J 53 ' 
A R I T 
H M E T I C.‘ 
Ex. i . W 1 at known parts 
of coin are equal 100-9406251. 
0-9406251. 
20 
j 8 -812500s. 
1 2 
9-75000od. 
_ 4 
3-000000 qu. 
Ex. 2. Wha f is the value 
of o- 5S04687 5 of a lb. troy ? 
•580468751b. 
12 
6-96-562500 ox. 
20 
19-3125000 dwts. 
24 
7 -5000000 grs. 
To reduce a finite decimal to its lead equivalent vulgar 
'fraclion. Place the decimal as a numerator Over 1, and as 
many cyphers as there.are figures.in the decimal, for a de¬ 
nominator; then reduce it to the lowed terms. 
Ex. Required the lead vulgar fractions equivalent to 
”3 o 6 5» ,2 5> ’5, '75> ' 12 5> and ’0625. 
Thuq 1 9 £ 5 - -< 3 _25.— 1 _5_—x —1 
J IllO, 100 00-2000* 10©-4* lO-2* 1 O -4* lOOO-8* 
T©3uo— tg> ll;e vulgar fractions required. 
CIRCULATING or REPEATING DECIMALS. 
When the denominator of a vulgar fraction, in its lowed 
terms, is not compounded of 2, or 5, or both, the decimal 
produced from fuch a vulgar fraction will be infinite; it 
is called a repetend, or circulating decimal, from a conti¬ 
nual repetition of the fame figures. 
A (ingle repetend is a decimal where only one figure re¬ 
peats, as -3333, See. -4444, &c. and thefe may be exprell'- 
e.d by putting a point over the fird figure. Thus *3333, 
&c. may be denoted by -3, and -4444, See. by -4. 
A compound repetend has the fame figures circulating 
alternately, as-507650765076, &c. -123123123, &c. and 
thefe may be didinglviihed by putting a point over the fird 
and lad repeating figure : thus, -50765076, &c. is written 
-5076, and • 123123, &c. -123. 
Pure repetends are fuch as have no figures in them but 
what belong to the repetend, that is, when there are no 
figures,or none but cyphers,betwixt the point and repetend ; 
as -3434, &c. -0044, &c. -046046, &c. -00303. 
Mixed repetends are fuch as have cyphers, or fignificant 
figures, between the repetend and decimal point, or fuch as 
have whole numbers to the left hand of the decimal point, 
as -344, &c. -304848, See. -0467272, &c. 4-375, &c. 
Similar circulating decimals are fuch as confid of the 
fame number of figures, and begin at the fame place, either 
before or after the decimal point; thus -33, &c. -77, &c. 
are fimilarcirculates, and thefe, 2-345656,&c. -004242,&c. 
Diflimilar repetends confid of an unequal number of 
figures, and begin at different places from the decimal point, 
as -354, &c. 2-7534, &c. 9-253. &c - °‘ 475 2 . & c - 
•Similar and conterminous circulates are fuch as begin and 
end at the fame place; 3558-2753, 81-9104, and -0613, See. 
A feries of nines infinitely continued is equal to 1 on the 
left-hand place ; thus -9999, &c. ad infinitum 2=1, -09999, 
See. =:-i, and 7-9999, &c. 22:8, if infinitely continued. 
Any number may be multiplied by 9, 99, 999, &c. by 
annexing as many cyphers to the right hand of it, and then 
fubtrafting it from itfelf thus increafed. Thus 236X9= 
2360 — 236—2124, 236 x 9922:23600—236—23364, and 
236X999— 236000—23622:23364, the products. 
In any circulating number, the whole circulating or re¬ 
peating part, running on for ever, is equal to a vulgar 
fraftion whofe numerator is the number repeating (or the 
repetend), and denominator as many nines as there are fi¬ 
gures in the repetend. As in 24-35076507650765076, &c. 
ad infinitum ; 5076507650765076, &c. =ff in 
file lowed terms. 
The circulation may be fuppofed to begin at any figure 
of the repetend'; and therefore 24-35076350763507, &c. 
— 24"3^J^. =24-35^^*-, =24- 3 5O | Jj | g — 24-3507^-^222 
24-35076^^!, Sec. Hence, if the repetend be divided by 
as many nines as it lias places, the quotient will be equal 
to the whole circulating part, or the figures of the repe- 
tepd repeated tor ever. Thus |°% j;— the whole circula¬ 
ting part. And, if the whole circulating part be multiplied 
by a number confiding of as many nines as there are pla¬ 
ces in the repetend conlidered as a decimal, the prodiuft 
will be the repetend. For ffff-'x99992=5076, and 5076 
X-99999=5076, the fird repetend. 
If a circulating decimal has a repetend of any number 
of figures, it may be confidered as having a repetend of 
twice or thrice the number of figures,"or anv multiple 
thereof. Thus in the number 4-137,37,37, having the 
repetend 37 of two places; it may be confidered as having 
the repetend 3737, or 373737 ; of four or fix places, &c. 
Hence any number of dilTimilar repetends may be made fi¬ 
milar and conterminous, by changing them into other re- 
petends which (halt each confid of as many figures as the 
lead common multiple of the feveral numbers of places, 
found in all the repetends. 
Di/Timilar. Made fimilar and conterminous. 
, 68-345 = 68-345454545 
9-621 2= 9-621621621 
Examples « 
0-242424242 
1 -500000000- 
0-266666666 
73-083 := 73-0.83333333 
_ 8-09 = 8-090909090 
REDUCTION of CIRCULA TING DEC IMA LS. 
To reduce a Tingle or compound repetend to a vulgar 
fraction, Place the figures of the repetend for a numera¬ 
tor, and as many nines for a denominator; then reduce it. 
to the lowed terms. If one or more of the left-hand pla¬ 
ces, in the given decimal, be cyphers, annex as many cy¬ 
phers to the right-hand of the nines in the denominator. 
If there are integral figures in the circulate, as many cy¬ 
phers mud be annexed to the numerator as the highed place 
of the repetend is didant from the decimal point. 
Ex. 1. Required the lead vulgar fractions equal to -3, 
•05, -123, 2-63, -0594405, -769230, and 1-62. 
Thus -3—2-63—2-|| = 2 -jJj. 
;°5 =^o=tV *059+405=^^^2=2% 
* j -j n ■ — 1 2 3 « ■ . 4 1 * "7 A n 7 7 O Z ft ^ ^ ^ 1 O 
1 -9 9 9-3 3 3 / U 7 X, J U -99 9 £ 9 9- 
Ex. 2. What are the lead vulgar fractions equivalent to 
•6, -162, -945, and-09? Anfwer, §, %, and -5%. 
To reduce a mixed repetend to a vulgar fraction. To 
as many nines as there are figures in the repetend annex 
as many cyphers as there are finite places, for the denomi¬ 
nator of the vulgar fraftion. Multiply the nines in the 
faid denominator by the finite part of the decimal, and tq 
the product add the repeating part for the numerator. Or 
find the vulgar fraction as before anfwering to the repe¬ 
tend, then join it to the finite part, and reduce them to a 
common denominator. 
Ex. Required the lead vulgar fractions equivalent to 
’53> '583. '138, and -5925. 
9 X 5+3 
48 8 
90 
9 ° 15 
5 SX 9+3 
525 
7 . 
’ 900 
900 
12 
, 13X9+8. 
__ 125 
5 _ 
" 59 2 5 
900 900 36 
5+999+925 5920 16 
999 ° 
9990 27 
y Anfvvers. 
Having a vulgar fraction given in the lowed terms, to 
find whether the decimal fraction equal thereto be finite or 
infinite; and, if infinite, whether it will produce a pure or 
mixed repetend ; and how many places the repetend will 
confid of ? Divide the denominator by 2, or 5, as often as 
podible; then divide 9999, See. by the former refult till 
nothing remains, and the number of nines made life cf 
will be equal to the number of places in the repetend ; 
which 
