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.
in D irect an d In vers e P r o ortion
p
br s
.
lb
c ts
10
A n d , as 7
Qu es t
m
.
42 5
.
5
gm ?
3
1—
5 4
c
.
.
Qu es tion s
s
d
.
13 1
do l l s
.
A s 17 5 0
P rom i s cu ou s
d ol
.
5 12 th at is , as 1
do l l s
.
mu s t s el l f or
1
A s 64
.
Qu es t
g al
.
.
l bs
.
1
lb
2
8m
.
.
X 2 0 16
.
2 0 16
lb
t
12
4
c ts
.
s
.
.
.
44
S ING L E R UL E
O F
P age 6 3
l
t
(
cw
4
l
.
N
qr
.
ow
,
0
lb
.
ch
.
e s ts
lb
.
.
.
»
ch
.
T H RE E
-
lb
es t
.
.
wh ol e weigh t
th e
‘
as 1 l b
11
b
11
b
1 1b
o
d lls
80
.
‘
90
10 5
12 5
.
.
.
.
c ts
.
.
100
80
.
113
40
12 6 l b
.
13 2
12 6 l b
.
15 7
3 0
50
c ts
.
126 1
b
c ts
.
12 6
c ts
.
c ts
.
lb
$ 5 04
Then
5
5 04 d o l l s
5 o z x 12 = 6 0
.
A
as
,
.
nd ,
d o z en
as 5
o f ro l l s
4
:
unce s
60 02
cw
.
as 17 92
,
o
.
t
.
o
d ll s
.
c ts
00
$ 99 1
3 0
Then
7
.
ho
80
rs e s
s
o
d lls
.
as 27 0
,
qr s
z
.
in a
.
1 d o l l ar
.
d o z en
of
w eigh t o f
th e
Cu
‘
5
r ol l s
flo ur in a
.
.
99 1
40
rs es
z
8
7
48
1
G e ts
.
00
70
60
c ts
c ts
3 0
mu s t s e l l
.
b bl
.
z
:
Hal f a to n = 10
ho
7
.
.
.
C os t
0
78
F reigh t
3 7
O ther charge s 3 0
G ain
143
bb l
A
480 z
2 24
d lls
6
b rea d
of
.
1 1b
.
00 th e amt
.
oz
Then
z
o
.
cw
1
3 d ol l s
t
= 40 q rs
.
675
.
fo r
4
—
7
c ts
.
.
l 4l b
8
8
wh ol e
.
7
:
the
.
=
i
n
111 inche s ,
a
n
d
9
f
t
3
Q
in n
inche s I n tha t d i s tance th e l arge
wheel wil l have ma d e 111 revo l u ti o n s an d th e s m all er 144
turn s
B u t 144
turn s o f th e l e s s m o re than th e
grea ter in tha t d i s tance N o w , as 3 3 turn s 15 984 inche s
7—
43 6 3 T 1 inche s ; thi s , when re d u
10 0 0 turn s
48
d i s tan ce
c e d to mi l e s , is 7 mi l e s , 5 furl o ng s , 3 4 yar d s , 1 fo o t,
u es t
.
.
.
.
.
.
'
.
inche s
.
S
INGL E RUL E
P ag
63
e
Q u es t
9
.
h rs
.
da
.
.
18
A s 15
yd s
10
A
.
T H RE E
O F
45
.
.
T hi s q u e s ti o n being invers e
h
15
18 15 X 3
rs .
12
o
d lls
.
12
.
c ts
.
= 2 2 ; d ay s
2
yd s
l
s
sa
y
,
.
6 7 d oll s
15
W ha t 15 yar d s c o s t; b u t g o f 6 7 d ol l s
am oun t tha t 15 yar d s s ell s f or
N o w 42 $ yar d s at 1 d ol l ar c o me s
w hol e gain
50
.
.
c ts
.
5O
c ts .
d ol l s th e
.
.
.
,
42 d ol l ars 5 0
to
,
cen ts
,
.
d
c ts
.
d
.
c ts
.
d
.
c ts
.
.
mu s t ge t in all
receive d f or 15 yar d s
19 1
Sub trac t
D i ff
27
%yards mu s t b ring
th e s u m t
hat the remainin g
.
B ut
.
yds
A
11
.
.
.
A
s
O
r,
r od
A s 40
.
.
6 40
40
o
c
.
rul e
f
.
.
d
.
1
.
r od
yd
.
s
b y th e general
yd s
ft
r od
12
c
d
.
S tate
yd s
.
rod
red
(
=l
A
.
red
.
r od
1
:
P ag
e
.
6 0 x 4)
.
16 0
t
44
:
r
.
4
:
r
.
1
z
od
s
od
s
.
16
.
16
64
.
B ecau s e th e l arge whee l h as 7 0 c og s , an d th e s m al l
o n e 5 2 , th e s mall whee l wi l l mak e 7 0 rev o l u ti o n s whil e th e
l arge o ne s make 5 2
B ut 7 0
rev ol u ti o n s that the
s m al l w h ee l wi l l gain in th e s ame time
Then s ay ,
13
.
.
.
SI N G L E RU L E
P ag
64
e
16
S tate d
.
A
mo
.
s
.
l
s
17
A
.
A
.
A
5
,
l l)
.
90
3
1b
.
z
3 6
s
o
.
5
18
5
d lls
.
.
3 6x5
.
5
:
.
s
15 24
:
l
yd s
d
.
d
163 5
19
s
.
.
qr
1 15
g
ga
d ay s
3 65
.
2
it wil l
c
.
.
inches
be
m
.
18
3 04 6
1
p o un d s
.
3 6
t
.
= 3 60
in
herefo re
an d
d ay
1
.
in
.
.
d
y
20
s
lb
18
19
o
.
12 7
in
.
general ru l e it wil l be
d ll
d ol l
b y th e
mo
m m
“
.
.
%
.
197
'
4
78
9
3 65
3 95 0
47 3 5
23 6 7
.
28
8000
4
"
o
*
12 )
c u
7 2 000
21
0 ) 6 0 0 l0
3 0 0l
d
y s
21
1]
.
A
sq .
But
nd
22
A
.
.
A s 6 00
.
840
,
.
qr s
x3 =3
f eet
Then
.
840
3 x 2 i = 7 4, 2 5
in a piece
10
q
.
f
sq
ft
sq
t
p
.
.
.
ze c e
sq
tn
5 0 10
tn
.
6 10
: 2
.
.
sq
.
.
f
%z
.
:
15 448
.
ft
.
ln
wal l s
th e
.
t
.
2
—
10 I T
75 6
1
50
ft
=8
4=75 6
40— 8
as
s
.
2
s
2n
.
u
ou
u
H e re m it m l tip l yin g b y th e thir d n u m b e r ,
l d im m e d iate l y h av e to d ivid e
b e a s e y ou w
by th e s am e n m b e r , to b rin g f ar thin gs
piece s
12 8
7 4, that is
tn
.
f
t
.
in
.
10
%
DO
UBL E
or
P ag
64
e
I
l
a
g
h pt
.
.
23
.
.
U B L E
D O
l
a
g
.
z
15
47
.
.
h pt
A s 16
.
T H RE E
UL E
R
.
O
R U L E
%gal l o ns
93
10 0
:
F
H
T
R E E
.
Bel ieving tha t it wi ll b e accep tabl e to s o me I here in s er t
tw o ru l e s f o r s ta ti n g th e D o u bl e R u l e o f Three in a dd i ti o n to
tha t in th e W C al cu l a to r
Pl ace th e three c o n d i ti o nal term s in th e f ol
R UL E F IR S T
l o wing o r d er : tha t which is th e principal cau s e o f gain l o s s
o r ac ti o n p o s s e s s e s th e fir s t p l ace ; tha t which d en o te s s pace
n d tha t which is th e
o f time o r d i s tance o f p l ace th e s ec o n d; a
gain l o s s o r ac tio n th e thir d ; then p l ace the o ther tw o term s
which m o ve th e que s ti o n un d er th o s e o f the s ame name an d
if th e bl ank s pace fal l s un d er th e third mul tip l y th e three l a s t
term s f o r a d ivi d en d a
n d th e tw o fir s t f o r a d ivi s o r : b u t if th e
bl ank fall u n d er th e firs t o r s ec o n d pl ace mul tip l y th e fir s t
s ec o n d
an d l a s t term s to ge ther f o r a d ivid e n d an d th e o ther
tw o f o r a d ivi s o r ; a
n d th e qu o tien t wi l l be th e an s wer
R UL E SE C O N D
W o rk b y tw o s ta teme nts o f th e Singl e
R ul e o f Three
,
,
.
.
.
,
,
,
,
,
,
,
,
,
,
,
,
,
‘
,
,
.
.
.
P R O P O R T IO N
DIR E C T
P age
Q u es t
.
T h en
ho
2
10 x 6 0 x 3 6
18
X 20
O
rs e s
A s 18
60
H o rs e s
d ay s
.
r
th
y
da
u
18
10 b u s h el s
20
= 10 x 3
x 2 = 60 b
bu
r,
60
.
3 6
u s hel s
z
10
:
1
.
Then
h or s e s
d ay s
.
by
h
A
s
tw o s tatin gs ,
o
r s es
18
d ay s
A n d , as 2 0
bu
h
.
10
bu
3 3
b u s hel s
18X ? ”
3 6
O
.
s,
s
20
66
.
o
r s es
60
33
%b u s hel s
d ag
/s bu s h e l s
.
%z
z
:
3 6
.
.
6 0 as b ef ore
.
.
48
P ag
66
e
S tat
e
d th
us
,
M en
3
5 6 1h
O
21
me n
3
d ay s
d ay s
Th en
X3
x3 x3
7 X 14
r
by
lb
m en
: :r
lb
s
d
un
o
6
3
p
14
As
lb
d ay s
d ay s
.
an d
15 8
,
.
56
21
7
16 8
3
3 6 p ound s
O
S tat d th
S tu d ents 8
us,
e
4
3
2L
s
tw o s tatin g ,
m en
As
s
lb
s
a
d
m
y
—
6
—
5
4
1
7
en
.
56
O
th u s ,
r
12
d oll s
4
3 8
.
10 mo
m o n th s 6
0
1
2
1
4X
38
L e
Then
8X 6
—
.
8
12
thu s ,
m o d ol l s
4
3 8
6
s tu
s tud
r
—10
e
,
O
sl u
A
by
r
tw
s tu
.
8
s
5
.
nd
576, a
s
s
l
ar
l
o
d
96 0
576
O
e
.
.
10
S tat
t
d lls
s
m0
6
o
,
d ol l
4
3 8
d ol l
12
mo
As
o s tati n gs
d th u s ,
d ol l s
20
40
.
cw
25
mil e s
10 0
50
cw
t
.
d ol l s
t
—2 5
.
-
.
50
2 01
40
—10 0
mil e s
s
l
ar
l
o
d
0
=
0
1
=
2
2
25 x x
40 X 10 0
r th u s ,
.
20 x 5 0
O
r
by
n gs ,
i
t
a
t
s
o
w
t
cw
t
.
cw
t
.
O
S
6
.
tate d th u s ,
D ol l s
.
7 00
mo n th s
d ol l s
.
40 0 d oll s
6 0 mo
l
s
l
o
d
d ol l s m
4
1
6
7 00
—
40 0 6 0
.
.
r thu s ,
o
—
.
DO U B L E
P ag
67
e
O
S tat
e
A
4
in v
.
er
Then
ac
1
by
c
a
r es
1x 2 0 x 1
1x 1
8%and ,
7
.
th u s
e
.
,
me n
10 0
0
0
1
7
x
12 x
U r
c
a
d ay s
ac
4
As 8
r es
0
10 0
-
5 d ay 3
n gs ,
i
t
a
t
s
o
tw
by
7
20
s
1x 1
4
2ox 8
’
s
e
r
c
a
d ay
4
8
2
1
—
'
acre s
5
<
1
>
1X
s,
m en
.
.
acre s
Th e n
inv
u
r th
O
S tat d
M e n inv
6
A
12
5
S
m en
m en
d ays
d ay s
m en
m en
r es
m e“
20
tin gs ,
o s ta
N
—100
5
s
84x 5
,
—
rs e
inve
d ay
5
As
5
7
.
2
1
<
0
>
0
1
7 x
d ay s acr es
84
12
m en
acre s
10 0
m en
d ay s
0
d thu s ,
th u s ,
r
r es
,
10 0
:
d ay s
%
an d ,
14
A
s
d ay s
m en
m en
20
7
%
14
:
5 d ay s
.
40
6 06 0
c ts
m
l bs
O
r
hu
t
mi
2 00
s,
—
40
c ts
O
s
l
e
i
m
60
t
in gs ,
a
t
s
o
w
t
by
r
A
mi
.
s
A
s
S tat d
e
3 20
th
w
u
6 06 0
lb
s,
d
d
n
a
,
0
6
60
.
.
2 02 0 0
m il e s
c ts
40
40
lb
.
40
0
4
1
)
x
1
0
2 02
c ts
1nv
.
.
0
6
0
6
0
X
40 X 2 0
T he n’
c ts
0
6
—
0
6
0
2 02 0 0
lb
.
il e s
cen ts
2 00
mil e s
s
l
e
i
m
0
6
6060
O
s
l
l
o
d
0
3 0
22 6
men
22w
v
u
l
n
e
m
12
g
a
M j
fl
fi
m en
d
5
12
6
s
eek
m
w
r th u s ,
—2 2
0
2 d ay s
d ol l s
—2 0 0
.
-
3 00
—48
24
= 12 0 acre s
l zx 8
O
r
b y tw
o s tarin
c
a
A s 12
24
w
2
c
t
w
.
w
.
h o r s es
}
18x 3 5
45 00 x 3
by
tw o s tatin s ,
g
cw
.
,
.
.
d ay s
r s es
18
6
h o rs e s
ho r
t
o
ho r
.
—
.
cw
.
t
cw
.
12 0
acre s
d ay s
t/
b bl
t
.
45 0 0
.
an d ,
t hhd
20
:
d ay s
cw
.
s
cw
3
t
hu s
t
r es
.
mx 6
,
cw
A
r
—6
a
3 5 00 c w t
3 d ay s inver s e
18
O
r
48 z
.
h
O
8
s
an d 9 d ay s
45 0 0 c w t
invers e 6 d ay s
Th 9 11
A
t
BO W
.
c
d ay s d ay s
r es
20 and ,
t
.
gs,
a
r es
0
cw
A
d ay s d ay s
3
s
6
:
hor
: :
hor
.
28
14
t
12 x 9 = 10 8
n d 2 , 5 x 5 0 = 12 5
, a
S tat d thu
10 8c w t
6 0 mi l e s
e
.
r
12 5
cw
3 00
mil e s
m il
.
.
108
cw
.
.
o
d lls
.
3 00
t
O
t
r
hu s
t
,
100
t
12 5
.
.
o
d ll s
o
.
.
5 0 0 an d ,
.
5 00
do lls
.
5 78
.
.
-
cen
t
s
%
d lls
mi
t
10 8 6 0
125 —3 00
= 5 7 8d o l l s 7 0
tw o s tatin gs ,
60
cw
As
.
10 0
by
m il
As
d ol l s
cw
10 8
x 60
O
.
s,
}
Phen ,
.
c ts
.
.
o
d lls
-
.
10 0
0
~
T h en ,
1 X 45 X 4= 2
1
By
yar d s
atin gs ,
tw o s t
lb
lb
.
.
an d ,
d ay s
12
9
.
n
e
m
4
2
Irfv
o
ng
l
t
f
0
0
2
.
.
7 2 0 m il e s
16 hrs inv
.
ays
fee t high
4 fee t thi e
6
8f t high
.
6 ft
.
.
T h en ,
4
6
<
X
—16 d ay s
6
8
X
U
6 X 2O X
M
E
PR A C T IC E
Pa
g
10
Then
.
Then
12
v
9 x5
,
d ol l s
I
+ 4)
12
.
%
o
d lls
12 X 6 ) 11
_
4
8X 10
5
.
6
Th e n
,
45
d ay s
9
by
2 2 , 5 x 17 , 3 x 10 , 2 5 x 9 x 8
, 2
,
3
12 3
P R A C
C as e 1
.
2
fee t Icing
fee t wi d e
fee t d eep
m en invers e
d ay s i nvers e
s
5 x 2 x 2 x 12
.
.
—
u
2,
d
c ts
111 3 7 5
.
.
I nvers e
I nver s e
c ts
per s ons
6 m o n th s
11 d o l l ar s
11
fee t lo ng
fee t wi d e
f t d eep _
men
h o urs
.
112 0 d ol l ar s
-
$
114 “
per s o n s
8m o n th s
45 0
8pers o n s
1 m o n th
10 d o l l ar s
11
.
l4
45 0 x
,
68
e
do ll s
pers o n s
m o n th s
9
5
.
53
.
IC
T
P ag
e
ca
n
d ay s
E
e lh n
d
t
g
'
.
69
.
.
c ts
.
25 0
yar d s
O
r,
25 0
.
s
5
50
25
87 5 0
12 5
3 50
*
‘
o
C an e l l in g
T hat is , w h e n the d iv is r an d d ivid e n d ar e th
e c o n tin u e d
p r d u t o f a n m b e r o f f a t r s , y o u c an d ivid e b th b th e s am e
v
d i is o r ;
y
and th e p e r ati n w il l b e re atl abr id e d
g
y
g
T hu s , in the ab v e e x am l e I b s e r ve that
p ,
g e s in t 45 tw i e ,
w ith u t a r e m a
in d e r ; th at
is
n ta
in e d t w i e , in
an d th at
9X 8
n e in t
6 X 12 , 3
, 2 g es
N o w , b y d ivid in g the n um e rat r
an d
d e nomin at r by th e s e d ivis r s w e o b tain L
,
as ab ve
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ec a
us e _
2 2 00
8
d
26
O U N T
.
18
5 7 5 0 X 10 0 X 3
—4
103 § x 3
.
112
.
i
n tere s t f o r th e
%
d
.
c ts
cl
.
18
57 50
1
8
d01
ti m e
o
c ts
.
17 9 1
8
.
20
03 1 1
c ts
7 91
8
2 O—
03 1 1
~
(
d oll s
h en
mo
5
%
.
7 x 2 ) + 10 0 = 114 d oll ars
4
T
.
83 %p er c e n t
.
18
.
100
z
15 18
.
(
1
12 int f or th e ti me
.
d
2 46 4
.
mo
3
.
A s 12
.
mo
(
1
.
e
.
8127 13
6
O R
E
Pa
mo
7
.
7773 3
8
3 113 4
d
.
,
.
as 114
d ol l s
.
14
d ol l s
.
65 0
6 5 0 ) 14
l l
:
d o ll s
=7 9
.
c ts .
,
d ons
P ag
mo
5
h en
cl
.
d o l ls
.
as 10 3
,
A
3
A
.
mo
T
hen
d ol l s
d ol l s
12
c e n ts
.
S
1)
Is l
o te
tu n e
d ol l s
.
= 5 000 p
d ol l s
.
.
w
.
495 0
5 0 00
.
.
ra
te
f o r th e ti me
d ol l s
.
0
7 28
.
0 th e d i s c o u n t
78
.
0 x 9 = 6 5 5 20
7 28
,
dol l s
d olls
.
12
mo
.
10 3
N
.
do l l s
.
8
:
as 112
,
Agai n
.
92
e
d olls
.
18 z
d ol l s
5 15 0 x 10 0
.
—
99
12
s
= a rate f o r th e
3
5 15 0
P ag
.
.
,
A s 10 0
mo
cl
.
d ol l s
.
10 0
g ain
.
8
A s 12 { 45
.
d ol ls
T
mo
.
91
e
i n tere s t at 6
i n tere s t at 2
er c e nt
p
.
i n tere s t at 8p e r
d i s c oun t
ub tr a
ct
c e n t.
D i ff e ren ce 3
N
dol l s
2
3
cl s
.
246 4 d o ll ar s
><
B A N K
c e n ts
g
d ay s
dol l s
(
60
O
17 X 2 5 0
r,
3 0
d ay s
90 + 4) X 6 00
.
60
O
r,
93
d i s c ou n t
= 2 26 6 d oll ars
O
.
U N T
88c e nts
.
.
= 1 d oll ar 41%
c e n ts
= 1 d oll ar 41%
ce n ts
do l l s
(
e
c e n ts
.
3 0 + 4) x 2 5 0
.
c e n ts
D IS C
Pa
3
.
.
8= 197 l 2
B u t 2 46 4 d o l l ars — 197 12
2
2d
16 5 0 x 3 = 495 0 = 49 d ol l ars 5 0
.
.
o te
.
9 d ol l ars 40
47 d ay s X 6 0 0 d oll s
.
c e n ts
= 9 d oll ars 40
c en ts
.
dol ls
d ays
4
c ts
60
4
0
0
6
2
1
2
3 X
O
r,
3 0
dol l s
d ays
5
c ts
60
d
2
.
cl
d ol l s
5 40 0
d
00 0
A s 18
or
.
3
10
(
1
.
olls
d ol l s
.
c ts
d
.
9
0
C
7
7
3
0
+
cl
c ts
.
z
2 0 46 7 5
d
.
.
c ts
cl
.
1 23
0 37
9 4O
00
:
z
z
.
’
’
i
n
a
g
s
6
s
6
13 5 0
(
l
.
.
d
.
,
.
B
cl s
.
h en
22 5 0 A
7 5 00
6 000
45 0 0
.
.
.
T
0
0
0
=
8
1
,
45 0 0
C
.
0
0
0
6
B
A 7 5 00 +
.
d ol l s
d
cl
.
0
1 0
c ts
96 8
3 7 5 27
5 3 2 05
17 1
.
%
d
.
=27 29
d
7
5
C O MPO U N D
F
g
Pa
10
ELL O W S H IP
96
e
.
5 d oll ars = 10 5
2 0 d oll ars + 8
.
(
I
N
as 10 5
cl
d
ow
nd
A
63
,
.
(
l
d
d
.
hen
C
,
d ol l s
as 12
d
.
21
.
20
’
M P O U N D
B pu t i n }
F E L L O
97
e
W
S
l
(
.
.
12 0 0 x 10 = 12 00 0
6 00
80 0 X 8= 6 400
6 00 x
112 0 0
100 0 x 12 = 12000
112 0 0
80 0
15 6 0 0
1920 0
18
00 X
§
I P
H
mo
.
200
A
i
n
u
t
p
.
.
.
gai n
s
50 C
(
l m0
= 48
6 00 x 8
00
.
g ai n
s
.
3 5
g
3 0
O
k
d
.
P ag
2
’
12 A
2O
d
T
s to c
cl
C
.
Wh ol e
d ol l ar s
63
d ol l s
.
3 6 00
B
A
B
C
15 6 0 0
7 20 0
46000
C
192 0 0
p
’
5 6 0 d ol l s A s
’
0 d oll s B s
78
’
96 0 d ol l s C s
1120 0
N
O
O W
,
r
as 46 0 0 0
20
1
.
3
As 9
.
15 6 00
1920 0
2 3 00
mo mo
d ol l s
.
6 00
12
g
Pa
mo
A
4
.
s
.
8
mo
12
d
.
z
:
mi
.
5
.
A
6 00 + B 6 O O
.
e
.
s ha
re
.
.
d ol l s
800
.
98
.
.
18
00
2 7 00 d ol l ars
mi
.
mi
.
m il e s
+ O 3 00 + D
6 00
6 00
00
1
L
—
:
6 0 d ol l s
.
A p ay s
60
B
10
C
D
.
PR O F IT
g
Pa
6
5
.
ox e n
8
75
.
98
e
.
m o n th s = 22 é
40
x 4é
x5
_5 8
3
x cg
9
L O SS
AN D
-
-
um
S
"
12 1
d
d
c ts
145 2 0
.
N
ow
7
,
as
12 1
22 $
.
40
58
02
A p ay s
B
48
:
58
%:
7 0 20
400 0 d ol l ars X l 2 m o n th s = 48
000
—
3 000
45 0 0 0
X 15
5 00 0
40 000
X 8
.
~
13 3 0 0 0
’
d
A
O
s
13 3 00 0
r
20 0
‘
8
.
48
0 00
45 0 0 0
40 0 0 0
.
6 65
1
2 40 d oll s
225
5f
.
.
200
58
0 d oll ars x 6 é m o n th s
1
58
0
X 92
87 0 I
x sg
'
’
A s
’
B s
’
C s
p art
=3 770
= 5 5 10
= 7 5 40
‘
,
16 8
20
d
A s 16 8
20
cts
.
26 3
.
90
F I T
P R O
2
.
15 0
cent s
12 0
c en ts ,
3 0
d
y
3
.
A
s
1
z
:
-
e
99
A
.
B
C
45
3 0
L O S S
.
.
f or
120
cents
d
.
30
.
D
86
118
15 c
.,
c os t
cts
.
cts
.
s ol d
,
120
,
7 5 40
cent s g ai n o n
c ts
Then a s
,
:
A N
P ag
5 9d
3 770
5 5 10
:
d
10 0
:
2 5 pe r
d
s
(
l
cts
y
42 : 2 7 7 2
S ub trac t 2 1 0 0
.
R em
.
.
.
c e nt
.
s ol d
c os t
6 7 2 gai n
f or
.
l ost
P R O FI T
76
A N D
P ag
4
gall s
32
.
g
A
5
kn if
A
a
l
g
.
cl
.
l
g
.
10 8
e
c ts
1
.
60
,
f or
loss
on
100
e
c ts
.
A
3
d
y
.
7
10 0
z
$ 200
z z
d
.
l
d
.
A
.
10 8c ts
.
z
12 0
100
s
—90
h at
c ts
c ts
.
l
100 d ol l ar s
18 w
.
cts
.
a
ll
on s x l I O
g
18 z
c ts
d
.
.
.
0 8m u s t s e l l f o r
gain per
h at
t
d oz
d
c e n ts
= l 26
50
5
00
.
ha s
.
9
115
.
1
= l 8c ts
.
1
.
d
.
90
:
N o w as
,
kn ife
2
z
d
8
e ach
.
51
s
d
y
per gall o n
43 2 0
:
149 d oll ar s
.
s ol d
cl
P ag
6
c ts
.
1
s
for
c o s t per gall o n
50
kn ives
.
s ol d
c ts .
.
cen ts —17 cents = 3 cent s
20
.
.
96
as 18
0
nd ,
18
0
gall o n s
d oll ar s = 10 8d ol l ars
d
.
as 192
,
.
,
+l2
g al l s
Then
99
e
arrel s = 192 gal l o n s bought
S u b t ract
12 gal l o n s l e ake d ou t
9 6 d o l l ars
nd
.
x6 b
.
R em
A
L os s
.
.
43 2 0
2 40
:
c ts
.
A dd
c os t
$ 13 1 5 0 m u s t s e l l f o r
‘
13 1é gal l s
13 1 d oll s 5 0 cts
B u t, as l d ol l
1 gall
th e num b er o f gall o n s that m u s t b e s ol d
C o n s eq u en tl y
gal l o n s — l l 5 gall o n s = l 6 é gall o n s o f wa ter
-
.
.
.
.
.
.
d ol l s
10
lb
.
cw
.
c ts
t
.
.
448
.
c ts
.
13 4 40
cos t
112 0 0
s ol d
22 40
lb
.
T he n as 448
,
o
d lls
.
c ts
.
2 2 40
lb
f or
W h ol e
l os s
.
1
5
cen ts per
lb
.
.
P ag
100
e
o
d lls
A s 12 0
11
.
z
100
y ar d s
Then
12
o
d lls
A
.
c ts
4
S
As
.
o
d l ls
.
90
12 0
d
N
1440
ow
.
as ]
,
d
z
s
bu
.
25
z
15 0
:
d
.
1
z
.
.
c ts
.
per b u s he l
1 25
c ts
.
.
3 0 b u s he l s
3 7 50
:
,
22 5
.
1
ct
o
1000
H
4d oll ars
Then
d lls
.
c os t
.
1
.
.
1440 W h ol e
yd
.
cen ts
d
13
o
$ 4 90
.
d ol l s
.
17 2 8
:
d lls
as 3 6 0
,
.
5 0 0 0 gal l s at 48cen ts = 2 400 d oll ar s
Inte re s t f o r 2 m o nth s
3 2
.
5 000
gall s
at 5 0
.
C o s ts $ 243 2 to pay c a
sh
cen t = $2 5 00 in tw o m o n th s
s
D i ff
6 8gaine d b y b o rr o wing
th e m o ney an d paying th e ca s h
.
.
h e ad d o l l s
15
12
.
o
d lls
.
2 40
,
c o s t p er hea d
d ol l
lb
C
2 0 th e
c ts
Then
.
lb
.
as 4
s
.
1
.
.
20
2
qr lb
.
.
l
:
B A R T
P ag
.
E R
.
10 1
e
.
3 20 b u s h e IS X $ 1
.
Subtrac t
ca s h
16 0
$ 22 4
c ts
Then a s
,
20
cl s
3
4
.
z
)
.
As
.
lb
.
.
1
.
70
z
224 d ol l ar s
c ts
s
25 0
A 3 10 0
.
28
12 5
3 19
z
l 20
.
ba
c ts
A
1120 l b
: :
%
d
c ts
1 25
7
c ts
.
23
.
5 0 bu s he l s
91
6
.
m
29
d
.
:
.
cl s
.
0
c ts
1 50
.
.
3 0
nearl y
.
P ag
e
d
6
d
c ts
0
2 8
AS 2 50
c ts
c ts
7
A
s
.
c ts
.
A n d , as 90
d
.
c ts
.
.
oz
d ol l s .
5
2
s
l
l
o
d
6
s
4
l
>
arre
b
25 0
.
.
lb
.
cw
c ts
t
.
A n d 448
6 4 lb
2 5 y at
1 3 44
s
l
i
l
m
5
4
1 0
lb
10
ms
-
99
95
-
c ts
8
c ts
.
d
2—
s
l
—
—
he
s
u
b
2 1 11
2 40
.
0X
2’8
c ts
1 20
ba
.
90
99
c ts
d
c ts .
102
at
.
3 0
00
.
.
C ash
2 0 6 10
0
5
2
0
6
$
50
ct s
.
E X
C H A N GE
P ag
10 4
e
d
3
s
.
d
.
d
.
79
.
.
s
.
d
.
L
.
L
.
s
.
d
.
.
.
£ 15 0 0 x 3
O
r,
.
.
.
93 3
5
8
4
g
8 d
A
s
8 An s wer
6
0
18
8
6
.
2 40
O
as b efo re
£ 18
0
60
r,
£ 2 40 as b efo re
(
1
d
.
5
.
AS
d
.
8)
56
O
8
.
.
3 60
72
2
£ 28
0
r,
3
80
:
ad d
£ 3 6 0 as b efo re
d
6
.
A
s
3
.
cl .
.
3
O
3 04
r,
6
15
3 3
15
46 8 19
4
.
3
5
d
.
.
s hil l in
p en
c
%
Pa
g
e
.
105
3
.
gs
s
'
Th e n
l
,
P ag
e
.
Then
.
80 12 0 +
108
.
,
07 %
22 17 d o ll s
.
4 d ol l s
ct s 5 m s
10 ct s
07
.
.
.
.
d ol l s
—
2 6 = l 5 63
.
c ts
.
8
5
°
d
.
d ol l s
.
.
9
-
.
.
s
.
,
d
C as e 3
5
6
e
75 0
.
%
10
%
Then
.
23 6
8 10%
£ 23 6
C as e 2
3 04
.
%
45
27 0
d
.
8)
56
72
s
d
.
c ts
90 = 2 00 1 3 3
25
.
%
80
ALL
IGA T I ON
P ag
d
.
108
e
c ts
.
.
.
P ag
e
3
112
.
0
2
.
16
18
22
20
quantity given
2
2
I
) )
24
2
lb
N
ow
,
.
3 4
of
g;
3
10
as 2
b
10 1
of
17 0
al l o y
c a ms
l
h m
i
24
C as e 4
2
(
.
zo
15
2
17
2
ll8
)
2
2
[
2
16
Oz
T hen
,
40
as 16
2
5
3
2 3g
0
1 T hen
,
as 6
of
5
1
s
l
a
g
‘
.
oz
25
10
3
15
of
70
N
ow
carats fine
22
ga
s
l
l
58
%
gal l s
11%
.
.
ne
ater
of W i
of w
,
144
co
m
O
meas ure
2 a
—= —
s b
r,
7
3
efore
3
75
'
50
75
50
(
((
5)
1
O
r,
5)
5
-
as b efo re
RE DUC T I O N
V
or
UL GA R FRA C T I O NS
P ag
00
48
0 0 ) 10 8
.
.
(
Then
2
5
m
.
{ 6%
12 0 0
,
96 0 0
co
.
114
e
4
83
‘
(
me as ure
O
48
00
12
48
r,
:
3
as b efo re
.
91
(
3
76
c om
.
m ea s ure
)
13
(
26
2
26
6
4
8888
88884
98
76 )
.
(
Therefo re
9
C as e 2
2
12
X 17 +
.
e
115
98
76)
(g
T
.
frac
ti o n require d
?
21
T
17
18
3 x 2l
P ag
.
,
9 87 6
m
73 8884
+5
o
3
l
l
”
o
48
l
H
21
5 14x 16 + 5
_ 8M2 9
O
16
frac ti o n require d
frac ti o n require d
I
)
.
42 5
68
4
I
3
27 3 7 03
Z
C as e 3
848+
.
3
3
.
2 46 5 —
2 7=
.
equiva
en
mi
l
t
x e d num b er
%
e qfi iva
3 5 2%
l e n t mi x e d numb er
5 6 % equival en t mi x e d num b er
5
—
—
5 141 6 equival en t mi x e d num b er
:
-
4
.
5
m re d
re q
18
3
21
2
frac tl o n
.
P ag 116
3 x5 x6
90 fir s t numera to r
9
4x 4>< 6 = 96 s ec o n d numerato r
10 0 third n u mera to r
5 x 4X 5
12 0 c o m m o n d en o mina to r
A n d 4x 5 x 6
95
i’
—
a
T h erefo re
nd
are
the
frac
i
t
s require d
on
0
12
} 23
C as e 4
.
3
:
O
:
:
,
e
.
84
RE D U C T I O N O F
P ag
3
116
e
.
.
firs t numerato r
s ec o n d
5 40 thir d
x 9
x l 5 = 112 5 fo ur th
c o mm o n d en o mina to r
><
5 40
l l ;
i
d
are
the
frac
require
a
nd
2 0 23
20 25
1x 5 >< 15 x 9
67 5
3 x 3 x 15 x 9= l 2 15
:
.
A
nd
4x 3
5
5
5
x
x3 x
:
3 x 5 x 15
Therefo re
6 7 5
20 25
,
,
.
C as e
2
ow
5
.
20
.
l
N
GA R F RA C T I O NS
VUL
2
3
5
1
5
x 5 12 0 c o mm o n d eno minato r
Then 120 c om d en om inato r
4x 3
,
x2
:
.
,
40
f 3
I
D ivi s o r s {
1
3
L
6X5=
20
Hence
’
W
s—
o
o’
m
>
| N ew
3 0J
‘
o
_4
__
45
—
12 0
’
an d
12 0
P ag
.
9
15
5
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e
3
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10 0
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,
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.
s ta
l rs
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3
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ll
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44
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A n d 40
.
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73
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m ill s
3
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,
s
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.
( ft
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,
.
6 in
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(
17 0
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.
X 16
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quare s
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11
52
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L as tl y
.
.
45
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e
s
6
3
.
L as tl y
8q u a‘es
ll
mi
c ts .
.
x4
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e et
10 0
,
8
.
n
s q aa
re s
m
f
.
ms
.
= 1O 10 6 2 5
10
Then
q
2 10
.
,
10 0
9 1 ft 9
_
.
'
t
.
10 0
(
quares
3
s qw
f eet
e
f
ee t
BR
12 0
13
(
(
7
3
(
IC K L A " ER S
ft
n
x
i
8
83 ft
) 0
6
f
t
ft 8i n )
F
a
x
7
n
X
ft 6 i )
.
A N D MA S O N
2
.
.
.
6
W
X7 X , 5
.
.
.
.
.
.
s
d
ya
r
quare
s
T h en
0
s x8
d
y
.
,
c ts
d ol l s 6 0
.
.
S
R
E
"
A
L
K
I
C
R
B
P ag
e
f
2
ind
d oo r
3
t in
57 3 )
(
.
in
.
.
f
e et
in
f
.
17 3
’
W
-
c ts
.
O R K
.
7
1—1
7
2 ,
3
m ill s
.
k
i
c
r
b
a
h lf
.
e et
x5=
5
7
3
4
7
2
0
,
2
e ss
t
ic
k
n
h
d
s tan d ar
i
et n
2
.
3
.
f
eet in .
(
6 ) x 12
2
2
1
x
i
5
53 , x
c
r od s
13 10 , 7 5 f e e t
12 2
PA V E R s
Pa
g
f
in
e et
.
9
7
(
(
17 7
e
.
.
xv
s qua
re
5_
_
i
y ard s
10 3
10 8
9
sq
.
d
s
y
.
x 12 c ts = 3 8
7 d
.
6 in + 2 4ft 9 in
.
WO R K
l0
5
3g
,
ft
.
feet
——
3 0
.
in
6 ) x 27 5
105
Then
f e et
.
’
e et
in
5
f
.
x 2 = 110
.
.
e et
r
o un d
)
s quare feet in the wal l s
in the c ei l ing
.
.
f
m
%
41 0 9
.
f t = 1105
.
x 2 43
.
quare yar d s
cents
s
2
O
r,
4 d ol l ar s 13
P A V E R S
’
e
a
(
3 5
2
.
m ill s
3
W OR K
.
.
.
)
3 5
l
s
qu a e y a d s
r
r
.
Then
3
3 m
O
ents
17 8
P ag
O
c
s x 28
d
sq
y
%
3 23
,
(
ft
27
.
.
10 in
.
.
(
— m ill s
c ts 8
3
d oll s 06
.
.
.
t 9
4
f
1
X
)
s
.
’
s
d
quare
yar
fi
—
s
d
Then
— x3
fi
-
,
4
.
63
ft X
.
8c ts
(
f
(
.
.
9
6 3 f t X 3 9 t 9 n"
'
9
=7
l 73 3 ,3 9
:
feet—5 feet
ft 3 in )
45
5
.
3
x5é
inches = 3 9 feet
feet
s q.
) = 7 x 3 92;
S
h en
,
3 6
%s q uare
a
rd s x 3 6
y
X3 0
S
um
ms
9
.
in c hes
s br o ad
d
y
.
q yds
.
s
c ts = 13
=8
3
.
.
3 3
c ts .
d ol l
T
c ts
.
.
c ts
s to
pe bbl e s
.
.
28
47 %
1
‘
$ 96 7 0 5
ne s
.
f
.
P A IN T E R S
P A IN T
E R S
Pa
g
2
7 4 ft 10
(
ft
i
6
m
7
(
6
f
8
i
(
22 ft 8in
.
t
(
.
.
.
4 in
X 3
(
ft
X l
.
(
ft
6
f
86 6 9
281
l 1 f t 7 in
3 ft 9 in
X
.
9
.
.
)
.
.
in
.
.
n.
.
W OR K
’
17 9
e
12 3
.
t
.
.
WO R K
’
=
x5
.
10
6 d oo r
4 W ind ows
8b reak s
111 1
13 2 2
2
in ) x 5 feet
113 8 3
3 3 9
:
4
0
chi m ney
f
’
1104 e
A
nd
Then
3
5
—
s
3 24
%
12 2
,
(
s x8
d
2
y
.
cts = 10
"
.
’
.
.
ft
3 ft 2
.
Then
(
.
.
,
=
.
S
65 7
9
um
%
s
9
ms
.
%feet fi re p l ace
55
quare yar d s
18
0
e
%
*
9
-
.
II
(a
24
(
c ts
feet W in d o w s
P ag
4
17
.
—5 5 3
7 13
.
feet r o un d
s quare feet
x 4:
in ) x 6 ft
d ol l s 43
.
.
4
.
1104g
g
:
quare yard s
2 0 f t + 14ft
6 9 ft X l O ft 4
.
(
s
-
4:
eet
(
ft
3
7 ft 9
.
.
f t = 103 9
feet
24
54
6
3 ft 6
1 ft 3
(
ft
.
6
l u.
.
60
.
5
6 0 d oor
3 0 W shut
0 0 b reak s
'
.
24
1
.
)
fe e b
x5
117 7 10
27
6
4
115 0
9) 115 0
g
12 7 23
T h en ,
c ts
T he
s
.
= 7 d oll s 6 6
.
o
am e as qu e s ti
n
c ts .
4, p l as te r
e
m
s
%
9
rs
’
w
.
ok
r
.
.
.
I 24
ME A
S U RE ME N T
O F
G L A Z I E R S
G
R OU N D
.
W OR K
’
.
P ag e 181
.
2
3
(
6
ft
(
6 5
3
.
Th e n
Th e n
6
.
,
.
.
.
(
.
‘
5 8d
:
.
+ 6 ft
.
6 1 c ts
.
10 d
.
6 8s q
.
.
s ec
.
.
X
ft 9
7 f t 10 in
3
ft 6
)
ct s
.
.
\
.
2
f t x 10
in
9
quare feet
c ts
‘
6
.
s
16 f t 9 m
s q.
,
.
.
)
quare feet
4in + 4ft 3
.
f t x 143
s q.
(
ft
.
,
14 ft
s
.
12 f t 6 in
.
.
ft 5
3
1 1 ih + 5
X , 5 x7=
(
4
.
.
1
5
ft
7
.
81n + 5 ft
.
.
.
f e et
.
,
c ts
ft
ms
.
10 in
.
6
.
.
sec
.
a 11 in ) x 9
4
.
r
s q ua
83 7 5 =
46
.
ms
.
.
.
3
:
3
x
% %
x 9=
19
119
—x
-
fi
,
6 99
%s quare
g
5 5
.
T
o
fi
nd
f e e tx 14 c en t s = 97 d ol l ars
the
35
.
f
Th e n
,
7 0 X 70
:
.
T
o
fi
nd
thecon ten t of
cal l ed
30 A
p er
ch
es
2 R
.
A
an
obl on g
a p ar all el og ram
P ag
e
12 0
f
o
.
ou n d
r
g
.
.
.
3
18
4perch e s
perc h e s x 8
16 0 p erch e s
.
R
.
20 P
.
2 3 23
.
2d
2
.
p erche s
490 0 s quare feet =
s q.
3
87 c ents
a s qu ar e p iece
con ten t o
x2 = 70
feet
%
G R O U N D
F
O
P age 182
2
6 99%s quare
:
‘
M E A S U R E M E N T
l st
3
4
2
Th e n
x 9=
d
i
ece
o
r
oun
p
f g
.
,
12 6
20
Then
3 3
3
x3 3
§ x 40
2 94
,
(
q
17 x 17 x 29
3 59
,
5
hen
.
18
%
53
3
81 =2 3
dia m eter
55
%
ons
a
l
l
g
=
18
—
18
)
g(
22
—
6
3
0
X
2
20 x
3
,
= 17 m ean
}
.
T
147
— 1
4
Then
T 37 3
-
3 59
3
62 X 62 X 3 1
3
X 3 5 ?)
dia
m
eter
ean
m
gall ons
P
R O MISC U O U S
P R O M IS C
U O
U S
3
.
146 8
I
.
5
of
3
0
1) = 146 8
+5 8
cents
of 100 ct s
10 0
%
4
46
Sum
50
%
22
.
.
cen ts
-
.
cents
the num b er requ ire d
.
6
7
.
14 7 6 =
—
146 7 6
4
1
6
7
6
g
4
5
.
1
.
.
IO N S
19"
e
2
127
.
U E S T
Q
P ag
Q u es t
U E ST I O NS
Q
6
95 2
2
Supp o s e o ne si d e of the s quare to consist o f 100 me n
= 1023 1 the W ho l e num b er of
Then
me n
2
A nd
10 1
the W hol e of the m en
,
74
the
A gain s upp os e
Then
,
,
A
nd
2
12 1
erro r too l ittl e
side O f the s quare to ha ve 120 m en
the W h o l e nu mb er O f men
the W ho l e nu mb er O f m en
3 4
E
Th e n
al ike
rror s
(
.
120 x 7 4
)
,
—
.
10
.
1
—=
}
,
,
as
i
12 6 0 d
8years
d ol l s
.
10 00
40
W
1
x2
22
:
.
1
10 0 0 d oll ar s
2 5 00 d o ll ar s
15W
2
L as tl y ,
a
.
v
hen
:
o ne s ide
= 16 6 11
t
z
3 5 00—25 0 0
3
on
me n
or
l l 3 X l 47 —2 l
Th e n
l ittl e
— 13 7 m e n
-
34
of
9
to o
.
L as tl y ,
8
erro r
5
5 per
cent
.
100 d oll ars
1
year
5 040
do ll s
.
128
Q U E s ri oN s
P RO M ISC UOU S
P ag
'
193
e
.
2 10 X 3
11
.
N
12
§
As
O W
z
2 10
zz
1
.
$ 3 15
2
el d er b ro ther
’
s
r
o
p
.
s
=
l
ar
o
l
d
90
1
8
2
3 15 x 3 x
,
(
10 1
18
4
=
=
(
1
0
2
X
5
4
00
X
.
.
.
year s
13
.
= 2 5 0 d ol l ar s C
10 0 0
14
.
d ol l s
Th en as
,
And as
,
15
16
40 0
=
,
.
16 0
As
6
And a s
Then 2 7
,
z
;
1
7
as 1
.
d oll s
t in
7 00
.
.
’
the captain s s hare
=
13 d
6 §
ha
re
s
i
cts
a
sa
l
or
s
60
’
.
.
97 2 d ol l s
(
1 in t
.
.
,
d ol l s
z :
3 50
z
’
16 2 00 d ol l ar s
10 0
d ol l s
.
A pu
80 0 doll s B s cl o th is worth
2
2 17
10 0
hare
.
d ol l ar s
T
s s
d ol l s
5 00
of
—
5 44
0
2
27
.
Then
d ol l s
25 0
2
1
.
’
.
90 0 0
.
i
r
her
o
t
on
p
16 2 0 0
s
l
l
o
ar
d
0
0
0
2
7
1
estate
1 year
of the father s
d ol l s
.
’
d oll s
15 0 0 0 0 the
.
father s estate
’
.
3 s oo do ll s
.
13 0
P
R O M ISC UOU S
P ag
e
Q
UE ST I O N S
.
194
.
mil e s = 2 5 02 0 mil e s ro un d the earth
0 2 0=
2 5—
—
1
5
days
2
1
Then as 2 0 m i l es 1 d ay
2 5 02 0
20
day s
L astl y 12 5 1 d ay s —
5 3 6 5 5 d ay s = 3 year s
3 6 0 d e g x 6 95
3 1
.
.
,
,
P ag
e
3 2
195
.
10 0 x 3 é = 3 2 5
67 5
15 0 x 43
2 0 4X 5 § = 117 3
.
:
45 4
m onths
da
20
.
3 3
A
.
7
S
And as
,
da
20
.
.
1
:
l
12
.
A
4
:
d ay s
ca
n
fi
1
d o alo n e in
th e
a
rt
p
on e
d ay
an d o in a d ay
=
Then 4+fi
t
h
e p art of th e w o rk that they
fi
84
can do in a day w o rking together
8
days
L astl y a s éé} w ork 1 d ay
1 w o rk
,
,
-
,
,
3 4
40 7 10 0 42 2 6 5 6 25
—1
.
,
3 2 5 6 d o ll ar s
0 r thu
s by h
t e tabl e ,
,
p age
X 40 0
86
X 400
05
80 cen ts
3
mill s
.
:
40 0 = 5 10 5 l 2
x 40 0=
X 40 0
X 40 0
x 400
X
.
:
:
:
m o d ol l s m o
A s 12 5 z : 4
.
.
3 5
A
s
3
.
g
5
(
o
and
%
d ll
100
= 10 1
U LE
ll
s.
o
10 0
A
.
ou
:
z:
(
R
Th e n
s.
7 00
25
t
R
—l
7 00
L
3
0
H
or
u
e re u
.
— 68
85 2 ‘15
= th e an nu ity R
co
.
,
‘
am
nt O f £
mp
1, o r $1, in o n e ye ar as in
=
tim e f or w hic h th e an n it is in arr e a
r
A
th e am
y
e nd of t
ye ar s
th e rati
th e
do
.
1
.
R
the
d
.
ou
o
:
d in te r e s t; t:
nnu ity a
un t o f a
t
n
P R O MI S C UOUS
Q UE
P ag
195
e
24
21
36
:2
.
S T IO N S
.
and
3%
1
;
13 1
.
s
t
ar
p
1
7
%
T h en , as
'
A Pa
nd
X 140 X 3
140
3
—
(
1 2 7 1 10
.
l l xg
8
do
d 11
A nd a s
,
37
ll
.
c ts .
s.
45
%
B
paid
Supp o s e the num b er to b e 12
Then 12 x 4+ 16 = 6 4 his m o ney
his m o ney
A nd
12 x 6
.
.
,
,
4 err o r
too l ittl e
A gain Supp o s e 2 0 to b e the n um b er o f b eggars
Then 2 0 x 4+ 16 = 96 hi s m o ney
An d 20 x 6 — 12= 10 8his m oney
,
,
E
Then
rr or s
u
nl ike
in
.
12 X 12 + 2 0 X 4
,
l
.
z
f
i
é
-
1
e
h
t
T1
C
n
ca
do in a d ay
p ar t o f
th e
work they can
do
—
18 11
i _
=
T
18
198
ow f—l —
do in a d ay
L astl y as T —Z
sg
,
th e p art B an d
.
A n d , as 11
1
l
a d ay , all working
N
l 4 b e ggar s
12 + 4
8 As
working to ge ther
3
erro r to o much
12
a
h
e
t
8
g
-
p arto f the w o rk that A e ap
2 8days 3 h rs
8
m
in
%
3
3 9
Supp os e tha t one d oll ar w as th e s u m p ai d by 4mem b ers
o f C o ngre s s —o r 5 m erchants —o r 8l awyers —or 12 citi z ens
1
1
,
1
2
.
25
.
.
.
(
l ol l
T he n , A s 4
5
viz :
16
:
1
H
:
1
”
d o ll s w o ul d then
.
d ol l
.
20
3 0
.
5
z
6
$ 16
b e th e
who l e b i ll
.
P R O M I S C UO US
13 2
Q u e s tio n 3 9
co
ntin
d
ue
d oll s d ol l s
.
N ex t
A
,
16
s
P ag
U E S T IO N S
195
e
.
z
6
3
:
f
f
~
41
acre s
per
200
.
40
c iti z ens
P ag 196
s quare inche s
= 5 60
28
x 20
14X 10 X 2 = 28
0
28
0
l awyers
3 6
24
e
.
.
b y he
2
40
.
.
192
z
Q
.
%a c re s
1
:
3 20 0 0
p
erc h e s =
b
*
is
The ru l e
9
+5
the l o
tlf e s h o rt
s
ng s i d e
id e
the l onger side
4
th e
Then J
12
,
A
12
nd ,
:
s
id e
V
‘
2
of
:
— 1=
as 4142 13 6
s quare in perche s
ay ,
,
quare mea d o w tO b e 1;
its diagon al
4142 13 6 what the diag o nal is m o re
th e
s
the si d e O f the
s quare th e S i d e
an d w e
20
,
.
,
a= the d ifi e re nc e
b
T
:
the s
i de s
an d
,
conten ts in p erche s
—
lie n m—
l a) x x the c ontents O f th e ground in perche s
That
By c o mpl e ting the
ate b b y the que s ti o n
th e
,
(
:
.
:
.
2
s
of
quare
8 ge t x + ax + %
2
W
conse q uently
+ b,
’
: :
rP u t x :
th e
(
side
of
2
b
+
;
g
and
e vo l ution
by
2
cc :
J ;
+b+§
the s quare
(
,
an d a==2 0
.
x + 2 ax + a by tran s
Then E ucl i d 1 47 )2 x
23 + a
)
2
2
d
osition
we
get
n
by
c
o
m
l
eting
t
h
e
s
quare
73
a
x
p
p
2
2
2
—
v
2
2 a by e o l ution x
a ./ a he n ce x
a+ a./ 2
2 ax + a
: the si d e O f the s quare
N o w b y re s to ring th e val ue O f a we ge t a
perche s
to b e
in th e side the s ame as ab o ve whence the c ontents m ay b e
read i l y found
,
.
2=
2=
2
~3
.
,
'
:
:
:
.
.
.
13 4
P RO M I S C UOUS
P ag
e
Q
UE S T I O N S
197
.
.
F irs t, b ecau s e tha t r16 o f the W h ol e c o n ten ts w as d rawn
o ut ea c h time , c ons equen tl y 1 o f the par ts w a
s d ra w n
T 0
50
.
-
.
Hence
gal l o n s o f W ine
s ubtrac t
5 00
,
1
0
fi
50
:
45 0
45
:
40 5
1
T 5
40 ;
:
%
3 64
3 28
i
l o
%
3 2%
=
2 95
9
—
4
53 0 0
295 5 5 ? gal l s
thu
A nd ,
01
'
.
gal l s wine
wa ter
.
s,
Here the s everal p o r ti o n s o f W ine d rawn
50
remain s 45 0
of
were
,
5 0 X 45 0
remain s
5 00
5 0 X 45 0
2
—
5 00
2
45 0
remam 8
45 0
3
re "
3
5 0 X 45 0
5 00
45 0
m
is
4
re
4
m ain s
5 00
5 0 X 45 0
2
5 00
5 00
m
5 00
so
3
3
5 0 X 45 0
4
2
2
5 0 X 45 0
3
2
s oo
45 0
45 0 2
5 00
5 00
5 O X 45 O
5 00
5 O X 45 0
4
4
45 0
3
5 00
2
45 0
4
5 00
3
45 0
5
5 00
4
5
T h e re f o re
éz
g
2 95 5 73
4
9 =
—
C o n s equen tl y 5 00 2 95 fi 6
,
A
n al
c
y ti al l y
afte r d raw in g
Then
t
%
5 0= d , 5 = t, an d
Put
gall s
2 04
x
of
wa ter
= w hat w in e
re
m ain s
e x pre s s e s the quan ti ty o f wine l eft after the firs t
c
_d
c
d
W
th e
c
t the s
out a
e c o nd drawing
,
and
c—d
quan ti ty o f W ine d rawn
(
—d >2
c
c
0
r
q u an ti ty
.
*
tim e s
’
d raw in oo ,
.
ga
l
o n s o f wine
l
,
4
of
wine l eft after the s ec o n d d rawing
.
the
P RO M I S C UOUS
P ag
2 x2 X4
5 1
O
c t.
.
April l s t princi p al 186 4 d oll a rs
=
3
s
d
t
b
1
9
l
7
l
m
u
ay
4
d
o
l
h
8
1
6
t
15
y
P ai d
18
17 ,
.
197
e
13 5
.
p ar t
4X 4x 8 12 8
.
52
16
S T I ONS
QUE
.
.
.
18J an l oth
18
.
x
8
7
X
12 6
6 7 2 08
l 42 5 49, 5
P ai d
M ay 16 th
P ai d
5
58
Aug l s t
I n tere s t
77
"
.
.
B al ance
I n te re s t
3 6 x 6 7} x 8
2
?
2 o47 s
l5
12 8
C on tin uatio n o f N
A l s o,
—d
ot
e on
13 4
.
.
2
c
.
p
—d ) >
c
d rawn
An
c
at th e thir d d raw
out
(
d
c
-
d)
(
2
—
d) X d
c
2
,
c
c
l e ft after th e third d ra w
—l
t
—2
c :
H
t
in g
d
2
c
in g ;
an d
t— l
—
d
()
2
th e
t
C
the
of
q u anti ty of
w
me
w
i ne
unive rs al l y
xd
—l
d
3
—
d)
c
(
2
c
.
.
c ord s
3 —th e quan tity
c
d
c
3 $i
25 6
quan ti ty of w i ne d raw
n
C
o ut a
t the
t d raw i n g
.
t
A nd
_2
_1
t
t
c
c
re main i n g after th e t d raw i n g
c d a
n d t i n th i s fo rmu l a w e
.
,
,
5 00—5 0
,
,
5
45 0
y
the q u n ti t o f
_l
t
C
a
o
By
s
ub s ti tuting
th e
me
o
w
val u es
of
e
t
g
5
a
s
ll
g
s
.
of W i
s a
b ove
n e th e same a
.
»
13
6
m s c vou s
P no
s 'r ro rrs
s
Pag e 197
54
A
.
.
56
.
A
d lls
.
18
s
c ts
.
%
d ol l
1
o
z
97
1
d lls
.
.
d ay
.
3 56 3 4
3 65
s
c ts
55
o
d ay s
c ts
.
I nvers e 8men
cen ts
o
d lls
.
25 64
:
.
6
mi
s
ll
%
c ts
.
.
13 6 7 9 7 6
6 4 men inver s e
10 0 d oll ar s
d ay s
3 0
25
3 0 x ex
57
I nver s e
.
3 0 x 25
4
6
2_
M
3
d ay s
86 men invers e
2 b ri d ge
m o n ths
m en
3
1 b ri d ge
6 X 3 4X g
12
T 1
m
o n th s
%
4
=
43
8
pay s
5
A
d oll ars
4 6
7 z
5 d ol l ar s an d B 7 d o l l ar s
A
.
s
d o ll ars
A nd ,
d ol l s
d oll s
2 40
.
L as tl y , as
7
: z
o
A
.
'
109
s
(
ms
198
3
7
.
.
h our
.
A nd ,
Then
,
.
z
.
.
o
t
85 6 4 20
1 hr
:
c s
.
.
o
d lls
6
.
c ts
.
2 5 + tax
gal l s 2 q t 1
gal l 3 q t 1
remaini n in th e ve s s el at the e n d o f
.
.
.
pin ts
6 7 6 pin ts
is 6 7 6
.
45 h o u rs 4minu te s
s
61
.
O
ne
T hen
62.
acre
12 10
,
.
g
84
%gall o n s
as 15 pts
.
d lls
.
4 gall s 2 q t )
pints gaine d, o r
60
= 15
t
p
one
c ts .
.
receive s
%
'
P ag
d lls
SB 5 3 3 3 %A
7 4 66 B
$ 112 0 0 C
5
.
e
59
C mu s t p ay w hen
50
bu s hel s
3 O
12
fee t
:
73
%p erche s
q
.
f
ee
t
8
13
P RO MI S CUO U S
S T I O NS
Q UE
.
198
P ag
e
.
45 %
f t —9 %
in = 5 46 in —9%
in = 5 3 6 %in ches
?
A n d 9%
in
6%
in = 2 z inche s gaine d in a d ay
10 7 3
I
N ow , 5 3 6 %
195 T T d ay s
xfi
5
Thatis , itw il l b e wi thin 9%inche s o f the top in 195 gT d ay s
C o n s equen tl y it will ge t to th e top in
d ay s
67
.
.
.
—
.
.
.
.
:
-
-
.
.
18
0
68
45
72
-
4o
337
.
nd
A
3 60
g ages 5
1
3
—
u
Then
3 60
,
2
A
s
-
ggfi
46
’
69
l
Supp os e
.
fi
x 3 60
23
2 00 d o l l s
94
= 7 2 0 tree s
A gain Su ppos e
.
,
v
10 6
of
10 6
o f 10 6
:
206
5 3 l en t
of
2 1 cl o the s
94 d e b ts
of
%
:
s um
$ 16 8
g
o f 20 0
18
0
-
9
T
3
‘
rror s
Th e n
70
a
l ike
.
%
3 l
8 s um go ne
of
error
e
%
3 1 to o mu ch
‘
4
3 00 x 11a
5
63 60
l 1g
.
20
P ag
—3
d oll s
e
199
.
4
-
.
T hen
,
as
l
T f
12
84 z
:
1
8
4X12=
10 08
Q
Here 3 —2 4 f t
3
12 fee t th e s e mi d i am e ter o f
thecircl e tha t th e o u ter w heel m ake s
C o n s equen tl y 12 X 2
2 4 fee tth e d iam ete r o f th e o u te r c ircl e
—4x 2 16 f ee t th e d iame ter of th e Inner c ircl e
A n d 24
=
l
2
4
e
l
t
1
4
4
h
u
t
e
h
fee
r
w
ee
3
1
6
5
3
8
t
b
t
o
9
7
x
s y
y
f ee t b y th e i nner w h eel
71
.
.
:
.
'
%
:
.
20 0 x 3 15
,
l en t
41 cl o th e s
94 d eb ts
3 23
rr
o r to o m uch
%
11
E
s
20 6
go ne
:
5
3 00 d oll ars
94
‘
.
‘
=
,
n
a
‘
,
,
,
P
RO MI S C U O U S
Q UE
P ag
Here
th e s i d e s
72
of
%
5
1
.
=
125
6
.
equal s i d e s
—
3 7 5 4 25 0
—
A nd
Th en
th e
on e o f
g
,
2
2
—5 =
the
15 6 2 5 X l , 7 3 20 5 0
3 6
3 6
2
6
.
«a
—
xi
-
1.
s um
hal f the
equal rem
12 5
«
15 6 2 5
s
,
4
6
3
l
o ne o f
12 5
2
13 9
.
199
e
*
S T I O NS
:
8
quare fee t
.
is o n e o f th e equal s i d e s
3=
U L E 11 thu s ; 12 5 —
2—
n d c on
n d b ecau s e the triangl e is equi l a teral a
o f the triangl e a
s equen tl y equiangul ar the incl u d e d angl e o f any tw o s i d e s 1s
wh o s e natural s ine is 86 6 02 5 4
8
6 6
2 5 4=
1
2
s quare fee t
Hence
1
8
5
1
5
5
9
6
7
7
x 39 x
3
O
by R
r,
,
,
,
,
,
,
,
H
e re
d atais giv e n
T he rul e f o r
s u ra
tio n , as f ll
s ol
o o
w
to
M ul tip l y the h alf
an d th e
R
the
the
s
a
u
q
re r
an d all
of
a s im il ar kin d , is b y me n
c
v
th e s u m o f thr e e s id e s su b tr a t e a h s id e s e e r a
lly
oo t
of
su
th e
m an d
the thr e e
o uc t
pr d
A n y tw o s id e s
t again b y h a
l f the
pr d
are a of th e trian gl e
U LE
o
s ti n ,
.
c
2
.
s id e s
s:
Fr om h al f
'
fi n d th e
ving this qu e
1
.
,
11.
o uc
il l b e
w
c
m ain d e r s o n tin u all y
th e ar e a r e q ire d
re
u
og
t
e the r ,
.
a trian gl e b e in g m ul tip l ie d tdge the r,
na
l S in e o f th e ir in l u d e d an gl e , w il l gi
tu r a
of
c
v
.
T hat is , A C X C B X n atural
.
s in e o
f
th e
an gl e C ==
tw ic e are a
.
e
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M E R CQ A N TFUL E
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