(9) (21) Answ. 23 C. 3 qrs. 21 lb. 9 oz.

2. In 15966720 Inches ; How many English Miles, &c. Answ. 252 Miles, &c. as occafion requires.

There are many useful Questions may be answered by the help of Reduction only: As the changing one fort of Coin for another; and comparing one fort of Measure with another, &c.

For Instance: Suppose one had 347 Rixdollars, at 4 s. 6 d. per Dollar; and defired to know how many Pounds Sterling they make,

347

54 = the Pence in one Dollar, viz. 4 s. 6d.=54 ch

Answ. 781. 1 s. 6 d. Sterl. are = 347 Rixdollars.

Quest. 2. In 645 Flemish Ells; How many Ells English?

Note, 3 Quarters of a Yard English make one Ell Flemish, and

14, or 5 Quarters of a Yard, is an English Ell.

Therefore, 645

3 = the qrs. of a Yard in 1 Ell Flemish.

qrs in 1 Ell=5) 1935 (387 English Ells for the Answer.

Quest. 3. Suppose a Bill of Exchange were accepted at London, for the Payment of 400 1. Sterl. for the Value delivered at Amsterdam in Flemish Money at 11. 13 s. 6d. for 1 Pound Sterl. How much Flemish Money was delivered at Amsterdam?

First, 11. 13 s. 6 d. = 402 d. the Value of one Pound Sterl. at Amsterdam.

Then, 402 d. x 400 = 160800 d. = 670 l, Flemish, and fo much was delivered at Amsterdam.

CHAP. CHAP IV.

Of Uulgar Fractions.

Sect. 1. Of Motation.

A Fraction, or Broken Number, is that which represents a Part or Parts of any thing proposed, (vide Page 3.) and is exprefsed by two Numbers placed one above the other with a Line drawn betwixt them:

Thus,

{3 Numerator,

4

Denominator.

The Denominator, or Number placed underneath the Line, denotes how many equal Parts the thing is supposed to be divided into (being only the Divisor in Division). And the Numerator, or Number placed above the Line, shews how many of those Parts are contained in the Fraction (it being the Remainder after Divifion). (See Page 29.) And these admit of three Diftinctions:

A proper, pure, or Simple Fraction, is that which is less than an Unit. That is, it represents the immediate Part or Parts of any thing less than the whole, and therefore it's Numerator is al

ways less than the Denominator.

An Improper Fraction is that which is greater than an Unit. That is, it represents some Number of Parts greater than the whole thing; and it's Numerator is always greater than the Denominator.

As for or&c.

A Compound Fraction is a Part of a Part, consisting of several Numerators and Denominators connected together with the Word [of].

As of of, &c. and are thus read, The one Third of the three Fourths of the two Fifths of an Unit.

That is, when an Unit (or whole thing) is first divided into any Number of equal Parts, and each of those Parts are fubdivided

subdivided into other Parts, and so on: Then those last Parts are called Compound Fractions, or Fractions of Fractions.

As for instance, suppose a Pound Sterling (or 20 s.) be the Unit or Whole; then is 8s. the of it, and 6 s. the of those two Fifths, and 2 s. is the of those three Fourths; viz. 25. = of of of one Pound Sterling.

All Compound Fractions are reduced into single ones, Thus,
RULE.

Multiply all the Numerators into one another for a Numerator, and all the Denominators into one another for the Denominator. Thus the of of will become. Oro. For 1 x 3x2=6 the Numerator, and 3×4×5= 60 the Denominator, but or ro of a 1. Sterl. is 2 s. As above.

Sect. 2. To Alter or Change different Fractions into one Denomination retaining the same Value.

IN order to gain a clear Understanding of this Section, it will be convenient to premise this Proposition, viz. If a Number multiplying two Numbers produce other Numbers, the Numbers produced of them shall be in the same Proportion that the Numbers multiplied are, 17 Euclid 7.

That is to say, If both the Numerator and Denominator of any Fraction be equally multiplied into any Number, their Products will retain the same Value with that Fraction.

That is, and . Orand. Or and are of the same

Value, in respect to the Whole or Unit.

From hence it will be easy to conceive, how two or more Fractions that are of different Denominations, may be altered or changed into others that shall have one common Denominator, and ftill retain the same Value.

Example. Let it be required to change and into two other Fractions that shall have one common Denominator, and yet retain the same Value.

According to the foregoing Proposition, if be equally multiplied with 7, it will become, viz. 2x7=14.

3×7

21

Again, if be

equally multiplied with 3, it will become 24, viz. 3x3=2.

H

And

And by this means I have obtained two new Fractions, and, that are of one Denomination, and of the fame Value with the two first proposed, viz and A=.

And from hence doth arife the general Rule for bringing all Fractions into one Denomination.

RULE.

Multiply all the Denominators into each other for a new (and common) Denominator. And each Numerator into all the Deno

minators but it's own, for new Numerators.

Example. Let the proposed Fractions be, and ..

140 168 32

360

are

+20

Hence 420 is the common Denominator; and 140.168.315.

360, are the new Numerators, which being placed Fraction-wife

423

420420

28 the New Fractions required.

Sect. 3. To bring mixed Numbers into Fractions, and the contrary.

MIX'D Numbers are brought into improper Fractions by the following Rule.

RULE.

Multiply the Integers, or whole Numbers, with the Denominator of the given Fraction, and to their Product add the Numerator, the Sum will be the Numerator of the Fraction required.

Example. 9 by the Rule will become +2. For 9×5=.
And, the improper Fraction required.
Again, 13 will become. For 13 × 15 = 4.

=

And+ And fo for any other as occafion requires. To find the true Value of any improper Fraction given, is only the Converse of this Rule. For if = 9, as before is evident:

Then

Then it follows that if 49 be divided by 5, the Quotient will give 9. And if 206 be divided by 15, it will give 13, &c. confequently it follows, that

If the Numerator of any improper Fraction be divided by it's Denominator, the Quotient will discover the true Value of that. Fraction.

EXAMPLES.

25. And=4. And=6. Or=3&c. When whole Numbers are to be expressed Fraction-wise, it is but giving them an Unit for a Denominator. Thus 45 is 92, and 25 is, &c.

Sect. 4. To Abbreviate or Reduce Frations into their Lowest or Leaft Denomination.

THIS is done, not out of any neceffity, but for the more convenient managing of such Fractions as are either proposed in large terms; or swell into such, either by Addition or otherwise: befides it is most like an Artist to express or set down all Fractions in the lowest Terms poffible; and to perform that, it will be neceffary to confider these following Propofitions.

Numbers are either Prime or Composed.

1. A Prime Number is that which can only be measured by an Unit. Euclid 7. Defin. 11.

That is, 3, 5, 7, 11, 13, 17, &c. are faid to be Prime Numbers, because it is not, possible to divide them into equal Parts by any other Number but Unity or 1.

2. Numbers Prime the one to the other, are such as only an Unit doth measure, being their common Measure. Euclid 7. Defin. 12.

For instance, 7 and 13 are Prime Numbers to each other, because they cannot be divided by any Number but an Unit. And 9 and 14 are also Prime Numbers to each other, for altho 3 will measure or divide 9 without leaving a Remainder; yet 3 will not measure 14 without leaving a Remainder; Again, altho' 2 will measure 14 without any Remainder, yet 2 will not measure 9 without leaving a Remainder, &c.

3. A composed Number is that which some certain Number measureth. Euclid 7. Defin. 13.

For instance, 15 is a composed Number of 3 and 5, for 5 x 3 = 15, confequently 3 or 5 will justly measure 15. Alfo 20