232

ARITHMETIC.

initiations except the least, find how many of the next lower de-nomination it is equal to, and add to it the small number (ifany), which was in this denomination before; then divide thesum.

Examples.

L. s. d. yds. f. in-s') 1 8 0 12) 150 1 7

9 6 12 1 1-h

See Practice. Rule 2.

Proportion, or the Rule of Three.

(23.) The Rule of Three is that by which a number is foundhaving the same proportion to a given number, which is betweentwo other given numbers. It is called the Rule of Three, be-cause in each of its questions there are given three numbers atleast. And because of its excellent and extensive use, it is oftennamed The Golden Rule.

Rule, for stating, or setting down the Three given

IS UMBERS.

1. Write down the number which is of the same kind w ith theanswer op number required.

2. Consider w hether the answer ought to greater or less thanthis number; then write respectively the greater or less of the.two remaining numbers on the right of it tor the third, and theother on the left for the first number or term.

3. Multiply the 2d and 3d terms together, divide the productby the first, and the quotient will be answer.

Observations.

1. When you can conveniently multiply and divide, as in com-pound multiplication and division, it is best to do so.

2. If not, reduce the compound terms to the lowest namementioned in them, and the first and third to the same name itthey be not so already ; then the answer will be of the same namewith the second term.

3. When there happens to be a remainder after division, reduceit to the name next below the last quotient, and divide by thesame divisor, so shall the quotient be so many of the said nextname; do this as long as there is any remainder, till you havereduced it to the least name, and all the quotients together will bethe answer.

4. If the 1st term, and either the'2d or 3d can be divided byany number, without remainder, let them be divided, and thequotients used instead of them.

(24.) There are four other methods of operation besides theabove general one, any of wliich, when possible, performs thework much sooner, viz.

1. Divide the 2d.term by the 1st, multiply the quotient by the3d, and the product will be the answer.

2. Divide the 3d term by the 1st, multiply the quotient bythe 2d, and the product will he the answer.

3. Divide the 1st term by the 2d, divide the 3d by the quo-tient, and the last quotient will be the answer.

4. Divide the I st term by the 3d, divide the 2d by the quotient,and theJast quotient will be the answer.

Example I.

To find the value of 14 oz. 8 dwt. of gold, at 31. J9s. 1 Id. anounce.

Explanation. Having stated the three terms by the generalrule as here annexed, the 2d term is reduced to pence, and the3d to dwts. these being their lowest denominations, as above di-rected. The 1st term is also reduced to dwls. that it may agreewith the 3d. The 2d term is then multiplied by the 3d, and theproduct divided by the 1st, according to the general rule, whenthe answer comes out 13,8u9 pence, and 12 remaining over;which remainder being reduced to farthings, and these divided byIbe same divisor 20, the quotient is 2 farthings, and 8 remaining.Lastly, the pence arc divided by 12 to reduce them to shillings,

and these again by 20 for pounds ; when the final sum comes out57/. 10 s. 9i/. 2ij. for the answer.

oz. 1. s. d. oz. diet.

1 : 3 19 11 : : 14 8 :

20 20 20

20 79 • 2S8

12

959

7672

7672

1918

2,0)27619,2

13809, pence, or 1

12)13S09</ 2±lqr.

2,0) 1150s. 9i/.

Answer, 57/. 10s. 9d. 2 jypExample II.

I low' many men must be employed to finish a piece of work in15 days, which 5 men can do in 24 days ?d m d15 : 5 : : 245

15)120(8 men. Answer.

120

Or, by the Third method :

5)15 : 5 : : 24

3)24

8 Answer.

Compound Proportion, or the Rule of Five.

(25.)This Rule is so called, because that in it there are five num-bers or terms given, to find a sixth. It is often named the DoubleRule of Three; because its questions are sometimes performed bytwo operations of the rule of three. Of the five given numbers,three contain a supposition, and the oilier two a demand ; one oftile terms of supposition being of the same kind with the num-ber required, and the other tw o of the same kind as the demand-ing terms.

Rule for Stating the Five given Numbers.

1. Write down the term of supposition which is of the samekind with the answer, for the middle term.

2. Take two of the other terms both of a kind, and place themaccording to the directions given in the rule of three ; do the samewith the other term of supposition and its correspondent demandingone, w riting the sign of multiplication between these and their ad-joining terms.

Let those terms which have the sign of multiplication betweenthem be multiplied, aud the five terms will be reduced to three,with which proceed as directed in the golden rule, [f the termsQinnol be conveniently multiplied as in compound multiplication,reduce the corresponding terms, that is those which are of thesame kind, to the lowest denomination, which must be the samein both. If either of the two first terms, and any one of the threelast terms can be divided exactly by any number, let them bedivided and the quotients used instead of them; which will shortenthe work. In some cases it will be easier to divide by the twofirst terms separately without actually multiplying them into eachother.

Example.

If 100/. in 12 months pa’n 61. interest, what principal will gain31. 7s. 6d. in 9 months r

It Z.6