Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

60

of the fame Article, may be reduced to. Also, if either the Numerator or Denominator have a Cypher, and the other end in 5, than may both Parts be reduced by dividing by 5 Thus this Fraction, 3, each Part being divided by 5, will be reduced to ; which, being again divided by 5, will be reduced to, which is its lowest Terms.

25

Another Method of reducing a Fraction is, if the Numerator and Denominator are even Numbers, take the Half of each; and, if they are yet both even, halve them again; and by this Means a Fraction may often be reduced fo low, that it may appear what Number will divide both the Numerator and Denominator, fo as to reduce them to their lowest Terms.

[blocks in formation]

Here 6 and 9 are divifible by 3, which reduces them to, their loweft Terms.

Sometimes it happens a Fraction may be reduced by dividing it by 3, 4, 5, or any of the other Integers, as in the following Examples; where the Figures that ftand over each Par-tition are the Numbers which divide both the Numerator and Denominator.

[merged small][merged small][ocr errors][merged small][merged small][merged small][subsumed][subsumed][subsumed][subsumed][merged small][merged small][subsumed][merged small][ocr errors][subsumed]

In the first Example, the Numerator and Denominator of

the Fraction

384

may be divided exactly by 2, which gives ; and this may be divided by 4, which gives 43; and dividing this by 4, it gives; and dividing this by 6, it

gives for a new Fraction, in the Room of 3. In the fame Manner is the other Operation done. And in this Method the Learner is at Liberty to try any of the Integers for a Divifor; only he muft carefully remember, that it must divide both the Numerator and Denominator exactly, without any Remainder. And, if he has an Inclination to fatisfy his Curiofity, to know whether he has done his Operation right, fay, As the given Denominator is to the given Numerator, fo is the new Denominator to the new Numerator; then multiply the first and fourth Numbers, and the fecond and third Numbers together; if thefe Products are equal, the Works is true. Let us take the laft Example:

As 1080 is to 810, fo is 4 to 3.

[blocks in formation]

In the fame Manner may any of the other Examples be proved.

Article 37. To reduce a Fraction of a lower Denomination to a Fraction of a higher; as the Fraction of a Farthing to the Fraction of a Shilling or a Pound.

Rule. Make it a compound Fraction, thro' all the intermediate Divifions between the given Fraction and that Denomination to which it is to be reduced, as in the following Examples; and then reduce this compound Fraction to a fimple Fraction, by Art. 3.

Exam. I.

Exam. 2.

Reduce of a Farthing to the Reduce of an Inch to the Fraction of a Pound.

[blocks in formation]

Fraction of a Foot.

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

2

I

20

2

2880

Anf. of a Pound Sterling,

H

Exam, I,

ΤΣ

ΤΣ

Exam. 1. A Farthing is of a Penny, whence of a Farthing are of of a Penny; and, because a Penny is of a Shilling, therefore of a Farthing are of of of a Shilling and, becaufe a Shilling is of a Pound, of a Farthing areofofofof a Pound: Which compound Fraction being reduced to a fimple Fraction, by Art. 3, is of a Pound Sterling, equal to of a Farthing.

2

[ocr errors]

2

3

Exam. 2. Becaufe an Inch is of a Foot, therefore of an Inch is of of a Foot: Which compound Fraction being reduced to a fimple Fraction, by Art. 3, is T2 of a Foot, equal to of an Inch.

9

120

Article 38. To reduce a Fraction of a greater Denomination to a Fraction of a lefs.

Rule. This is the Reverfe of the former; and is performed by multiplying the Numerator of the given Fraction, by all the intermediate Divifions of the Integer, for the new Numerator; under which place the given Denominator, and this will be the Fraction fought.

[blocks in formation]

Thus have we done with the most difficult Part of Arithmetic, the Vulgar Fractions; only it may be neceffary to fhew the Learner the Manner of working an Example or two in the Rule of Three direct, in Fractions.

Article 39. To refolve Questions in the Rule of Three in Fractions.

Rule. After the Question is ftated according to the ufual Manner in the common Rule of Three, multiply the Denominator of the firft Term into the Product of the Numerators of the fecond and third Terms, for a new Numerator; then multiply

multiply the Numerator of the firft Term into the Product of the Denominators of the fecond and third, for a new Denominator; and this new Fraction is the Answer fought.

[blocks in formation]

Exam. 1. The 5 Yards are reduced to the improper Fraction , by Art. I; then multiply the Numerators of the fecond and third Terms, and that Product being multiplied by 4, the Denominator of the firft Term, it gives 140 for the Numerator of the fourth Term; then the Product of the Denominators of the fecond and third Terms being multiplied by 3, the Numerator of the firft Term, it gives 24 for the Denominator of the fourth Term: So is 140 the Anfwer fought; the Value of which, by Art. 35, is 5. 165. 8d. If the Learner has the Curiofity to examine this Question by common Arithmetic, he may turn it into whole Numbers thus; of a Pound is 175. 6d. then if of a Yard coft 17 s. 6d. 5 Yards will coft 57. 16s. 8d.

6

20

[ocr errors]

6

Exam. 2. The 6s. 8d. are changed into the Fraction of a Pound Sterling thus; becaufe 20 Shillings make a Pound, therefore 6 Shillings are of a Pound; and 12 Pence making Shilling, and 20 Shillings 1 Pound, therefore 8 Pence are of of a Pound, by Art. 37: Whence of a Pound, being added to of of a Pound, will be in Fractions the fame Thing as 6s. 8d. But of being reduced to a fimple Fraction, by Art. 3, are ; then and being reduced to Fractions of the fame Denomination, by Art. 4, are 1 and ; and the Sum of thefe Fractions, by

4800

ΤΣ

160

[ocr errors]

20

H 2

8

20

2

Art.

[ocr errors]

4800

Art. 11, is 100: Laftly, this Fraction being reduced to its lowest Terms, by Art. 36, is; whence 21. 6s. 8d. is in Fractions 2; which being reduced to an improper Fraction, by Art. 2, is . Now ftate the Question in common Arithmetic, and proceed according to the Directions, and the Answer will be 75 of a Pound, or 21. 8 s. 7 d. 4.

D

Of Decimal Fractions.

ECIMAL Fractions are a modern Invention, tho'

of great Use: It is faid to be the Invention of Regiomontanus, for the Calculation of his Tables. Some object against Decimal Arithmetic, becaufe, in Divifion, it does not always give the true Quotient, only very near, which will be the Cafe while there is a Remainder; whereas the Vulgar Fractions give the Truth. But this Objection is of no real Force: For if it be confidered, that, tho' you can't have the true Quotient in fome Cafes, yet, in all Cafes, you may come as near the Truth as you pleafe, fo as to be within the Icoooth Part of a Farthing, or a Grain, or whatever else the Integer is. Add to this, that, in a Decimal Fraction, we can fee, by bare Infpection, without any Trouble of Reduction, the Value of it to Half a Farthing in Money, to of an Inch in Foot-Measure, and to as fmall a Part in other Things, as will be fhewn hereafter; which, in most Cases, is near enough, Whereas, in Vulgar Fractions, after you have been at the Trouble of reducing them to the known Parts of Coin, Weights, Measures, &c. by Art. 35, you often have a fractional Part of the leaft Denomination into which the Integer is ufually divided, as of a Farthing in Fractions of Money. But the fuperior Excellence of Decimal Fractions will be readily acknowledged, by all who are acquainted with the Facility of their Operation, compared with the Difficulty of Vulgar Fractions, tho' these have their Ufe in fome Cafes.

The Definition and Notation of Decimal Fractions.

When Integers are divided, or fuppofed to be divided, into 10, 100, 1000, &c. equal Parts, any Number of thofe Parts are what are called Decimal Fractions: A Decimal Fraction,

therefore,

« ΠροηγούμενηΣυνέχεια »