arithmetic?. 
fraBion . Multiply the whole number by the denominator 
of the fraction, and to the prOdijcft add the numerator, and 
the fum is a new numerator, under which'write the given 
denominator. 
Ex. Reduce 4, 4 |, j|, and 7f, each to an improper 
fra&ion. 
2X24-1' 5 4X3+ 2 > 4 . 5 X 4+3 2 3 . 
Thus 
and 
7 X 443 
34 
4 ' 
Problem V. To reduce an improperfraBion to a whole 
or mixed number. Divide the numerator by the denomina¬ 
tor, the quotient is the whole or mixed number. 
Ex. Reduce \ 4 , ■§-, y, and y, each to a whole ot» 
-mixed number. 
Thus 24-7-6—4; 54 - 2 = 4 ; 144-3=4!r 2 5 ~r 4 = 5 l> 
and 394-5=74 j the numbers required. 
Problem VI. To reduce a fraBion to its lowcjl terms. 
Divide both terms of the fraction by the greateft common 
meafure or divifor, and the quotients will be the terms of 
tile fraction required. The greateft common meafure is 
found thus; Divide the greater term by the lefs, and 
this divifor by the remainder; and fo on, dividing always 
the laft divifor by the laft remainder, till nothing remains ; 
then is the laft divifor the greateft common meafure requi¬ 
red. But, if the laft divifor be 1, the fraction is already 
in its loweft terms. 
Ex. Reduce and iff to their loweft terms. 
168)240(1 144)560(3 
168 43 2 
72)i6S(2 
144 
Common mea. 24)72(3 
72 
128)144(1 
128 
Common meafure 
16)128(8 
128 
and 
14 4 .. 
5 60 
16—+ in their,loweft 
—To* 
terms. 
Any fraction may be reduced by a continual divifion of 
both terms by 2, if they end with an even number or a 
cypher. 
Ex. Reduce fff to its loweft terms. 
Thus fH 4 - 2 =f! 4 - 2 =H 4 - 2 =H 4 - 2 =fi- in itS loweft 
terms. 
If the fum of the digits of any number can be divided 
by 3, the number itfelf may alfo be divided by 3. 
Ex. Reduce to its loweft terms. 
Here 7+442 = 12, the fum of the digits in the numera¬ 
tor. And 9+5+4=18, the fum of the digits in the deno¬ 
minator ; each being divifible by 3 ; therefore the numbers 
741 and 954are alfo divifible by 3. 
Then -^4-3—in its loweft terms. 
When both terms end with 
5, or one with 5 and the 
other with a cypher, divide both by 5. 
Ex. Reduce ||-, f-°, and to their loweft terms. 
Thus |-| 4 ~ 5 =tV* tt=5=t 5=3=!> an ^ 7 o~5=i’ 4> 
in the loweft terms. 
If both terms end with cyphers, caft off as many as are 
common to both ; then reduce the fignificant figures to 
lower terms, by fome of the foregoing rules. 
IV v Rprtnrp 3500 1 5ooo 
rsx. tceciuce t^ooo) ,35000) 
Thus 
183 
by 
Ex. 2. Alfo ?X 5 X 2 X 3 X i XiX 7 == 4 Xi = 
5X7X2X5X3X9X6 5X3 
omitting the common terms and dividing by 2. 
Problem" VII. To reduce a complexfraBion to a Jimple 
one. When the the fractional part is annexed to the nume¬ 
rator, multiply the numerator of the fra 61 ion by the deno¬ 
minator of the fractional part, and to that produff add the 
numerator of the fractional part for a numerator; then 
multiply the denominator of the fraction by the denomi¬ 
nator of the fractional part of the numerator for a deno¬ 
minator. 
-1' p 4 ., 
—, , and — r , each to a fimple fraftion. 
712 6 
Ex. Reduce 
Tims 
and 2S 
6 X 4 
and to their low- 
7X4. 2 S 12X5 6a 
—3 -; the fimple fractions required. 
When the fractional part is annexed to th.c denominator : 
multiply the denominator of the fraction by the denomi¬ 
nator of the fractional part, and to that produft add the 
numerator of the fractional part for a new denominator. 
Then multiply the numerator of the fraftion by the deno¬ 
minator of the fractional part, for a new numerator. 
3 
3. 
Ex. Reduce -=, 
4 i 
3X2 6 . 2 
Thus -—=—r 3 =-» 
4X2+1 9 3 
15 
4 , and each to a fimple fraction, 
and -+*+- 
5 X 4+3 23 8X3 + 2 
2=2—; the fractions required. 
26 
Problem VIII. To reduce a compound fraBion to a fu¬ 
gle one. Multiply all the numerators together for a new 
numerator, and all the denominators together for a new 
denominator, of the fingle fraction. If any part of the 
compound fraction be a whole or mixed number, reduce 
it to a fraction by the foregoing rules. 
Ex. 1. Reduce f of ^ of f of f to a fingle fraction. 
Thus 
2 X 3 X 4 X 5 2 
3 X 4 X 5 X 7 
by Problem VI. 
7 
Ex. 2. Reduce f, of f, of •£, of -4-, of 6£, of 7 
7 i 
to a fingle fraction. 
,2 
A of __ : 
0,01 6 
and _1—.UL—2. 
ansa . - 30 - 5 , 
Firft-i-=^=f; 6 |=V; 7 |=Vi and ^ 
li . 6 
now the compound fractions are reduced to of 4 , of f, 
off, of y, of V, Of i = , X 5 XSX»X S 7 X 67 X._ 
2 X 27 X 67 1 X 9 X 67 
2 X 4 X 6 X 3 X 4 X 9 X 5 
67 —the anfwer. 
4 X 6 X 4 X 9 2 X 2 X 4 X 9 2 X 2 X 4 
Problem IX. To find the value of a fraBion in known 
parts of the integer. Multiply the numerator by the integer, 
and divide by the denominator; multiply the remainder 
by the next inferior denomination, and divide by the de¬ 
nominator ; and fo on till you come at the loweft denomi¬ 
nation, as in Compound Divifion. 
Examples. 
2. What is the value of 4 of 
in their low¬ 
eft terms. 
eft terms. 
Wi 
5 —1 50 . 3—To'J 
When the fame numbers are both in the numerator and 
denominator, they may be omitted or caft out of both. 
Ex. 1. Reduce to its loweft terms. 
4 X 8 X 3 X 5 X 6 
rrM. 3 X5X 1 X4X3 3 21 , 
Thus--222-—-2=-=—; by omitting 
4X8X3X5X6 8 X 6 8 X 2 16 ’ 3 B 
the common terms 3, 5, 4, and then dividing by 3, 
. ■ t 
1. What is the value of 
of a pound fterling? 
7 
20 
ll)l 40 (l 2 S. 
8 
1 2 
11)96(8d. 
4 
i*) 3 i( 2 -£H rs - 
10 
Anfwer, 12s, 8 d. jAf-qcs. 
a guinea ? 
21 
_4 
j)84(i'6s. 
4 
I 2 
5 )48( 9 d. 
3 
_4 ’ 
5)i2(2fq r s, 
2 
Anfwer, 16s. 9 d. 2|-qrs. 
3. What 
