-2:36

ARITHMETIC.

Examples.

1. What is the 4 of 2/.

2 /.

6 s.4

2. Required the value of -j of 2 l.202

5)9 4

3) 40(13s. 4rf. Ans.

A ns. 1/ l0.s. Qd. 2-|y.

1

12

13(4ff.

(37.) Rule VIII. To reduce Fractions to other equivalent Onesof a diferent Integer; a certain number of the Less Integerbeing contained in one of the greater.

Consider how many of the less integer make one of thegreater; and by that number multiply the numerator if tire reduc-tion be to a less integer, or the denominator, if to a greater.

of tire subtrahend fronr the denominator, and to the differenceadd the numerator of the minuend ; and carry one to the integerin the subtrahend.

Examples.

1. What is the difference between -|t and 4f ?

13 11 253-242 13

-=-=-Ans,

22 17 17x22 374

2. What is the difference between 5 J and 2-/ r ?

5| — 2 jZj. = 5£$ - 2 Lf = 2f|.

Multiplication of Vulgar Fractions.

(41.) Rule. Reduce mixed numbers, if there be any, to frac-tions ; then multiply all the numerators together for the numera-tor, and multiply all the denominators together for the denomina-tor of the product required. A fraction is best multiplied by aninteger, by dividing the denominator by it if possible, but if thatcannot be done, multiply the numerator by it.

Reduce —/. to the fraction of a shilling.

9

2 2x20 40

— I- =-= s. l

9 9 9

40

Reduce — s. to the fraction of \l.

9

40 40 2

—s. = -

9 9X20 9

If a compound whole number he proposed, reduce it to thelowest denomination mentioned, and proceed as before.

(38.) Rule IX. To reduce Fractions to equivalent ones of a dif-ferent Integer, zvhen a certain Number of the less is not ex-actly contained in the greater.

1. Bv the last, reduce the given fraction to an equivalent one ofsuch an integer, of which a certain number are contained in theinteger to which the fraction must be brought, or which shall con-tain a certain number of this. 2. By the last also, reduce thisfraction to an equivalent one of the. integer required.

Example.

2

Reduce — of 1 /. to the fraction of a guinea.

7

2 2x20 2X20 40

— 1. = - s. =- gui. - guinea.

7 7 7X21 147

Addition of Vulgar Fractions.

(39.) Rule. Reduce compound fractions to simple ones, andall to the same integer and denominator if they be different; thenadd all the numerators together, and set the sum over the com-mon denominator, for the sum of the fractions required. Whenseveral fractions are to be collected, it is commonly best first toadd those two together which most easily reduce to a common de-nominator, then their sum and a third, and so on.

Example. What is the sum of I\, and ^ of | ?

* ■+ n -H of i = i- + 7* + i = | + 7* +1 = 7-7- = 8| thesum.

Subtraction of Vulgar Fractions.

f40.) Rule. The same preparations being made here as in ad-dition, take the difference of the numerators and set it over thecommon denominator, for the difference of the fraction required.In subtracting mixed numbers, when the fraction in the subtra-hend is greater than that in the minuend, subtract the numerator

Example.

What is the product of 3), 5, and 4 of 4 ?

2 3 3 2x 13x3x3x3 39

— x 3| x 5 x - of—- =-■= — = 4f Ans.

3 4 5 3x 4x 4x 5 8

Division of Vulgar Fractions.

(42.) Rule. Having prepared the terms as in multiplication ;take the quotient of the numerators and of the denominators if theywill exactly divide, for the numerator and denominator of thefraction required ; but if that cannot be done, multiply the di-vidend by the reciprocal of the divisor, for the quotient required.By the reciprocal of a fraction, is meant the fraction got by invert-ing its terms: so the reciprocal of -} is j, and of 5 or 4. is 4. Afraction is divided by an integer by dividing the numerator by it,if possible ; but if it will not exactly divide, then multiply thedenominator by it.

Examples.

25 3

1. What is the quotient of — by — ?

9 3

25 5 23—1^6 5

-1-=-= — = If. Ans.

9 3 9—3 3

5 2

2. What is the quotient of — by — ?

9 15

5 S 5 15 5x 15 25

■-;-= — X — =-= — = 4i. Ans,

9 15 9 2 9X2 6

Decimal Fractions.

(43.) A decimal is a fraction whose denominator is 1 with acypher or number of cyphers annexed ; being always one or more10th parts, lOOdtli parts, lOOOdth parts, 10,000dth parts, or thelike.

Decimals are written down without their denominators, thenumerators being so distinguished as to evince what the denomi-nators are; which is done by separating by a ]>oint so many ofthe right hand figures from the rest as there are cyphers in thedenominator; the figures on the left side of the point being in-tegers, and those on the right decimals. 'I’hus 44 is written 1.3,and named 1 and 3-ft ntns; and is written -25, and named25-hundredths or hundredth parts. But if there be not a sufficientnumber of figures ill the numerator, cyphers are prefixed to sup-ply the defect. So Tea is written -01, that is 1-hundredth; andthus -015, that is 15-thoman itiis.

The use of cyphers in doc. in.d , as well as integers, is to bringthe significant figures t„ their proper places, on which their valuedepends. As cyphers, when pi. ced on the lett hand of an inte-ger, have no signiii ation, but when placed on the right hand, in-crease the value ten times cacti; so cyphers, when placed on the

right

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