. s/^ _ 8/ 15 .
%  % 5.
~ 5 ll. 6'.
*. 7V> subtract fractions had it f/ neither
numerators nor comnton denom
inators.
RULE. Mu I tiply ench numerator into the other
(If/loin iii'i r<>rs, take the difference un<l i^rite it over
t In pnnl a rr of the denominators.
I.', l:j
EXAMPLE. % 3/ 7  _
EXERCISES.
107. 1. Solve : 6/ ? _ 5/ 8 . .;.
. To subtract tni.red numbers.
RULE, ^rihrract u'h.ole numbers and fractions
^l v, uniriu.g
32 SHORT UKTUOUX
NOTE. If the traction of the .subtrahend is i>naUr than tliaT of the min
uend subtract a unit from the minuend and add it to tin traetion before
taking the difference.
EXAMPLE. 8% 5%
8 5 := 3
% ~ % Vlo
11  8  3
IVa ~ V 2 %
3%.
EXERCISES.
Solve: 22% 16%. 5. 89%  35%.
75% _ 48%. ^ 9 5 i/ 6 __ 7434.
XOTE. A g\.od method is to take the complement of the diiference of th