Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

Chap. X.

A Fraction exprefs'd by greater Num bers, is reduc'd to an equivalent Fracti13. The Re- on exprefs'd by lefs Numbers, by any duction of Number that will divide both the Nume rator and Denominator of the Fraction Terms. given. Thus,, because 16)

a Fraction

to leffer

14.

on of Inte

6

1 6

(;

or thus, 2). So 3A is re

A

29

6A

duced to by dividing both Terms by 3A. This Reduction is of vaft Use, it being appparent, that Fractions may be much more easily work'd in leffer than in greater Terms. The Method of reducing a Fraction into the leaft Terms it can be reduc'd into, is here omitted, as being very frequently no lefs troublesome than to work by the Fractions given.

An Integer is reduc'd into an equivaReducti- lent Fraction of a given Denomination, gers inte by multiplying the Integer into the DeFractions nominator given, and taking the Product of a given for the Numerator. Thus, the Integer 2 nation. is turn'd into a Fraction, whose Deno

Denomi

minator is 4, namely, 2

4 x 2

8

13

4 4

And

this Reduction is neceffary, as often as a Fraction is to be added to, or fubftracted from an Integer. Thus, 3++!

20

4

5

Lafly

Laftly, A common Fraction is turn'd in- Chap. X. to a Decimal, by dividing the Numera- 15. tor, increas'd with Cyphers (as occafion Reductirequires) by the Denominator. Thus, on of com

ctions into

15. For 2) 10 (5. So=125; for mon Fra4) 100 (29. And 2175; for 4) 300 (75. Decimals. Alfo 1333. For 3) 1000, r. (333,

c. That is, the common Fraction can't be reduc'd into a Decimal exactly equivalent thereto: However the Difference is (or may be by further dividing) render'd so small, as to be inconfiderable. Namely, there is not wanting to make the Decimal 1333, exactly equivalent to the common Fraction. And this is fufficient to our Purpose concerning Reduction.

CHA P.

Chap. XI.

T. Proportion, what.

2.

CHAP. XI.

Of Proportion, and more especially of the Rule of Three, or Golden Rule.

A

NY one

N

Number dividing any other Number, the Quotient thence arifing, fhews the (*) Proportion of one to the other. Thus, because 2)6(3, therefore the greater Number 6 is said to be in a triple Proportion to the leffer 2, and 2 is faid to be in a Sub-triple Proportion to 6.

Four Numbers are faid to be proporProportienal Num. tional, when the two first dividing one bers,what. the other, give the fame Quotient, as the two laft. Thus, because 2)6(3, and alfo 4)12(3, therefore 2 and 6, 4 and 12, are faid to be proportional one to the other; which their Proportionality is wont to be thus express'd, viz. 2:6:: 4 : 12, or 6:2::12: 4.

3.

Of four Proportionals, fome two are A Remark always the Products or Quotients of the other two, multiplied or divided by fome

concerning Proporti

anals.

one

This is by Mathematicians properly call'd the Ratio. But we use the Word Proportion in the fame Senfe in common Speech.

one and the fame Number. Thus, in Chap. XI. the fore-going Example, 6 and 12 are the Products of 2 and 4 multiplied by 3; or 2 and 4 are the Quotients of 6 and 12 divided by 3.

4.

The Foun

the Rule

Hence, if four Numbers be proportional, the Product of the two Extreams, dation of ́i. e. of the 1ft and 4th Terms or Num- the Golden bers) is always equal to the Product of the Rule; and two Means, (i. e.) of the 2d and 3d for proTerms. Namely, in both Cafes, the ving Num Factors are in effect the fame ; and con- Proportifequently, the Products must be the fame, onal. For Inftance; 2: (2 × 3, i. e.) 6:4: (4 × 3, i. e.) 12. And therefore, 2×12= 6×4, because 2 × 4×3=2× 3 × 4.

[ocr errors]

bers to be

5.

den Rule,

or Rule of

On the fore-going Property of Proportionals, is founded the Rule of Pro- The Golportion, which from its great Use in all Arts and Sciences, is commonly call'd the Three diGolden Rule; as alfo the Rule of Three, rect. from the three Proportionals given to find the Fourth. The Rule ftands thus: Of the three Terms or Numbers given, let. the Second multiply the Third, and the Firft divide the Product; the Quotient will be the fourth Proportional fought. For Inftance: Let the three Numbers given be 2, 4, 6; and for the Fourth fought, put Q. Wherefore, 2: 4 :: 6: Q. Wherefore, by the fundamental Rule (given, S. 4.) 2×Q=4x6. And therefore, for

afmuch

2× Q

Chap. XI.

afinuch as

2

Q. it will follow also,

6.

the Terms

in the Golden Risle.

4×6

that =Q, that is, 2)24(12, the fourth

2

Proportional fought. For 2:4:: 6 : 12. If the Proportionals be of two several Of placing external Denominations, then they must be fo plac'd, as that the Firft and Third may be of one and the fame Denomination, and likewise the Second and Fourth. For Inftance Suppofing it be demanded, how much will be spent in 365 Days, at the Rate of spending five Shillings in feven Days; the Terms must be d. s. d. S,

plac'd thus: 7:5:: 365: Q

7. And if the Proportionals be exprefs'd Further,of by more than two external Denominatithe fame. ons, then they are to be reduc'd to Two,

8.

Direct

what.

Thus, fuppofing it be demanded, how much Money one shall spend in a Tear, at the Rate of five Shillings a Week, the four external Denominations here given, (viz. Money, Shillings, Tear, and Week,) must be reduc'd to Two, viz. Shillings and Days, as in the Inftance afore.

If the Proportion runs fo, that the greater the Third Term is, fo much the Proportion, greater must be the Fourth; or the less the third Term is, fo much the less must be the Fourth; then it is call'd direct Proportion; and the fourth Proporti

onal

« ΠροηγούμενηΣυνέχεια »