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1.8
J >IPPLIED IIVMGE
inc
1653 East Main Street
Rochester, New York 14609 USA
(716) 482 - 0300 - Phone
(716) 238- 5989 -Fox
■- 'I'
THE WONDERS
M'
OF
AKI7HMETIC
OR
The Art of resolving, usinfr only one figure, or by
simple Addition, all rules of Interest or Dis-
count, simple or complex, without having
to divide, acquired in ten minutes
study.
STOCK OPERATIONS,
The Four Boles proved by a simple AdditioOi
as quick as thought.
BY
J. M. DANIAX7D.
fc'^S'
MONTREAL :
BUSllBE SENfiCAL, PRINTER and PUBLISHER,
6, 8 and 10 St. Vincent Street,
1880
Entered according to the Act of Parliament of Canada,
in the year one thousand eight hundred and eighty
m the office of the Minister of Agriculture by J. M
0«^4j^4.^8
'Canada,
d eighty,
by J. M.
CHAPTER I.
Showing how all operations of Interest or Discount
can be made without a Division.
If you want to find the interest on any sum, for one
year, you shall multiply the capital by the rate, either
simple or complex, that is with decimal fractions.
The rate is called simple when it is not followed by
any fraction, as 1,2, 3, 4, 5, 6, 7, 8, 9, etc.
The rate is called complex when it is followed by
fractions, as 1.25, 1.75, 2.20, 2.25, 2.75, etc.
When you have multiplied the capital by a simple
rate, you point off two decimals, by which you have
divided by 1 00. The dollars stand to the left of the point
and the cents to the right.
If you have a complex rate, you point off four deci-
mals and the dollars stand to the left of the point, as
above, and the cents to the right.
What is the interest on $4842 at 6 o^o for one year ?
Example : $4842
6 o;o
$290.52 Answer: $290.52 cents.
What is the discount on $542 at 4 o^o for one year ?
Example : $542
4 0^0
$21.68 Answer : $21.68 cents.
What is the interest on $6424.10 at 5o?oforone
year?
Example: $6424.10
5 ojo
321.2050 Ans.: $321.20 cts. 50 on i.
What is the interest on $248 at ^ o/o for one year ?
Example : $248
4.50 0/0
992
11.1600 Answer: $11.16 cents.
V
— 4 —
$23,223
8
0/0
1857.84 Answer: $1857.84 cents
oneVay ? '"""" °'' "''"""' "" «3«2 at i o;o for
, Example: $3422.00
•art.^, ■ ... ■ ^80221 Answer $0.38 cents
471332
21 —days
471332
942664
B
you
$9.9
oper
not
Proc
will call a«; «q Qn^l^^^? • ^"^wer : $9.89 79 that I
Prooi by Division :
4242
4 a/o
16968
21 — days
16968
33936
356328 I 360
3232 1 «
35':8 9.898
2880
0000
2r
Yc
oddi
Pr
Answer : 9.90 cents.
Oe
— 5 —
X, can be
one after
1 it, begin-
m right to
each add-
s if every
It 4 0/0 for
Its.
il days?
79 that I
usandths
But as you llnd yourself with more than i cent rest,
yo" may add a unit to the cents and have the result :
$9.90 cents. This result is the same as by the other
operation which is much shorter and in which you do
not have to divide.
What is the interest on $548.10 at 4 o^o for 4 days ?
Example: 54810.00
6089998
4 •-days
^ . . ^ 24.35.99.92 Answer : $0.24 cents.
Proof by Division :
548.10
4 0/0
219240
4— days
876960 1 360
1569 '
1296 2436
2160
^ . 0000 Answer : $0.24 cents.
2nd example :
What is the interest on $4842 at 4 o;o for one dav?
4842.00 '
537998 Answer : $0.54 cents.
You strike off, as before, the thousandths, and you
add a unit to the cents.
Proof by Division:
• 4842
4 0/0
19368 1
1368 1
ooon
0000
360
CO o
OO.O
Answer : $0.54 cents.
5ents.
On the same capital at 4 o/o for 15 days
y
.^
— 6 —
Example: 4842.00
537998
15 — days
2689990
537998
\
Proof by Divi^fonT' ^"^^^^ ^ ^^.07 cents.
4842
4 0/0
19368
15— 'days
96840
19368
290520 } 360
2520 I
0000 8.07 Answer:
On the same capital at 6 o;o for one dav •
Example : 4842,00 *
537998
268999
.07 cents.
^ _ 806997 /.nswer
Proof by Division :
4842
6 0/0
.81 cents.
29052 J
2520 1
0000
360
80.7 Answer: $0.81 cents.
Same capita! at 6 o/o for 20 days.
Example : 4842.00
537998
268999
$0699?
20— jours
16.139940 Answer: $16.14 cents.
Pre
San
Proof by Division :
— 7 —
4842
60/0
29052
20 — days
581040
2210
504
1440
Other proof by Division :
4842
20— days
96840
36
8
24
360
16.14
Ans. : $16.14 cents.
16.14
What is the interest on
Example : 490.00
54443
108886
45^days
544430
435544
Answer: |M4 cents,
at i 0^0 foi 45 days ?
4.899870 Answer: $4.90 cents.
Same capital at 5 0/0 for 120 days :
Example : 490.00
54443
27221
13610
5 0^0
68050
120— days
136100
68050
8.166000 Answer: $8.17 cents.
~8 —
Proof by Division;
490
8 0/0
3920
45— days
Proof by Division:
490
5
19600
15680
2450
120
49000
2450
360
' 176400 I
3240 I
0000 4.90
Answer : $4.90 cents.
Two more examples :
Capital ;— .
6848.00
IoS???~"* o?o interest for
380443—2 070
190221—1 oyo
95110— io;o
47555— i 070
23777-i 0/0
360
(t
((
294000 I
600 I
2400 8.166
2400
240
Answer: 8.17 cents.
Capital : —
20000.00
one day— 2222222— 4 o70
1111111-2 070
555555-1 070
277777-1 0/0
138888 — J oJO
I' 69999— J ojo
ih^iiS^^? ^""^ represent the interest for one day on
the respective capitals of $6848 or 20000. ^
„n !hl operation shows what is the result of addini?
ttre^Cesfn "«"' " '«"' ^' ™ny «™ "a!
'^^'ttiJ^^L'^^l 3m« Sus7?KS
flrqt- iqn99i"^T?'' "' " '^^"' ^'^ ^^ isonehaliofthe
T ' ,^^"''21 must represent 1 070.
In the example on 20000.00, there being only one
y Division;
I
)
J. 17 cents.
.00
i22— 4 070
i 11-2 0/0
>55— 1 o;o
^77-1 (^0
188 — i 0/0
'99— i 0/0
le day on
of adding
*■ times as
J capital,
decimals
ied off, to
3 right of
up from
^presents
. then re-
alfofthe
aaly one
figure, 2, you cannot do «,• mise than to write 2
down under the ciphers and .vuu have 2222222, for the
interest at 4 o/o for one day ; ! 1,1 1 1 1 1 for the interest
at 2 mo, and so forth.
It follows that when you have added up all the fi-
gures of a capital sum, as many times as there are
figures, and adding to the figure next following on the
left hand the amount carried from each addition, the
result will he the interest at 4 o/o for one day, one half
of such result will be the interest at 2 o/o, and soforth.
ffl.oiL*^®" y°^ ^^°* *^ ^'"^ o^t ^^6 interest at 4 o;o on
|20000 for 50 days, multiply the result of the addition
from right to left of 20000 by 50 days and the product
will give you the interest, after you have pointed off
six decimals, the dollars standing tc the left of the
pomt and the cents to the right.
Example : $20000.00
50 jours 2222222 4 0/0 for one day
111. 11110 Answer: $111.11 cents.
The following is the way in which this adding up of
the capital should be done to find out the interest at
4 0/0 for one day :
Suppose you have to add upthe capital 4785. You
begin by putting down the two necessary ciphers • then
you add up all the figures in the capital, one after
fi?^, 5'.,.*^^^"^ ^^^^ *° ^^^ *o '^e following figure to
the left, the amount carried from each addition and
the sum will give you the interest at 4 o/o for one day.
Example: *4785,00
531664 — 4 0/0 for one day
The sum 531664 has been obtained as follows : 5 and
8 are 13 and 7=20 and 4=24, 1 set down 4 and carry
^ ^^i^^l^*^^ '° *^® ^' I say : 2 and 5 are 7 and 8 = 15
and 7=22 and 4=26, 1 set down 6 and carry 2 which
I say : 2 and 5 are 7 and 8=15 and 7=22 and 4=26
1 set down 6 and carry 2; 2 carried and 8 are 10 and
/«17 and 4=21, 1 set down 1 and carrv 2; 2 carried
— 10 —
X
dW, there wSh/ .1 ««»'« only, the result for one
jnultiply th^whole t'm; 4heuevTf ZT' ^^^ "
than one day's interest. ^"^^^^^ yo« have more
»TOTHBR EXPLANATION OP THE ADOITION.
it .?n"ecra'X"StU°t.^f:7';£ 'i-^-P""".
on account of the iw„„C *'?.^* '" "'e last figure
la the same manner wLnfc'^°,"''^'°8.'''« «
body of numbers, you must eohJr.^^P''"''' '" '"«
the number of cvohern ft.. %?'''' ""'=* """"s 'ban
them on the left Cd siSe. ^™ "^^^ '''""""•'8
2o524'2!'l wUl put dodi° ^ ""'•"l-'P «■« '•""•wing:
Kt?s^s-E£S??v"^^^^^^^^^
I add the flgi-es 2 I A I' iv k''* T^"*^ *« »Pi«al,
aught and clrrv I '.'.:' ^' y'"'=i'„*''^ '«■ I set down
ana -^ are 3 and 2=5 and '4-9ar,rt T'Vi ' i ' ,*=*"''««*
•»'' carry 1 ; , carried andYa^sUliis'aZ
\
id carry 1 ; l
res in the ca-
4 0/0 interest
.53 and some
iber of days
in one day.
?andths must
on by num-
the number
5st, because
esult for one
' for a consi-
necessary to
have more
TION.
the capital,
3 last figure
the capital,
ters in the
more than
t following
(following:
er the ca-
?oing back
one of the
^hich it is
the num-
he capital,
set down
1 2=5 and
1 canied
set down
2=5 and
— 11 —
il7?'fK^®*o1?,!^'V^.'^.°*^ ^*^^ *"^^*5 I ^egln again
with the 2 (the first figure of the capital) and 1 say: 2
&na I are 4, 1 set down 4 and carry aught ; as I have
nothing more to add for the ciphers, I set down 2 for
each of them and also lor the first figure of the capital.
The sum of this addition is 22249110. And if I point
off SIX decimals, I will have 22.24, which I will put
down as $22.25 cents interest for one day, leaving off
the thousandths unless I should multiply by the num-
ber of days my capital has been bearing interest.
11 1 want to find out the interest at 6 070 on this ca-
pital which gave me as interest at 4 070 : 22249110 I
shall take one half of that number, and add it up to the
same and that will give rae the interest at 6 070 : it
will then only bo necessary to multiply by the number
ot days stated and having pointed off six decimals, I
will have the $ to the left and the cents and thou-
sandths to the right of the point.
Example: 22249110— 4 070
11124555
33373665
70 — days
2336.156550
Answer: $2336.16 cents.
«io^« i«*^® }^ ^^^^ ^^y' ^^ ^^^ interest asked for
*i . :u 2^^^^^ ^^^ ^°°^® thousandths which are left
off in the final result of the operations.
It is well to strike off the thousandths, and add one
to the cents, (as I said before) because in the final re-
sult only cents need be accounted for.
The addition of the capital can be made in various
ways ; it can be made by putting down the capital five
times echellons by respecting every figure,putting down
fave times that representing the units, five times that
♦u^"i:'''"i'"5 ^^''^ iciiiiis, live limes liaai representing
the hundredths, etc., etc.
The sum of the addition will every time represent
the mterest at 4 070 for one day.
— 12 —
What is the interest on $845 at 4 070 for 40 days ?
Example: 845
845
845
845
845
9388795--4 o;o for one day
40— days
3.7555. iToo
^ Answer : $3.75.55 i. e. $3.76 cents.
55555
44444
88888
9388795—4 070 interest for one day
40— days '
same
3.75551800
Answer : $3.76 cents.
hn^if!/"'"" '' *^® '^°*® for both operations. With
both these methods, it is necessary to point off ei^ht
certfth: ^^T' '' '^^« *^« ^ '« tL leVan t
cents to the right of the point. Here is the proof of the
two proceeding rules by addition of the capTtel
Example; $ 845.00
^^^fn~"i °^° ^^^ °°® ^*^y
■IV — -uuys
3.755480 Answer : $3.76 cents.
for 40 days ?
day
} of the same
)r one day
$3.76 cents.
lions. With
nt off eight
left and the
proof of the
pital.
'3.76 cents.
Proof by division :
— 13 —
845
4 0/0
3380
40 — days
135200 I 360
2720 '
2000 3.75
W41.41, *!. . J^^^ Answer : $3.76 cents.
With the method of adding up the capital, the oper-
ations are considerably shorter ; and when you have
acquired the habit of adding up the capital as we have
just been doing, you will be able to calculate interest
or discount a? quick as thought.
We have seen that the operations can be made by
adding up all the figures of the capital in order to find
the inteirtst at 4 o/o for one day. We propose now to
show how the interest can be found for the number of
days the capital has been bearing interesi. If you
niultiply the capital by the number of days and if you
add up the figures of the result, in the same way as
above, you will have the interest for the number of
days given.
Example of the proceeding rules ;
•845 Proof: 845.00
40 days 93887
liiSo !!"^"y'
3.7654 Answer : $3.76 cents. 3.755480
..,, , . , Answer : $3.76
Other example with a capital of 3.241 at 4 070 for
12 days : '
3241 Proof:
12 — days
6482
3241
38892 addition.
4.3210 Answer : $4.32 cents.
3241.00
3601.10
12~days
720230
360110
4.321320
Answer : 4.32
— 14 —
Proof bv division ;
3241
4 0/0
25928
12964
155568 I 360
1156 I
768 4.32
48
Answer : $4.32 cents.
the adSn multiplication takes place before
Example :
845
40 days
33800
33800
33800
33800
33800
3.75551800
Answer
$3.76 cents.
rivld «t '^r^^t'jilP!™"""' the sam'e result is ar-'
cessary" to point oFeieMdP^t™'^'?'-'"*' '} '^ ""^y^ "«-
* .0 th^e 4 a"„Vti.^irsrft"gh''tirtii: xr"
other example
— 15 —
845
40 days
33800
00000
88888
33333
33333
4.32 cents.
Jr of days, four
1 the result, on
> foregoing re-
ary to addsthe
5s place before
result is ar-
s always ne-
' to have the
the point.
3.75551800 Answer : $3.76 cents.
The result remains the same, after pointing off eight
decimals.
AH these rules give the interest at 4 o;o ; and when
the interest at 4 o/o on any capital is once arrived at,
it is easy to get at every rate. Take one half of 4 o/o
you have 2 o;o ; take one half of 2 o/o you have I o»o.
When you have the 1 op rate, you multiply by the
rate given and thea by the number of days ; and you
get the interest asked for after pointing off four deci-
mals, if you have not added the two ciphers.
Every time you add those two ciphers to the right
of the capital, you must point off six decimals.
But if the rates are complex, you must point of^two
more decimals in the result.
If you want to operate on any rate of interest, you
can do so by getting first the * o;o interest which is i
of the 4 070 i. e. of the sum of the addition from right
to left, which represents the 4 o;^o rate.
Example : 3421.00 h 5 oio for 21 days
380110
190055
95027
5 0^0
475135
21— days
475135
950270
9.977835 Answer : $9.98 cents.
\
— J« —
rate; takeTof'the Lsuft a^add ?,'^"^'' ^^^" ^^ '^'
as there are figuresfn u • TatfJre "^ f ^ •'"''"y *'°»es
sum and you will have th^«Tfn^^^^
to the right of the pofnt * ^ ^^" ^^'^ ^^« ^^^^s
What is the intere^f on 8422 at S n-i« fn„ aq i
Example : 8422 cSpiiar "^^^^ ^
48— days
67376
33688
404256
I 3 o;o
1212768-totaI.
, . 606384
to be added u p 303192 -^i of the total,
33.6878 Answer : 33.69
^ Proof: 842200
935776^4 o;o
467888
233944— unit or i
3 0/0
701832
48 — days
5614656
2807328
33:68:78: 'a"S7s' added Ul"'.''^'''?*''' '.""^ ''°'<»>'>'
sandths left out. ° "■* "*'«' and the thou-
'ay: multiply
J, then by the
8 many times
icimals in the
md the cents
r 48 clays ?
)9
liplied by
■esult has
addition
um. fiirn-
e amount
the thou-
- 17 —
Other example on the same capital.
8422 Other example :
48
8422
48
67376
33688
404266
3
1212768
1347517
673758
33.6879
0^0
67376
37688
404256
449171
224585
112292
33.6877
Answer : 33.69. Answer : 33.69.
In this operation, the capital has been multiplied by
the number of days ; then by the rate, and the result
was a total sum of 1212768. This result has been added
up from bright to left, and the answer has been one
fourth of the sum of this addition.
Proof by division :
8422
3 0^0
25266
48 days
202128
101064
1212768
1327
2476
3168
2880
0000
360
33.688
Answer : 33.69.
The rule to be followed as to the number of decimals
to be pointed off is the following : For the capital, two,
for the simple rate, two, for the two ciphers that fol-
IffW ihf\ npnitfll iwn nnH if iVtCk Y>afo ia rtnmrAav f/\w
the decimals in the rate, two, making all told eight
decimals, if are combined in the operation all these
conditions.
— 18-.
Sj^ me»thod of operattng by the unit,
sample. 7424 d 7.50 for 50 Jays.
7424.00
824887—4 o;o
412443—2 o!o
206221-1 0/0 or unit.
7.50
1031 loT"
1443547
154665750
50-— days
, 77.33287500 Answpr • 77 qo ..„#
We Will give two examples of that below.
What IS the interest on 743 at 5.25 for 43 days?
oJr"" 632300
41277 ^^2554
20638-.unit 175638-unit
43-days 9 ^^^
Pro<
Oth(
Wh
Ans
61914
^2552
887434
5.25- -rate
1580742
22— days
San
Exa
4437170
1774868
44371 rn
3161484
3ioi484
34.776324
Answer : 34.78 cents.
I
4.6590?8o. Answer : 4.66.
Ans
Proof by division
— 19 —
5.25
3715
1486
3715
390075
43— days
1170225
1560300
77.32 cents,
lit, let the
)8l for one
by number
I represent
lapital has
'equired to
days?
nit
ays
.78 cents.
I
16773225 I 360
2373 I
2132 4.6592
3322
825
105 Answer : 4. 66.
Other examples :
What is the interest on 4842 at 4 ojo for one day ?
484200 Proof by division :
0.537998 4842
Answer : 0.54 cents. 4 o;o
19369 1360
1368 I
2880 538
0000
Answer : 0.54 cents.
Same capital at 4 o?o for 15 days :
Example : 4842.00 Proof by division :
537998
15 — days
4842
4 0/0
2689990
537998
19368
15— davs
8.069970
Answer : $8.07 cents.
96840
19368'
290520
— 20 —
290520 I 360
2520
0000 8.07
Answer : $8.07 cents*
Same capital at 6 o/o for 20 days
Example: 484200 Proof by division.
0^7998 — 4 o;o '"'"
268999—2 o/o
4842
20 — days
806997
20— days
16.139940
\ Answer: $16.14 cents.
96840 I 6
36 I
8 16.14
24
Answer : 16.14 cents.
What is the interest on $490 at 8 o/o for 45 days?
Example : 490.00
54443—4 o;o
108886— double or 8 o;o.
45— days
544430
435544
^
4.899870
Answer : $4.90 cents.
Proof by division :
490
8 0/0
3920
45 — days
19600
15680
irromn l nnn
3240 I
0000 4.90
Answer : 4.90 cents.
»20 I 360
.20'
00 8.07
r : $8.07 cents.
hy division.
20 — days
14016
I6T4
24
: 16.14 cents,
or 45 days?
0.
ats.
rs
: 4.90 cents.
— 21 —
Same capital at 5 070 for 120 days.
Example: 490.00
54443
13610 — unit or both together 5 0/0.
68053
120~days
1361060
68053
8.166360 Answer : $8.17 cents.
Proof by division : 490
5 o;o
2450
i20-~days
4900
2450
360
294000
600
2400 8.166
2400
_., , , ,^ . , 240 Answer: 8.17 cents.
What is the interest on 2333 a 5 010 for 22 days ?
Example :
2333.00
259221—4 070
129610—2 o;o
64805—1 op or unit;
5 0/0
324025
22— days
648050
648050
7.128550
Answer: $7.13 cents.
Proof;
2333
5 0^0
11665
22— days
23330
23330
256630 I 360
1030 7.128
3100
220
Answer: $7.13 cents.
V
— 22 —
What is the interest on 745 at 7 o70 for 15 days?
Example :
tt)§3 ^ 745.00
id 5 & 82776—4 070 '
^«^ 41388-2 0/0
^ 5 20694—1 o/o
144858
15 — days
724290
144858
5215
15— days
2172870
Answer: 2.17 cents.
Other example at
48948.20.
26075
5215
78225
622
2625
105
360
2.17
Ans.
2.17 cents.
0/0 for 20 days on a capital of
«50|3 48948.20.00
"" - I 543868885—4 o/o
271934442— 2 070
135967221—1 o/o
951770548
20— days
^7i o
CO <»^
Proof;
48948.20
7 0/0
34263740
20— days
190.35410960
Answer: 190.35 cents.
Ans.
685274800
3252
1274
1948
1480
400
$190.35 cents.— 40
360
190.3541
You may see that the result arrived at is always the
same as with a division, and the operation is much
shorter as it is easier and more convenient to add un
and to multiply than to divide. ^
With this method you will save much time, because
you won t have the trouble to find out how many times
„„.. , ^^^.j.. ^jjjj ^^^^^^ you have ac-
quired the habit of adding up the capital from right to
left, you Will be able to do all rules of interest or dis-
count as quickly as you can write down the figures
\
>r 15 days?
17
s. : 2.17 cents.
n a capital of
roof;
8.20
7o;o
1740
20— days
800 I 360
190.3541
8
80
400
-40
is always the
tion is much
3nt to add up
Lime, because
V many times
you have ac-
from right to
erest or dis-
he figures.
— 23 —
RULE AT 6 o;0.
In all rules at 6 op, where the number of days is
divisible by 6 without leaving a fraction, the capital
may be multiplied by the figure obtained as quotient.
Example :— What is the interest on 745 at 6 o?o for
42 aays ?
745 745 for 12 days
7 2
1.490
5.215
Answer : 5,21 Aus. : 1.49 cents.
What is the interest on 4323 at 6 o/o for 48 days ?
Example: 4323
48
Proof:
34,584
4323.00
480332
240166
Answer : 34.58
Proof: 745.00
82776
41388
720498
48
5763984
2881992
34.583904
Answer: 34.58
Proof by division ;
745
6 0/0
4470
42 — days
8940
17880
124164
12— days
248328
124164
1.489968
Answer : 1.49 cents.
By divisor 6.
745
42— days
360
187740
774 I
540 5.21
180 Answer ;
1490
2980
31290
12
9
6
5.215
30
Answer : 5.21 cents.
5.21
— 24 —
The same rule applies to the 4 o;o interest where the
Sw^ ^^ ^^ divisible by 9. Examples are given
It is clear that, as 42 divided by 6 gives 7, 1 multiply
745 by 7 and pomt off three decimals in the result in
order to have the $ to the left of the point and the
cents to the right.
What is the interest on 745 at 4 o70 for 36 davs?
Example: 745 ^
4
2.980 Answer : 2.98
Other example at 4 o?o for 72 days on a capital of
3248
i 8
Proof by addition of
the capital :
745.00
8277b
36 — days
496656
248328
25 984 Answer: 25.98
Proof by division
745
4 0^0
2980
36 — days
2.979936
Answer
2.98
17880
8940
107280 J
3528 I
2880
0000
360
2.98
R«foa^o„ , u . Answer: 2.98 cents.
Rates can also be computed with the capital.
What is the mtorest on $840 at 6 o;o for 70 days ?
Example: 840— 4o70) . . ^
420—2 op / ^^^^°fif together 6 o/o.
1260
Tn A ,
88200— to be added up
9.7998 Answer : $9.80 cents.
Brest where the
iples are given
IS 7, 1 multiply
n the result in
point and the
ir 36 days?
a capital of
f by division :
45
4 op
80
36 — days
80
30 j 360
?0 2.98
30
2.98 cents.
ipital.
•r 70 days ?
fether 6 o/o.
— 25 —
Proof by putting down the capital in echellons :
Proof:
1260
1260
1260
1260
1260
13999860
70-
9.79990200
840.00
93332—4 6?n
46666—2 0^0
139998
70— days
-days
Answei ; 9.80
9.799860
Answer: 9.80
Other example at 7 o/o on a capital of 840, for the
same number of days :
Capital— 840— 4 o/o )
Half— 420— 2 o?o }• Together 7 o/o
Quarter— 210— 1 o/o J
1470
70— days
Proof :
102900— to be added up
11.4332 Answer: 11.43 cents.
840.00
83332—4 0/0 )
46666 — 2 070 > together 7 o70
23333—1 0/0 )
163331
70— days
80 cents.
11.433170 Answer: 11.43 cents.
In the 6 o/o rule we have taken one half of 840,
which we have added to the latter sum ; and we have
multiplied the total 1260 by the number of days given,
the result was 88200, we have added up the figures of
this result as many times as there are figures in it; we
— 26 —
V
have pointed off four decimals (as the two ciphers
?'?QOQ"?i Jd^ied) and we have had for an answer :
y. /y98 that we called 9.80 leaving out the thousandths
and adding one to the cents.
r.„?f"' ^® ^°°,^ *^^' *h« addition can be made by
puttipg down the capital in echellons in the manner we
r^h^f 1. /"^S^fi?- r^^'^ '^ tantamount to adding up from
right to left the figures in the capital. © i' "^
Some examples will be given here :
EXAMPLES BY ECHELLONS.
What is the interest on 840 at 7 o?o for 70 days ?
840—4 0^0 ■)
420—2 o?o (.together 7 070
210—1 (^oj '
1470
1470
1470
1470
1470
16333170
70— days
01
Otl
11.43321900 Answer: 11.43 cents.
Pre
Other example :
00000
77777
44444 .
mil
Proof:
840.00
93332—4 070
46666-2 0/0
23333 — 1 0/0
16333170
70 — days
163331
70 — davs
11.43321900
Answer: 11.43
11.433170
Answer: 11.43 cts.
V
the two ciphers
i for an answer :
i the thousandths
;an be made by
in the manner we
) adding up from
i.
for 70 days?
' 7 0^0
cents.
Proof:
0.00
5332—4 0/0
3666 — 2 0/0
J333 — 1 0/0
1331
70 — days
!170
wer: 11.43 cts.
— 27 —
Other proof:
840— Capital
70 — days
58800— to be added up and struck off
65331 — 4 0/0
32665—2 0/0
16332—1 0/0
11.4328 Answer: 11.43 cents.
Other example at 8 o/o for 41 days on a capital of
42231.00
4692332
2— Multip. or doubling the sum
9384664
41 — days
9384664
37538656
384.771224 Answer: 384.771224
Proof by division:
42231
8 0/0
337848
41— days
337848
r 51392
13851768
3051
kill
2776
2568
360
48 Answer: 384.77 cents.
— 28 —
In all these rules by putting down the capital by
echellons it is necessary to point off eight decimals in
order to have the $ to the left of the point and the
cents to the right.
OPERATION BY ONE FIGURE ONLY.
All operations can be made by one u^ure only
either with the addition or with the division.
What is the interest on 840 at 4 o/o for 70 days ^
\ Example :
840
7— regulating figure
5880— sum to be struck off
6531— addition
1680 —double capital
23331
4 0/0 rate
93324
70 — days
6.532680
Answer : 6.53 cents.
Other example a 8 o?o on the same capital for 50
days.
840
7— regulating figure
5880— sum to be struck off
6531 — addition
1680 —double capital
23331— unit
50 — days
1166550
9.332400
Answer : 9.33 cents.
Ihe capital by
fht decimals in
I point and the
FLY.
e iigure only,
sion.
3r 70 days ?
'e
koff
)r : 6.53 cents,
capital for 50
— 29 —
Proof by addition:
840
70 — days
58800
6.5331
Answer : 6.53 cents.
0/0
cents.
Proof by addition of the figures in the capital, at 8
840.00
93332
186664
50 days
9.333200 Answer : 9.33 cents.
Other proof : 840
50 — days
42000
4.6666—4 o;o.
2
9.3332—8 0/0.
Answer : 9.33 cents.
cJhe'Sil^li^T ^' r^"Ja«"g figure, because it
can De made use of m all operations In fart if fn».
wJj], ? ":?" ''™'=* <"f «'>'l 'h« sum of th^aZmon
pUa.r2 i? ^S^C^ ^S"" -«P'-«<>- of '^-■
ine poiui auu ihe cents to the richt "" "" "' '"" "'
Otiier example on the same
for 82 days . •
capital of840 at 5.25 0/0
— 30 —
840
7— regulating figure
5880— to be struck off
6531— addition
1680 —double capital.
23331— unit
5.25 — rate
116655
46662
116655
12248775
82-.days
24497550
97990200
10.04399550
Answer : 10.04 cents.
PROOF.
84000
93332—4 0/0
46666—2 0/0
23333—1 0/0
5.25 — rate
840
5.25
U6665
46666
116665
12249825
82— days
24499650
97998600
10.04485650
Answer : 10.04 cents.
4200
1680
4200
441000
82— days
882000
3528000
36162000 J 360
1620 I
180 10.04
Answer : 10.04 cents.
— 31 —
♦v.}T® must here point off eight decimals on account of
the two decimals on the rate with this method.
Ifota. — Whenever the rate is simple, six decimals
are pointed off, and eight decimals whenever the rate
IS complex.
.r.^^ *^u 5"'? ^' ,^ °/o '8 of the most frequent occur-
ence with traders, I will give some more examples of it.
What is the interest on 4800 at 6 o/o for 50 days ?
Example: 4800.00
533332--4 o;o
266666—2 o;o
799998
50— days
39,999900 Answer : $39.99.
.^^u""^ r.?^" ^*" ^^^' leaving out the thou-
sandths and adding one to the cents.
What is the interest on 2323 at 6 op for 16 days ?
Example :
232300
258110— 4 0/0
129050—2 0/0
387160
16— days
2322960
387160
Proof;
2323— capital.
16 — days
13938
2323
37168 — to be added up
41295—4 070
20647—2 0/0
6.194560
R6ponse: $6.19 cents.
R 1Q/.9
Answer : 6.19.
The 6 0/0 rule can be made in the following manner :
— 32 —
^^^^fn ' ^" * ^'^P^^^^ °^ ^^^ ^^'^ ^2 days.
^fi „, Proof by division:
^—0/0 840
5040-to be struck off __82-d8y8
5599--addition leso
o40 — capital 6720
13999
82 — days
27998
111992
68880
8
28
48
. Answer : 1 1 .48 cents.
1147918 Answer ; 11.48 cents.
The operation has been done this way : the canital
has been multiplied by 6, the result was 5040, ^this
ire fl'.urrinTt -^Z' '^^'^ "L?.?."^^"^ times as there
are ngures m it; the sum was 5599, under this we hnvA
put down the capital, beginning inder the te^l^^II
we have seen done with the regulating figure 7 with
this difference that this time we do not double tCca-
pital ; then we have multiplied hy the number of davs
given and we have pointed off five decimals (this is f
general rule for this method) ^
The same rule by the divisor 7 is given below :
Example: 840 p^oof : 840
I 932—4 o;o
5880 _466-2o/o
ifi«n^ ^398
_^ 82-days
23331
6 o;o
139986
82
279972
1119888
•11.478752
2796
11184
11,4636
Answer: 11.46 cents.
Answer: 11.48 cents.
lays.
of by division ;
840
82— deys
680
20
880 I 6
18
r: 11.48 cents.
y: the capital
as 5040, this
times as there
' this we have
the tenths, as
Sgure 7, with
louble the ca-
imber of days
als (this is a
i below :
40
32 — 4 o;o
36 — 2 0/0
)8
J 2 — days
)6
(6
: 11.46 cents.
11.48 cents.
— 33 -
nn!^!!^i!^'^ kind of proof, it would be necessary to add
ord.r»n^'?T.f "'" "^^^'^'^" ^''^"^ '•ight to left, in
given befow "'""^ '^^''^^' ^^^le examples are
What is the interest on 450 at 6 o;o for 50 days ?
Example :
450
7
3150
3499
900 — double capital
12499
6 0/0
Proof:
450
499—4 0/0
74994
50— ^ays
3.7500
Answer : 3.75 cents.
3.749700 Answer : 3.75 cents.
For the modus operandi see first rule by the divi-
fo ]^^^^ F^°°^' ^J?! a,<^^ition has been made from right
i nin J""!? S?n ^^^^^ ' ^ ^*d i" this way 500, as the
LTi.f 7.n^?' ?n ^ T ^"t'^^est which gives 750 ; X
tW J^^.^y ^^^°^ ^ ha^« for an answer 3.75.
f«n*c ?? r?^/"^® ^^ generally unsed by accoun-
tants; It 18 to hud out the 6 0/0 interest for 60 days on
any capital, by pointing off two decimals.
What is the interest on $450 for 60 days at 6 o;o.
Answer : 4.50 cents. j a.* u u^u.
Proof by division : 450
60
■li{j\j\}
30
I
4.500
Answer : 4.50 cents.
Other proof:
— 34 —
45000
49999
24999
74998
60
4.499880 Answer : 4.50 cents.
What is the interest on 450 at 4 o;o for 90 days ?
Answer: 4.50 cents.
Proofby division : /»50
)0
40500
45
4.500
Answer : 4.50 cents.
This, as you may see, gives a correct answer, but
these two last rules, by pointing off two decimals, caii
only be applied to interest for 60 or 90 days. When
the mterest at 6 om for 60 days is found, you mav
easily find it for 36 days, for 15, for 7i, etc. If for more
than 60 days, you may add the 7i, 15, or 30 days in-
lerest, etc.
r.J^J}^Ju^° ^^^ ^ ^t^ ^"'^s, three decimals are
Pf ihi,ri^° k' ^|J*"^L*.^^P*^6^ *s left out in the divisor;
U should be 60 or 90 instead of 6 or 9 ; to equalize the
matter it should be necessary to strike off one figure in
the Gividend, if this has not been done, there is one
decimal more m the quotient, three instead of two.
It follows tbat the capital had better be multiplied
«y.?!>"S???.«!: S^ .^^.y^ and divided by 6, if it is at
V -„Yu, \ji uiviuuu vy y, 11 it ia ^i 4 o«q^
This last method may be applied to any number of
— 35 —
: 4.50 cents.
30 days ?
: 4.50 cents.
nswer, but
cimals, cari
lys. When
, you may
If for more
30 days in-
icimals are
-he divisor ;
qualize the
le figure in
here is one
of two.
multiplied
6, if it is at
number of
BULE FOR MAKING ALL OPERATIONS WITH THE
DIVISOR 9.
What is the interest on $840 at 3 o;o for 50 days ?
Example: 840 Proof: 31500
3 0/0 3.4999
Answer : 3.50 cents.
2520
50 — days
total.— 126000
half.— 63000
quarter.— 31500 I 9
45 I
3.500 Answer : 3.50 cents.
OTHER PROOF.
84000 840
93332 3 070
46666 '
23333 2520
3 0/0 50— days
69999
50 — days
126000
1800
0000
3.499950
Answer : 3.50 cents. Answer : 3.50 cents.
In these operations with the divisor 9, the result of
the multiplication by the rate and by the number of
days, should be divided first by 4 and then by 9.
Remark.— The addition of the capital from right to
left, as we have previously done, is equal to the nine-
tieth part; if this ninetieth part is divided by four, we
have the three hundred and sixtieth part, or the unit,
or the interest for one day.
If we should multiply the capital by the number of
days before dividing by 9, we will have as quotient
the interest at 4 o?o, for the number of davs the cauital
has been multiplied by.
If the capital is divided before the multiplication by
the number of days, we shall follow the division up to
three decimals ; and after the multiplication by the
— 36 —
number of days, we shall point
der to have the $ to the left and
of the point.
What is the interest on 450 at
Example : 450
50
22500
45
Answer : 2.50.
2.500
What is the interest on 5555 a
Example : 5555
80
Answer :
5555
80
444400
84
34
70
70
7
49.38 cents
off six decimals in or-
the cents to the right
4 070 for 50 days ?
Proof : 45000
49999
50
2.499950
Answer : 2.50 cents.
4 0/0 for 80 days ?
Proof : 555500
617220
80
49.377
49377600
Answer : 49.38 cts.
444400
49.3776
Answer : 49.38 cents.
Proof by addition.
5555
5555
5555
5555
5555
61721605
80
Answer : 49.38 cents.— 49.37728400
What is the interest on 752 at 4 ojo for 50 davs ?
"^^^ 50 83554
50
37600
4.1776
Answer : 4.18,
50
I
16 > - — 4.177700
70 4.177 Ans. 4.18 cts.
70 Ans. 4.18.
nmals in on-
to the right
)0 days ?
45000
49999
' 50
2.499950
2.50 cents.
iO days ?
555500
617220
80
49 377600
: 49.38 cts.
— 37 —
It will be seen from the proofs we have made of the
proceeding rules, that the addition from right to left,
is equivalent to division by 9.
Everybody knows that the general rule is to com-
Eute interest for a certain number of days ; it is only
y exception that it is computed for a month or for i,
i or I of a year.
It is therefore necessary to use a division in order
to make all operations of interest or discount.
With this system, division is abolished, and advan-
tageously replaced by the addition from right to left ;
it is always easier and more convenient to add up than
to divide.
CHAPTER II.
iddition.
555
55
5
605
80
iOO
days?
addition.
00
54
50
00
4.18 cts.
RULES FOR THE STOCK OPERATIONS MOST IN USE.
If the 5 o;o bonds are at $75, what capital may
be necessary to acquire an income of $650 ?
Example:
6500
9100
9750,0 Answer: 9750.
This operation is done by doubling the income
xirartiaA nnA «V.^» -^..itl-^i..: i jt. _ "...-^
'"Ir"-'" ■=■■"■- liSwii !iiui;ip:jr'i:;j^ i?y iiio tjUulSuuti.
One decimal is pointed off in the result, which is di-
viding by ten, and the $ stand to the left and the cents
to the right of the point.
2
— 38 —
Proof by the rule of three.
Example : 5 : 75 : : 650
75
X
3250
4550
48750
37
25
Ai
9750
Answer: 9750.
In this operation, the 650 income has been muItiDlied
by the quotation, 75, and the result divided by 5 this
means that as $5 is the income derived from '$75
what capital is the $650 to be derived from ? '
Tho above operation is made by the rule of three
simple, and it reads thus :
5 is to 75 as 650 is to X.
We will now reverse the preceeding rule and ask :
11 7.-? give 5 per cent, income, how much will 9750 giv e?
Example : 75 : 5 : : 9750 : X
5
48750 I 75
375 I
000
650 ;
Tf -re • ij o ,. Answer: $650.
U 75 yields 8, how much will 5625 yield ?
Example : 75 : 8 ; : 5625 : X
8
45000 I 75
000 '
8 : 75 : : 600 : X
75
Snnn
4200
45000
600 Ans. : 600.
To
thef
sum
only
ber.
add
dowi
^adde
whin
up ai
of al
must
— 39 —
: 9750.
lultiplied
>y 5 ; this
'om $75,
! of three
md ask :
^50 giv e?
1650.
;oo.
/
45000' 8
50 I
20 5625
40
00 Answer: 5625.
PROOF OF THE FOUR RULES.
Addition: 324—9 Subtraction; 84216—3
632—2
784—1
1740—3—3
Multiplication : 432—9
17—8
32214
52002—3
3024—9
432
7344—9
Division =.3 =4787 I 34 = 7
138 I «. =3
4787 270 14079=3
14 320
14
4773
PROOFS EXPLAINED.
To prove the addition, it is necessary to add up all
the figures of every number once, and reduce each
sum to one figure. The first number gives 9, as 9 is
only one figure, we will put it down opposite the num-
ber. The second number is 632, which gives 11 : we
add up tbis two figures and have 2, which we put
down opposite The third number is 784, the figures
added up give 19, we reduce to one figure and have 1,
■' "•-• i--«.-- Yi-Fvciw. iug inree uKures », ",c. i add
up and reduced
of all the figures
roust also give the figure 3
„_, , .-, - added
to one figure give 3. And the addition
in the sum, after reducing to one,
y
— 40 —
This proof may also be made by adding up the
figures of every number as if they were written in one
horizoi^tal line and forming but one number, passing
over the figure 9 whenever it is found and reducing
the sum to one figure.
The proof of the subtraction is made by adding up
the minuend and reducing to one figure, as for addi-
rion, we have therefore 21, which reduced to one
figure, gives 3. Then the subtrahend is added up with
the remai-'^er and we find again 21 which reduced to
one figure, .daves again the same figure 3, the figure
being tljie same the operation is proved to be correct.
In the multiplication, the multiplicand 432 is added
up and gives 9, which is only one figure and needs no
reducing ; the multiplier is also added up and gives 8,
I multiply by 8 and I have 72, I reduce lo one figure
and I have 9. The result of the multiplication is then
added up and gives 18, which reduced to one figure
gives 9, the two figures being equal, the operation is
proved to be correct.
In the division, the dividend is added up and if
there is a remainder it is deducted from the dividend.
Example 4787, the remainder 14 deducted, we have
4773, add up once every figure in the latter number,
you have 21, reduce to one figure, you have 3. Add
up the divisor 34, you have 7, add up also the quo-
tient 14079, you have 21, reduce to one figure, you
have 3, multiply 7 by 3, you have 21, which reduced
to one figure, gives 3.
The figure given by the dividend being the same as
the one given by the multiplication of the two others
numbers, the operation is correct.
It is not necessary to divide the sum of additions
by divisor 9, to prove an addition : all thai is wanted
is to reduce to one figure, that being equivalent to the
remainder after a division by number 9.
Example : add up every figures of 789654, you
have 39, reduce to one figure, you have 3.
1
2
3
4
5
6
7
8
9
10
g up the
en in one
', passing
reducing
riding up
i for addi-
1 to one
\ up with
iduced to
he figure
correct.
is added
needs no
1 gives 8,
ne figure
n is then
le figure
ration is
p and if
lividend.
we have
number,
3. Add
the quo-
:ure, you
reduced
same as
others
iddilions
J wanted
nt to the
354, you
— 41 —
By division, if vou divide 39 by 9, you have 4 • 4
limes 9 is 36, which deducted from 39, leaves 3.
Thus, without division, the same result is arrived at,
and operations can be proved as quick as thought.
In adding up, the figure 9 may be passed over ; that
IS to say, it is not necessary to add it up with the other
figures.
Multiplication Table.
2... 3... 4... 5... 6... 7... 8... 9... 10
4... 6... 8. - 10... 12... 14... 16... 18... 20
6... 9... 12... 15... fS... 21... 24... 27... 30
8... 12... 16... 20... '..... 28... 32.. 36... 40
10... 15... 20... 25... 30... 35... 40... 45... 50
12... 18... 24... 30... 36... 42... 48... 54... 60
14... 21... 28... 35... 42... 49... 56... 63... 70
16... 24... 32... 40... 48... 56... 64 .. 72... 80
.n I i?- ^^- ^^- ^^'" 5^- «3... 72... 81... 90
10 1 20... 35... 40... 50... 60... 70... 80... 90... 100
1
2
3
4
5
6
7
8
9
At 4 0/0 Multiply the capital by the number of days,
add up from right to left and point oflTfour decimals.
At 5 010 Multiply the capital by the number of days
add up from right to left, add one fourth.
At 6 0/0 Multiply the capital by the number of days
add up from right to left, add one half.
At 7o;o Multiply the capital by the number of days,
add up from right to left, add onne half and one fourth.
At 8 0/0 Multiply the capital by the number of days,
ade up from right to left, multiply by 2.
At 9 0/0 Multiply the capital by the number of davs.
add up from right to left, multiply by 2, add the
At 10 0/0 Multiply the capital by the number of
days, add up from right to left, multiply by 2, add one
— 42 —
Ex.— 250— -4 0/0
30- -days
7500
0,8332
At 5 o;o--250
30
At 7 o;o— 250
30
7500
8332
4166
2083
7500
8332
2083
10415
At 6 0/0— 250
30
7500
8332
4166
12498
14581
At 8 o;o— 250
30
7500
8332
At 9 0/0 --250
30
7500
8332
16664
2083
18747
Table of numbers represewting the rates.
hJ^ilT^ multiply the numbers representing the rat^^
tin finW?KP'^-*l ^"^ ^^^'" l>y the UmJt)ero^f days von
If you want to find out interest at 9Z «-•« toi^^ .i.
number 5556, to wdich add 94 qn in J ^?' !^^® ^^®
and you^have the mtereV^'aS '"''" '^'"'"""^
— 43 —
1
Hates.
i 0/0.
i o;o.
i 0/0.
i 0/0.
3 0/0.
I 0/0.
' 0/0.
0/0.
H 0/0. ,
H 0/0.,
II 0/0..
2 0/0..
2io/o..
2J 0/0..
2| 0/0..
3 0/0..
31 0/0..
3i 0/0...
3i 0/0...
4 0/0...
H 0/0...
4i 0/0. .
The SIGN 0/0 means per cent.
numbers.
.... Ul
1041
694
1389
..... jHfoO
2430
1736
2778
3472
4166
.... 4861
.... 5556
.... 6250
.... 6944
.... 7639
.... 8333
.... 9028
.... 972'>
...10416
...lllll
...11805
...12500
Hates.
43 0/0.
5 0/0,
51 0/0.
H 010.
H 0,0.
6 0/0.
61 0/0.
6i •JO..
63 0/0.,
7 0/0 ..
71 0/0..
7i 0/0..
0/0..
0/0...
0/0...
0/0..
0/0 ..,
0/0...
0/0 ...
0/0
71
8
81
Si
81
o
H
9i
91 0/0.
numbers.
13194
13889
14583
15277
15972
16667
17361
18055
18750
19444
.....20139
....20833
....21528
....22222
.. 22916
....23611
....24305
...25000
....25694
.. 26388
...27083