What a Fraction is, and how read.

AN Unite, or Integer, is one whole thing, as, one Pound, one Yard, one Gallon, one Hour, &c.

A Fraction, or broken Number, is a part, (or parts) of an Unite, or Integer; and is generally represented by two Numbers set one over the other, with a Line between them, thus, ½. ⅔. ¾. 5/7, &c. The upper number is termed the Nu­merator; the lower the Denominator, they are read, or pronounced, thus, ½ is one half, ⅔ is two thirds, ¾ is three fourths, 5/7 is five sevenths, and so of any other, naming the Numerator first, and the Denominator last; the Denominator [Page 2]shewing the parts into which the Unite, or Integer, is broke, and the Numerator, the part, (or parts) of the Denominator that is to be taken, or used.

Of the Varieties of Fractions.

OF Fractions, or broken Numbers, there are four sorts, viz. Proper, Improper, Mixt and Compound.

A proper Fraction is, that whose Nume­rator is lesser than the Denominator, as ½. ⅔. ¾. ⅘. &c.

An Improper Fraction, is, that whose Numerator is greater than (or, at least e­qual to) the Denominator, as 4/3 5/2 6/6 4/4 &c.

Mixt, are whole Numbers and Fracti­ons set together thus, 2⅓, 3⅚, 7¾, &c.

Compound Fractions are known, by ha­ving the word [Of] betwixt them, and are written thus, ⅔ of ¾, also ¼ of ⅚ of ⅞. They are likewise called Fractions of Fra­ctions.

Now before we can pass to the Rules of Addition, Substraction, Multiplication and Division in Fractions, they must be pre­par'd, and made fit for such Operation, and this preparation is perform'd by Re­duction, of which there are five kinds, as follows,

Reduction the First.

TEacheth how to reduce a whole, or mixt Number, into an Improper Fraction, which Fraction shall be equal in value to the said whole, or mixt Num­ber: And contrary, that is, it teacheth to turn an improper Fraction into its Equi­valent whole, or mixt Number.

CASE I. If it be a whole Number, the Rule is, multiply it by the assign'd Denominator, seting the Product there­of for a Numerator over the said Deno­minator, so shall this Fraction be equal to the given whole Number.

Example. Reduce 7 into an Improper Fraction, whose Denominator shall be 4.

[...]

Note, That these two Lines =is the sign of Equality, as 7 =28/4 shows that 7 is equal to 28 Fourths.

Reduction the Second.

TEacheth you how to reduce a com-Fraction to a Simple one, which shall have the same value with it.

Reduction the Third.

TEacheth how to abreviate a Fraction, or to find a Number that shall re­duce it to its lowest Terms at one Oper­ation, yet still keeping the same value it had at first.

Reduction the Fourth.

TEacheth how to bring Fractions of divers denominations into Fractions of one denomination, yet still retaining the same value.

Reduction the Fifth.

TEacheth how to alter, or change, a Fraction into another equal in va­lue that shall have any assign'd Deno­minator.

How to find the value of a Fraction.

This Reduction is the most useful of all others, for by it the value of any [Page 13]Fraction is found in the known parts of Coyn, Weight, Time, &c. And contrary, that is, any part of Coyn, Weight, Time, &c. is turn'd into a Fraction; the method of doing which is as follows,

Multiply the Numerator by the parts of the next inferior Denominator, that are equal to an Unite of the same, that the Fraction gives the parts of; the Product divide by the Denominator, the Quote gives the value in the parts you multi­ply'd by: If after this Division any thing remain, multiply it by the next inferior Denomination, dividing the Product by the Denominator, as before. Thus proceed, till you can bring it no lower, so will the several Quotients give the required value of the given Fraction.

How to turn any part of Coyn, Time, Weight, &c. into a Fraction.

THis is but the Converse of the for­mer, and therefore (from a little consideration of what foregoes) may be easily effected: For if you do but consider that 1 Shilling is the 1/20 of a Pound sterling, and 1 Penny the 1/12 of a Shilling, and 1 Farthing the ¼ of a Pen­ny: The Names of a Shilling, a Penny, and a Farthing, being only Denominati­ons given them, for the Vulgar in our Nation to know them by, the more uni­versal way of expressing them, being to call a Shilling the 1/20 of a Pound, and a Penny the 1/12 of a Shilling, also a Far­thing the ¼ of a Penny: And this way of expressing them (supposing the value of the Pound known) would be intelliga­ble to all Nations that have the know­ledg of Numbers; so that if it were re­quir'd to know what part of a Shilling 9 d. is, I answer, that 'tis 9/12, or when abreviated ¾. In like manner, if it were required to know what part of a pound 4 Shillings, it is evident that 'tis 4/20 or ⅕. But if it were requir'd to know what part of a Pound 1 Penny is, here I must [Page 17]consider that 1 Penny is the 1/12 of 1/20 of a Pound, and therefore if by Reduction the 2d. I reduce it to a simple Fraction, 'twill be 1/240 of a pound Sterling, so a Farthing is ¼ of 1/12 of 1/20 of a pound, and therefore the 1/960 of a pound: Lastly, if it be required to know what part of a Pound 13 s. 5 d. ¼ is, then reducing all into Far­things, it gives 645; and likewise find­ing the Farthings in one pound which is 960, and seting the former over the la­ter fraction-wise gives 645/960, the part of a pound that 13 s. 5 d. ¼ is.

After the same manner may any part of Weight, Time, Measure, &c. be expres­sed Fraction-wise.

Addition of Vulgar Fractions.

IF the Fractions to be added have not like Denominators, they must be re­duc'd to a common Denominator by Re­duction [Page 19]the 4th. Then add the Numera­tors together, and set the Summ for a Numerator over the common Denomina­tor, so shall the Fraction thus found be the Summ of the given Fractions.

Substraction of Vulgar Fractions.

THE Rules delivered for reducing and making Fractions fit for Addition, are in all respects and cases to be ob­serv'd in Substraction; so that whether they are Mixt, Compound, or simple ones, they must be reduc'd to a common Denominator; then take the Numerator of the Substractor, or Fraction, to be Substracted, from the Numerator of the Substrahend (or Fraction from which we are to Substract) and set the remainder over the common Denominator, so is this new Fraction the remainder or difference sought.

Multiplication of Vulgar Fractions.

IF they be simple Fractions to be Mul­tiply'd. Then Multiply the two Nu­merators, together, for a Numerator, and the two Denominators for a Denomina­tor, so shall the Fraction formed by these two Numbers be the product Required.

Division of Vulgar Fractions.

IF the Fraction to be divided, and also the Fraction by which we divide, that is, Dividend and Divisor, be both simple Fractions, then Multiply the Numerator of the Dividend by the Denominator of the Divisor, and set the product for a Numerator; multiply also the Denomi­nator of the Dividend, by the Numera­tor of the Divisor, and take the product for a Denominator; the Fraction thus form'd is the Quotient.

The Rule of Three Direct in Fractions.

THE Directions given, both for stat­ing and working Questions in the Rule of Three in whole Numbers, holds also in this of Fractions; so that having framed your Question, as is there direct­ed, 'tis but Multiplying the Fractions in the 2d. and 3d. place together, and di­vided the product by the first, according [Page 25]to the preceding Rules given for Multi­plying and dividing of Fractions, the Quotient is the Answer to the Question.

The Rule of Three Reverse in Fractions.

HEre also, as in that of whole Num­bers, you are to Multiply the se­cond Term by the first, and divide the pro­duct by the third, the Quotient answers the Question.

Reduction of Decimals.

WHat is here to be done is no more than what was shown in Reduction the 5th. of Vulgar Fractions, only here I shall more largely comment upon what I there but hinted; and show in the first place how to reduce a Vulgar Fraction to a Decimal, and then how to find the Value of any Decimal in the known parts of Coin, Weight, Time, &c. and that with as much brevity and clearness as I can.

To Reduce a Vulgar Fraction to a Decimal.

THe proportion for reducing a Vul­gar Fraction to a Decimal, is, As the Denominator of the Vulgar Fraction to its Numerator, So is 10, 100, 1000, &c. or any assign'd Denominator, to its Numerator, that is to the Decimal required.

To Reduce the known parts of Money, Weight, Time, &c. to a Decimal Frac­tion.

FRom what precedes, 'tis evident how the known parts of Money, Weight, Time, &c. may be turn'd into a Decimal of the same Value, or Infinitely near it, for if in Money, a Pound Ster­ling be an Integer, whatsoever is less [Page 41]than a Pound, is either a part or parts of the same; and when you know what part or parts thereof it is, you may easily Reduce it to a Decimal of the same Va­luc, from what was taught in the last.

Exam. What's the Decimal of 9 s. That is Reduce 9/20 into a Decimal consisting of two places.

[...]

Here working according to what has been before directed, I find the Decimal of 9 s. to be .45

So if I would know the Decimal of 9 d. here I consider that 9 d. is 9/12 of 1/20 of a Pound or 9/240.

Working therefore according to the preceding Rule, I find the equivalent Decimal to be .0375.

Again, if I would know the Decimal of 3 Farthings, here I must consider that 3 Farthings is the ¾ of 1/12 of 1/20 or 3/960 of a pound, and therefore working as be­fore, I find the Decimal to be .0031 near.

Lastly, if it were requir'd to find the Decimal of 7 l. 08 s. ¾ that is 371 Far­things, or 371/900 here repeating the like o­peration, I find the Equivalent Decimal to be 3864, the like is to be understood in reduceing to Decimals, the known parts of Weight, Time, Measure, Mo­tion, &c.

To find the Value of a Decimal Fraction in the known parts of Money, Weight, Time, &c.

THis is but the Converse of the for­mer; and therefore the Rule for finding the Value of a Decimal is groun­ded upon the same reason as that of turn­ing any part of Coyne, &c. to a Deci­mal.

For 'twill hold as the Decimal Denomina­tor is to its Numerator, So is the parts in the next Inferior Denomination to the Nu­merator, or Number of such parts contain'd in the Decimal.

And hence comes this Rule. Multi­ply the given Decimal by the parts of the next inferior Denomination, that are equal to the Integer the Decimal gives the parts of, and from the product cut [Page 43]of so many figures towards the right hand as there are places in the given Decimal, the remaining figures on the left side are the value of the said Deci­mal in the next inferior Denomination: If any thing remain, it is the Decimal of an Integer in the Denomination last found, and may be reduced as low as you please by the same Rule, and after the same manner, as it was in Reduct. 5th. of Vul­gar Fractions.

A TABLE, showing the Decimal of any part of a Pound sterling, & contra.

Addition of Decimals.

AS to the manner of adding, 'tis the the same as in common Addition, the business being only to see that they are rightly plac'd, according to the man­ner of their Notation, which thing is easily effected, by seting the point prefixt to them under each other; for then the rest of the places will fall right, whether they be whole Numbers and Decimals, or all Decimals.

Substraction of Decimals.

THe Operation here is in all respects like to that in Vulgar Substraction, the main thing (as in the last) being on­ly to see that they are rightly placed, which is done by the direction given in the foregoing Rule of Addition.

Multiplication of Decimals.

IN Multiplication of Decimals, both the manner of placing and multiply­ing is in all respects and cases, the same with that of placing and multiplying whole Numbers, the business here being [Page 51]only to find the value of the Product af­ter the Operation is ended, which to do take this general

Of Contraction in Multiplication of mixt Numbers.

THere is in this kind of Multiplica­tion, a certain way of Contraction, by which you may get the product, to as few, or many places as you please, without the tedious Multiplication of the whole; the Method of which is as follows.

As Suppose 9.58 was to be Multiplyed by 8.79, here 'tis evident the decimal [Page 53]will consist of 4 places, and only two would be sufficient.

Set down the bigger of the two Quan­tities for the Multiplicand, and then set the place of Unites in the Multiplior, under that place of parts in the Multi­plicand, you would have in the pro­duct, and then invert the Order of all the other places in the Multiplior, that is, set the place of Tenns, where Primes should be, and the place of Primes where Tenns should be, and so on with the invertion of the rest; then let each Figure of the Multiplior, Multiply that of the Multiplicand, which is just over, Remembring to add, what would have been brought thither from the following places; then add up all together, and from the Summ cut of two Figures (in this Example) next the Right Hand, and you have your desire, all which by the following Examples, compar'd with this direction will plainly appear.

Division of Decimals.

THE manner of working Decimal Di­vision, is in every thing like to that of Common Division, and therefore no regard as to their place and Nature is here to be had, any more than what was in Division of whole Numbers; the Mi­stery of this lying first in their prepara­tion, when need requires. Secondly, In finding the true Value of that Quote after the Division is ended.

First, Therefore when it happens, that the Divisor has more places than the Di­vidend, you must put to the Right Hand of the dividend, (whether it be a whole Number, mixt or Decimal Fraction) a certain Number of Ciphers at pleasure, by which it is made fit for Operation.

As suppose 14 was to be divided by 361, 'tis evident here is an absolute Ne­cessity of prefixing Ciphers to 14 the Di­vidend, before you can divide by 361 the Divisor.

The Dividend being thus prepared take Notice, that there must be as many decimal places in the Divisor and Quo­tient as are in the Dividend; for the di­vidend is in effect the product, and the Divisor and Quotient the Multiplicand and Multiplior. And therefore for the finding the value of the Quotient this is the

Of the Use of Decimals.

TO show the use of these Numbers in all Solutions, where they might be applied in Expediting an Operation, would be endless, they being of great use in most parts of the Mathematicks; but particularly, and principally in Gauging, Surveying, and Measuring, Calculating the Tables of Interest, raising of Loga­rithmes; as may be seen in most Books, that have Writ of these Subjects, I shall therefore forbear giving Examples of use­ing them in any of these parts, Except I had Treated distinctly of each of them; and shall close rhis Paragraph with the Collection of a few easy Questions, which are very speedily and easy solv'd by these Numbers.

By Multiplication.

IN 756 Pistoles, at 18 s. each, How many Pounds Sterling. Answ. 680 l. 08 s.

In 439 Guinea's, at 22 s. 6 d. each, How many Pounds sterling. Answ. 493 l. 17 s. 6 d.

If I spend 4 s. 6 d. per Day, How much is that for one Year. Answ. 82 l. 2 s. 6 d.

If a yard of Cloth is worth 6 s. 9 d. What comes 59 Yards to at that Rate. Answ. 19 l. 18 s. 3 d.

If a piece of Paving be 34 foot, 6 in­ches long, and 24 foot, 9 inches broad, What's the Content in square feet. Answ. 853 foot .875

If one Man's share in the Cargo of a ship come to 38 l. 14 s. What was the whole worth, supposing there was 158 Men in the ship. Answ. 6114 l. 12 s.

If the Interest of 500 l. for one Day is 2 s. 3 d. What's that for a Year. Answ. 41 l. 1 s. 3 d.

By Division.

IN 680 l. 8 s. How many Pistoles, at 18 s. each. Answ. 756.

In 493 l. 17 s. 6 d. How many Gui­nea's at 22 s. each. Answ. 439.

If I spend 82 l. 2 s. 6 d. in one year. What is that for one day. Answ. 4 s. 6 d.

If 59 Yards of Cloth cost 19 l. 18 s. 3 d. What Cost one Yard. Answ. 6 s. 9 d.

If the Content of a piece of Paving be 853 Foot .875 and the length be 34 foot 6 Inches. What's the true breadth. Answ. 24 Foot 9 Inches.

If the whole Cargo of a ship be 6114 l. 12 s. and there be 158 men in the ship. What comes each man's share. Answ. 38 l. 14 s.

If the Interest of 500 l. for a Year is 41 l. 1 s. 3 d. What is that for one day. Answ. 2 s. 3 d.

I conceive it needless to meddle with the Rule of Three, it being in all kinds and respects performed like that in Vul­gar Fractions; I shall therefore leave the Exercise of Questions of this nature to the Ingenious.

A Specimen of the Demonstration of the Operations of Vulgar and Decimal Fractions.

I Shall first begin with that of Vulgar Fractions, and in order to the more clear apprehending thereof, I must make a short Repetition of what has been al­ready declared in the first page, viz.

That a Fraction is a part or parts of some divisible Integer, and is represented by two Numbers, the one above the o­ther beneath a Line thus ⅖.

The Number placed beneath the Line is called the Denominator, and shows what parts the Unite is divided into.

The Number placed above the Line is called the Numerator, and shows how many of those parts are to be taken in the Fraction.

As the Fraction ⅖ denotes two such parts as the Integer contains 5.

From this method of expressing Frac­tions it follows.

That every Fraction is to its whole an Unite, as the Numerator is to the Deno­minator, and consequently.

First, That if the Numerator be

• greater
• equal
• less

than the Denominator, the Fraction is ac­cordingly greater equal or less than its whole an Unite. The first and second of these kinds are called Improper Fractions, the last are termed Proper.

Secondly, That Fractions are not to be estimated by the greatness of their Num­bers by which they are express'd, but by the proportion the Numerators bear to the Denominators.

Thirdly, That Fractions, whose Nume­rators to their Denominators bear the same proportion, are equal as ½. 3/6. 10/12.

Fourthly, That every Fraction is the Quotient of the Numerator divided by the Denominator.

This being granted, I propose the fol­lowing Lemma.

Reduction the First.

This is so clear from the nature and manner of expressing a Fraction, as also from the first consequent, that it needs no farther Demonstration.

Reduction the Second.

The Proof of this is a consequent from that of Multiplication, and therefore I re­fer it to that place.

Reduction the Third.

This teaches to abbreviate a Fraction, by dividing both Numerator and Deno­minator by any Number that will divide both without a remainder.

As suppofe 6/8 is given to be reduced to its lowest Terms, here dividing 6 by 2, and 8 by 2, there arises ¾; now since 3 and 4 multiplied by 2 produces the same Numbers, viz. 6/8. therefore 3:4::6:8 and therefore the Fraction ¾ is equal to 6/8 by the Lemma, and third Consequent.

Reduction the Fourth.

This teaches to reduce Fractions of divers Denominations into one Denomi­tion, having the same value, for doing of which the

Reduction the Fifth.

This is proved in the Reduction of a Vulgar Fraction to a Decimal.

Of Addition and Substraction of Fractions.

THe Fractions whose Summ or Diffe­rence is required must be reduced to equal Fractions, having the same Deno­mination. And then according to the Rule the

• Summ
• Difference

of the Numerators placed over the common Denominator is the

• Summ
• Difference

of the Fractions requir'd.

Exam. What's the Summ and Diffe­rence of ⅜ and 2/7.

The Fractions reduced to others equal to them, and of the same Denomination are 21/56 and 16/56 consequently 21 ± 16 / 56 is the

• Summ
• Difference

of the Fractions, that is the Summ is 37/56, the Difference 5/56, which was required.

Of Multiplication of Fractions.

For finding the product of any two Fractions, this is the

Of Division.

For working of Division the

Of Decimal Fractions.

DEcimal Fractions (as I have else­where noted) are only Fractions, whose Denominators are an Unite with Cyphers, as 10, 100, 1000, &c. And are conveniently written without their Denominators; for if the Numerators consists of places equal in Number to the Cyphers in the Denominator, then prefix a point before the Numerator, and [Page 70]omit the Denominator; but if the Deno­minator do not consist of as many pla­ces as there are Cyphers in the Denomi­nator, then you must supply that defect by puting Cyphers before the significant figures of such Numerator, with a point on the left hand of such Cypher, or Cy­phers,

As 3/10 78/100 86/1000 54/10000 are thus express'd .3 .78 .086. .0054

To reduce a Vulgar Fraction to a Decimal.

SAy, As the Denominator of the Frac­tion proposed, is to its Numerator, so is 10, 100, 1000 &c. that is, an Unite with as many Cyphers as I intend my Decimal shall have places, to the Nume­rator of a Decimal equal to it.

To Add or Substract Decimals.

Multiplication of Decimals.

MUltiply Integers and Decimals to­gether, as if all were Integers, and then cut off as many places from the Product towards your right hand, or Decimals, as is the Number of Decimal places in both Multiplicand and Multi­plior.

Division of Decimals.

DIvide the Dividend by the Divisor in all respects, as in whole Numbers, only observe that so many Decimal pla­ces as there are in the Dividend more than there are in the Divisor, so many must be cut off from the Quotient to the right hand for Decimals.